keiths, argues from and like an asshole

Now keiths sez:
No, that's just an ignorantbased opinion. And anyone can just as easily say that the idea of infinity is silly.
And keiths, your onetoone correspondence has been shown to be bullshit as the alignment problem crops up. The onetoone correspondence relies on arbitrary mapping.
OK, so here is what my opponents have to do:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
If you cannot do that then you whole "argument" is bullshit.
Might as well relabel infinity as NeverLand.
And if Cantor was such a genius, then he must have been correct about the universe being Created. LoL!
But anyway, as I said, we CAN count the elements in an infinite set. We count and count and count FOREVER. And the set of nonnegative integers will always be twice the count as the set of positive even integers (given my last train scenario). And that goes on FOREVER.
Now asshole sez:
That is nothing but your ignorant opinion, keiths. How is infinity logically coherent when no one can demonstrate that it exists?
Then I said The onetoone correspondence relies on arbitrary mapping.
keiths' reply:
Only an ignorant ass would say that and here is keiths. My alignment is natural number to number it is natural in that the number line remains as is.
keiths tries to answer my challenge:
How is that mathematically rigorous?
LoL! What is the value of Ln? The point being is it sounds EXACTLY like my LKN, which YOU sed was silly. Also Ln+1= Ln, dumbass.
keiths sez:
He was, dumbass. Just look what happens in stars and supernovae.
But anyway, as I said, we CAN count the elements in an infinite set.
We count and count and count FOREVER.
Exactly! And one set will ALWAYS have more elements than the otherALWAYS.
You don't have to finish and the answer is the cardinality of the set of nonnegative integers is greater than the cardinality of the set of all positive even integers.
Even if LNET goes to infinity? It looks like keits has finally acknowledged that I am correct.
Excepot Cantor didn't "show" any such thing. He couldn't even show that infinity exists.
And BTW assface, I never said, thought nor implied that infinity was a number. Heck it only exists in some people's minds. And that makes it totally useless.
Now keiths sez:
As has been pointed out with much mirth here and elsewhere, the idea of an LKN is silly.
No, that's just an ignorantbased opinion. And anyone can just as easily say that the idea of infinity is silly.
And keiths, your onetoone correspondence has been shown to be bullshit as the alignment problem crops up. The onetoone correspondence relies on arbitrary mapping.
OK, so here is what my opponents have to do:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
If you cannot do that then you whole "argument" is bullshit.
Might as well relabel infinity as NeverLand.
And if Cantor was such a genius, then he must have been correct about the universe being Created. LoL!
But anyway, as I said, we CAN count the elements in an infinite set. We count and count and count FOREVER. And the set of nonnegative integers will always be twice the count as the set of positive even integers (given my last train scenario). And that goes on FOREVER.
Now asshole sez:
Infinity is logically coherent. The LKN isn’t.
That is nothing but your ignorant opinion, keiths. How is infinity logically coherent when no one can demonstrate that it exists?
Then I said The onetoone correspondence relies on arbitrary mapping.
keiths' reply:
So does your “method”
Only an ignorant ass would say that and here is keiths. My alignment is natural number to number it is natural in that the number line remains as is.
keiths tries to answer my challenge:
In any case, it’s easy to meet your challenge:
1.Definition: Set A is infinite if every finite subset of A is also a proper subset of A.
How is that mathematically rigorous?
2. Proof that infinite sets exist:
a. Assume that infinite sets do not exist. By the definition above, that means that every set has at least one finite subset that is not a proper subset.
b. Consider all finite subsets S1, S2, etc. of the set N of natural numbers. Every such subset contains a greatest element Ln (if there were any elements greater than Ln, then one of them would be the greatest element instead of Ln).
c. Ln is a finite natural number. Therefore Ln+1 exists and is also a finite natural number.
d. Ln + 1 is not a member of Sn, but it is a member of N. Therefore every Sn is a proper subset of N, thus contradicting a).
Assuming that infinite sets don’t exist leads to a contradiction. Therefore infinite sets do exist.
LoL! What is the value of Ln? The point being is it sounds EXACTLY like my LKN, which YOU sed was silly. Also Ln+1= Ln, dumbass.
keiths sez:
And if Newton was such a genius, then he must have been correct about alchemy. LoL!
He was, dumbass. Just look what happens in stars and supernovae.
But anyway, as I said, we CAN count the elements in an infinite set.
Oh, really?Really, see my train example.
We count and count and count FOREVER.
Sure, you can count forever.
Exactly! And one set will ALWAYS have more elements than the otherALWAYS.
But you never finish, which means you never get an answer.
You don't have to finish and the answer is the cardinality of the set of nonnegative integers is greater than the cardinality of the set of all positive even integers.
Yes, every finite set {1,2,3,…,LNET} will have twice the cardinality of the finite set {2,4,6,…,LNET}, and that will be true no matter how large the LNET gets.
Even if LNET goes to infinity? It looks like keits has finally acknowledged that I am correct.
But you haven’t answered the real question, which is “does the infinite set {1,2,3,…} have twice the cardinality of the infinite set {2,4,6,…}?”Yes.
The answer to that question, as Cantor showed, is clearly “no”.
Excepot Cantor didn't "show" any such thing. He couldn't even show that infinity exists.
And BTW assface, I never said, thought nor implied that infinity was a number. Heck it only exists in some people's minds. And that makes it totally useless.
205 Comments:
At 10:36 AM, socle said…
Surely you don't think all this hasn't already been done? A glance at the wikipedia page for "Infinite Set" answers both your challenges.
At 10:46 AM, Joe G said…
I glanced at that page and didn't see any answers to what I asked.
Perhaps you could be so kind as to repeat those alleged answers here take your time I am going to the lake...
At 11:52 AM, socle said…
But haven't you acknowledged that infinite sets exist? For example, the natural numbers {1, 2, 3, ...}? If so, you have already conceded that infinity exists.
If you are asking whether there are infinite numbers, then the answer is yes again. Stephan Kulla's blog post, which you seem to embrace, explicitly refers to them. In fact, there are infinitely many infinite numbers in the set of hypernatural numbers.
Stepping back a bit, I suspect most mathematicians would respond to your challenge of "Demonstrate infinity exists" by saying "Infinite what? Sets, numbers, or some other sort of structure?" The word infinity has many different meanings in math, so you need to focus your question so that it refers to specific mathematical object(s).
At 12:58 PM, socle said…
One more note:
OK, so here is what my opponents have to do:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
You opponents would only have to provide definitions for terms that they have used in this argument; if you believe a term has been used by one of us without adequate definition, then you should be able to identify a particular instance. IOW, please supply a quote showing that this has happened.
At 1:29 PM, Joe G said…
OK, so socle cannot:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
At least IDists can demonstrate that CSI exists even if evoTARDs don't like the definition.
At 1:56 PM, socle said…
OK, so socle cannot:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
At least IDists can demonstrate that CSI exists even if evoTARDs don't like the definition.
If Joe cannot show that any of his opponents used the term "infinity" in a poorly defined manner, how is that a problem for us?
At 3:11 PM, Joe G said…
If not any of my opponents can:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
Then it is clear that infinity is arbitrary.
At 4:19 PM, socle said…
If not any of my opponents can:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
Then it is clear that infinity is arbitrary.
Does that mean you would have to retract these statements?
You like to change labels, swap LKN for infinity same thing, they are both neverending, yet allegedly one label gives a decidedly different answer to the cardinality question.
I said infinity goes on forever. I have said that many times.
and even your entire post, entitled:
How to Determine the Cardinality of sets that go to Infinity
At 5:19 PM, Joe G said…
Forever is not mathematically rigorously defined. And no one knows if it exists.
However I can point to CSI it exists.
At 5:34 PM, socle said…
Forever is not mathematically rigorously defined. And no one knows if it exists.
Agreed.
At 1:47 AM, Unknown said…
"1 Provide a mathematically rigorous definition of infinity"
The limit that a function is said to approach at x = a when f(x) is larger than any preassigned number for all x sufficiently near a.
(One of many definitions. You can substitute sequence for function which will match our discussions better.)
"2 Demonstrate infinity exists"
Can you demonstrate that 2 exists? You can point to sets of 2 things but does 2 exist? Does root 2 exist? How about Pi? Or e?
If showing two things is sufficient to 'prove' the existence of 2 then demonstrating that sets of infinite size exist should 'prove' infinity exists.
{1, 2, 3, 4 . . . . }
Or, based on the above given definition, how about a function that increases without bound?
f(x) = 1/(x2) as x approaches 2 from above
or
f(x) = tanx as x approaches pi/2.
or
f(x) = 1/x^2 as x approaches zero.
At 7:19 AM, Joe G said…
So Jerad sez infinity is a limit yet others say is doesn't have any.
At 8:42 AM, Unknown said…
"So Jerad sez infinity is a limit yet others say is doesn't have any.'
Are you sure you even took a Calculus class? Limit is used in more than one way in mathematics. Although we often say 'unbounded' for something that diverges.
I tried answering your questions honestly. But if you're just going to snear and waste time then I'll not bother in the future.
But, no response on my part doesn't mean you're unapposed. It might mean that I and others can't be bothered to argue if we think you're presenting challenges insincerely.
At 8:46 AM, Joe G said…
Don't blame me because you cannot define infinity in a mathematically rigorous way.
And don't blame me because you cannot demonstrate that infinity exists.
As for wasting time that appears to be all you are doing. You sure as hell won't respond to what I posted that demonstrates countable and infinite sets are not necessarily equal in size...
At 9:06 AM, Joe G said…
Jerad,
The point being is your having one train double its speed in order to keep up with the count of the first train, means that you have conceded my point that the set of nonnegative integers will always have a greater cardinality than that of the set of positive even integers.
At 9:08 AM, socle said…
Joe,
This must be your most incoherent post ever.
Also Ln+1= Ln, dumbass.
Did you read keiths' definition? Each Ln is a natural number. Do you know of any natural numbers n with the property that n + 1 = n? Hopefully not.
You're also freely using the term "infinity" here, which is odd to say the least. When you say:
Even if LNET goes to infinity? It looks like keits has finally acknowledged that I am correct.
and when you carefully instructed us on how to "How to Determine the Cardinality of sets that go to Infinity" what did you mean by "infinity"? What is your rigorous definition of that term?
At 9:11 AM, Unknown said…
"Don't blame me because you cannot define infinity in a mathematically rigorous way."
I did do that actually.
"And don't blame me because you cannot demonstrate that infinity exists."
Well, what do think the sequence 1, 2, 3, 4 . . . gets to then?
Or what value does f(x) = 1/x^2 get to as x gets closer and closer to zero? Can't be the largest known number 'cause if you tell me what it is I'll beat it.
"As for wasting time that appears to be all you are doing. You sure as hell won't respond to what I posted that demonstrates countable and infinite sets are not necessarily equal in size…"
I did respond by giving you the mathematically correct way of dealing with that situation. I spelled it out in several different examples. You can't even use your method to compare the cardinalities of some sets. So, not much of a demonstration on your part. And you've got no academic work or research to back up your idea.
At 9:16 AM, Joe G said…
socle,
keiths sed Ln was the largest number. Therefor Ln + 1 = Ln, ie the largest number.
And infinity, to me, is not mathematically rigorous just like forever...
At 9:19 AM, Joe G said…
Jerad,
No, you did not define infinity in a mathematically rigorous way. And the sequence 1,2,3,4,... will get as far as it gets until we die, then it stops, just like infinity.
And no, you did not respond to the train scenario except to imagine one train doubling the speed of the other.
And cantor cannot compare cardinalities. He just throws his hands up and sez they are equal or not.
At 9:24 AM, socle said…
Joe,
keiths sed Ln was the largest number. Therefor Ln + 1 = Ln, ie the largest number.
And infinity, to me, is not mathematically rigorous just like forever...
Each Ln is the largest element in a finite subset of the natural numbers. For example, the set might be {1, 2, 5, 9, 13}, and in that case, Ln would be 13. Likewise, all the Ln are finite.
At 9:25 AM, Joe G said…
13 + 1 = 14, ie the Ln
At 9:28 AM, socle said…
13 + 1 = 14, ie the Ln
But you said Ln + 1 = Ln. If Ln = 14, then your equation says 14 + 1 = 14.
At 9:30 AM, Unknown said…
"No, you did not define infinity in a mathematically rigorous way. And the sequence 1,2,3,4,... will get as far as it gets until we die, then it stops, just like infinity."
You are a finitist.
From the Wikipedia article about Finitism:
"The main idea of finististic mathematics is not accepting the existence of infinite objects like infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic."
At 9:31 AM, Joe G said…
LoL! The Ln changes, If Ln=14, then Ln + 1=15, which becomes the Ln.
At 9:39 AM, Joe G said…
And you, Jerad, are an illusionist and a nonrealist.
At 9:40 AM, socle said…
Joe,
Write down the largest number in the set {1, 2, 5, 9, 13}.
Wait 5 minutes and write down the largest number in the set again.
Did it change? Lol.
At 9:45 AM, Joe G said…
socle,
Write down the largest number in the set {1, 2, 5, 9, 13}.
Add 1 to it. Did it change? LoL!
At 10:21 AM, socle said…
Ok, gotta go to work. Have to take a break from all this "Intelligent Reasoning" for a while. Maybe you should do an OP analyzing keiths' proof? Heh.
At 10:50 AM, Unknown said…
"And you, Jerad, are an illusionist and a nonrealist."
Gee, I thought you'd be pleased that some mathematicians have agreed with you to some extent. Granted, mostly in the past but still.
At 11:03 AM, Joe G said…
LoL!
Here ya go:
2. Proof that infinite sets exist:
a. Assume that infinite sets do not exist. By the definition above, that means that every set has at least one finite subset that is not a proper subset.
OK.
b. Consider all finite subsets S1, S2, etc. of the set N of natural numbers. Every such subset contains a greatest element Ln (if there were any elements greater than Ln, then one of them would be the greatest element instead of Ln).
OK
c. Ln is a finite natural number. Therefore Ln+1 exists and is also a finite natural number.
Ln + 1 would be the new Ln. You just covered that in "b".
d. Ln + 1 is not a member of Sn, but it is a member of N.
No, it would be a member of Sn. Ln+1=Ln, just as YOU said in "b"
Therefore every Sn is a proper subset of N, thus contradicting a).
Not if one follows YOUR directions.
At 11:15 AM, Joe G said…
Hi Jerad,
It would be interesting to see how a real mathematician responds to one train, with two counters, one counting all nonnegative integers and one counting positive even integers, as it travels along the number line.
It would be interesting to see what is said about the fact that one counter will be doubling the count of the other, FOREVER, and yet just because the count never stops, the two counts are considered equal. And then he or she had better be able to tell me what that provides for us how it helps us in any way.
At 1:38 AM, Unknown said…
"It would be interesting to see how a real mathematician responds to one train, with two counters, one counting all nonnegative integers and one counting positive even integers, as it travels along the number line."
Why don't you ask Dr Dembski then? I would say you are mistaking the density of the sets with the size. Just because one set's members come up more often under one particular, arbitrary scanning procedure doesn't mean the sets are the same size.
"It would be interesting to see what is said about the fact that one counter will be doubling the count of the other, FOREVER, and yet just because the count never stops, the two counts are considered equal. And then he or she had better be able to tell me what that provides for us how it helps us in any way."
There are other ways of counting the sets aside from your arbitrarily chosen method. I've pointed out to you several times fields of mathematics that have Set Theory as part of their foundation. And you just say you doubt it and, apparently, don't bother to check up on the information.
At 7:23 AM, Joe G said…
Jerad:
Just because one set's members come up more often under one particular, arbitrary scanning procedure doesn't mean the sets are the same size.
There isn't anything arbitrary about my counting procedure.
There are other ways of counting the sets aside from your arbitrarily chosen method.
There isn't anything arbitrary about my counting procedure.
However it is very telling that you cannot properly respond to my comment.
I've pointed out to you several times fields of mathematics that have Set Theory as part of their foundation.
And I have pointed out that ypou are an ass for even bringing that up as it has NOTHING to do with the case in point the cardinality of infinite sets.
So here we have Jerad, too much of a coward to actually respond to my comment, too much of a dolt to understand what "arbitrary" means, thinking his cowardice and ignorance refutes me.
Strange...
At 11:07 AM, Unknown said…
"There isn't anything arbitrary about my counting procedure. "
When there are other methods available but you insist on only using yours then . . .
Maybe arbitrary isn't quite right. But you picked it for God knows what reason and you refuse to look at other methods.
"And I have pointed out that ypou are an ass for even bringing that up as it has NOTHING to do with the case in point the cardinality of infinite sets."
Part of Set Theory is about the cardinality of sets!! If that wasn't included then Set Theory would be less foundational.
"So here we have Jerad, too much of a coward to actually respond to my comment, too much of a dolt to understand what "arbitrary" means, thinking his cowardice and ignorance refutes me."
Why don't we just stick to the mathematics as name calling proves nothing.
I think my mathematics refutes you and I have no need to resort to inflammatory language. Besides, there are lots of questions you've dodged. Is that cowardice on your part?
At 11:14 AM, Joe G said…
Jerad,
My method is the natural counting method. THAT is why I chose it.
Part of Set Theory is about the cardinality of sets!!
The cardinality INFINITE SETS you clueless wanker.
And you don't have any mathematics. That much is obvious.
At 11:28 AM, Unknown said…
"My method is the natural counting method. THAT is why I chose it."
Matching two sets up to see which is larger seems pretty natural to me. And it works in places where your method doesn't work.
"The cardinality INFINITE SETS you clueless wanker."
Part of Set Theory is dealing with the cardinality of infinite sets. And Set Theory provides part of the foundation for aother areas of mathematics.
"And you don't have any mathematics. That much is obvious."
Well, aside from disagreeing with you on the cardinality of infinite sets what else would you like to compare mathematically?
At 12:26 PM, Joe G said…
Matching two sets up to see which is larger seems pretty natural to me.
Matching? Cantor's method doesn't match. A 1 doesn't match a 2. A 1 matches a 1.
Part of Set Theory is dealing with the cardinality of infinite sets.
Except it doesn't deal with it. It just throws up its hands amnd sez "they're equal, except when they ain't".
And it has becomne quite obvious that saying that two countably infinite sets have the same cardinality is totally useless and meaningless.
At 3:54 PM, Unknown said…
"Matching? Cantor's method doesn't match. A 1 doesn't match a 2. A 1 matches a 1."
The first element of the set of positive integers matches to the first element of the multiples of 2. The second positive integer matches/maps to the second multiple of 2. Etc. What's so hard?
"Except it doesn't deal with it. It just throws up its hands amnd sez "they're equal, except when they ain't"."
That's not true at all. Not a fair representation of the mathematics. In fact, Cantor's continuum hypothesis is a very difficult subject.
"And it has becomne quite obvious that saying that two countably infinite sets have the same cardinality is totally useless and meaningless."
It depends on what you're working on doesnt' it? Certainly for most people it is useless. But in mathematics it's an important distinction.
Are you agreeing that there is a class of countably infinite sets then?
At 6:11 AM, Unknown said…
"Matching? Cantor's method doesn't match. A 1 doesn't match a 2. A 1 matches a 1."
An amusing misinterpretation of my use of the word match.
How about 'pair up' or 'map to' then.
"Except it doesn't deal with it. It just throws up its hands amnd sez "they're equal, except when they ain't"."
Obviously the method has eluded you. Oh well, I tried.
"And it has becomne quite obvious that saying that two countably infinite sets have the same cardinality is totally useless and meaningless."
Obvious only if you've refused to follow up on the links that have been left where you can find discussions of the usefulness in mathematics.
Well, I've done my best to lead the horse to the water. Whether he drinks is his decision. But you shouldn't say it hasn't been shown when you've not followed up on references you've asked for.
At 7:20 AM, Joe G said…
The first element of the set of positive integers matches to the first element of the multiples of 2. The second positive integer matches/maps to the second multiple of 2. Etc. What's so hard?
Arbitrary "matching".
"Except it doesn't deal with it. It just throws up its hands amnd sez "they're equal, except when they ain't"."
That's not true at all.
Yes it is and you cannot demonstrate otherwise.
"And it has becomne quite obvious that saying that two countably infinite sets have the same cardinality is totally useless and meaningless."
It depends on what you're working on doesnt' it?
Nope.
Are you agreeing that there is a class of countably infinite sets then?
Nope. I doubt infinity exists and no one seems to be able to demonstrate that it does.
At 7:26 AM, Joe G said…
Matching? Cantor's method doesn't match. A 1 doesn't match a 2. A 1 matches a 1."
An amusing misinterpretation of my use of the word match.
Spoken like an ignorant ass:
match:
.
a. One that is exactly like another; a counterpart.
b. One that is like another in one or more specified qualities: He is John's match for bravery.
2. One that is able to compete equally with another:
Except it doesn't deal with it. It just throws up its hands amnd sez "they're equal, except when they ain't"."
Obviously the method has eluded you.
Obviously you are just an ignorant ass.
"And it has becomne quite obvious that saying that two countably infinite sets have the same cardinality is totally useless and meaningless."
Obvious only if you've refused to follow up on the links that have been left where you can find discussions of the usefulness in mathematics.
Fuck you loser. Obvioulsy you are too stupid to tell me about any use.
Well, I've done my best to lead the horse to the water.
Well your "best" sucks.
At 10:51 AM, Unknown said…
"b. One that is like another in one or more specified qualities: He is John's match for bravery."
Exactly. Like when you 'match' up elements of two different sets. The specified quality is the position in the sets when listed in a particular order. So, listed in ascending numerical order (comparing the positive integers to the positive even integers):
first elements: 1 and 2
second elements: 2 and 4
third elements: 3 and 6
"Fuck you loser. Obvioulsy you are too stupid to tell me about any use."
What an insightful and intellectual riposte.
I would give you a series of lectures in Set Theory but it would be difficult to type them all in the blog comment boxes and I'd have trouble with some of the symbols. Anyway, if you really want to know, you know where to look.
"Well your "best" sucks."
Again, so thoughtful and though provoking. It's hard to know what to say in response.
At 7:01 AM, Joe G said…
I would give you a series of lectures in Set Theory...
And they would all be IRRELEVANT to what we are talking about.
Again you have proven to be an ignorant ass, Jerad.
At 9:15 AM, socle said…
Joe,
I'm afraid it's your concept of cardinality that is useless rather than Cantor's. Here's why:
1) Cantor's definition works with any sets. Your Einstein Train (ET) definition only works to compare cardinalities of finite sets or particularly closely related subsets of N. Apparently {1, 3, 5, ...} and {2, 4, 6, ...} cannot be compared via ET.
Kulla's definition handles those cases, but his definition makes use of the axiom of choice in such a way that you can't actually tell which, if any, set is larger. And again, it only works for subsets of N, which is a huge limitation.
Which of {x1, x2, x3, ...} and {y1, y2, y3, ...} is larger? Cantor says their sizes are the same Joe's ET says?
2) Cantor's definition simply says that all infinite countable sets are equivalent (have the same cardinality) and leaves it at that. Your system is more complicated (and again only applies to N). What use can you demonstrate for saying that {0, 1, 2, ...} is larger than {1, 2, 3, ...}? Do you know of any problems this solves?
At 10:07 AM, Joe G said…
1) Cantor's definition works with any sets.
What do you eman by "works"? In what way does it "work"?
Apparently {1, 3, 5, ...} and {2, 4, 6, ...} cannot be compared via ET.
Sure it can. Are those numbers on the number line? Then the train counters can count them.
Which of {x1, x2, x3, ...} and {y1, y2, y3, ...} is larger? Cantor says their sizes are the same Joe's ET says?
Same size.
2) Cantor's definition simply says that all infinite countable sets are equivalent (have the same cardinality) and leaves it at that.
And that is useless. Do yoiu know of any problems that solves?
What use can you demonstrate for saying that {0, 1, 2, ...} is larger than {1, 2, 3, ...}?
The same use that saying uncountable and infinite sets have a greater cardinality than countable and infinite sets.
At 1:54 PM, socle said…
What do you eman by "works"? In what way does it "work"?
You can use Cantor's definition to compare cardinalities of any two sets. It's not clear what your ET method would say about comparing {1, 2, 3, ...} with {x0, x1, x2, ...}, or about comparing two uncountable sets, for example.
Which of {x1, x2, x3, ...} and {y1, y2, y3, ...} is larger? Cantor says their sizes are the same Joe's ET says?
Same size.
How about {x1, x2, x3, ...} and {y2, y3, y4, ...}? Same size, or different?
One last numerical example: What does ET say about the sizes of the sets A and B, constructed in this way:
Put 1 in A
Put 2, 3 in B
Put 4, 5, 6, 7 in A
Put 8, 9, 10, 11, 12, 13, 14, 15 in B
Put 16 through 31 in A
Put 32 through 63 in B
and so on. If you count the numbers collected by the A and B trains, the "lead" switches back and forth. For example, at various points:
A leads B 1 to 0
B leads A 2 to 1
A leads B 5 to 2
B leads A 10 to 5
A leads B 21 to 10
and so forth. How do you decide which set is larger? Cantor says they have the same size.
At 2:36 PM, Joe G said…
You can use Cantor's definition to compare cardinalities of any two sets.
Mine also. I just question the way Cantor "compares".
It's not clear what your ET method would say about comparing {1, 2, 3, ...} with {x0, x1, x2, ...},
Mine method would say that "x" is meaningless and only count the numbers down a number line (meaningless in that my method would say "x" is the name of the number line).
or about comparing two uncountable sets, for example.
My method would be OK with just throwing up our hands and saying they are equal even though I would prefer to say we just don't know.
How about {x1, x2, x3, ...} and {y2, y3, y4, ...}? Same size, or different?
Unless the letters are variables then they can be dropped. If they are points on a graph then you measure them from 0,0
And then there is still the issue with infinity...
Cantor says they have the same size.
Seeing that it is meaningles to say such a thing, he is welcome to say it. To me it sounds like giving up.
But then again I say that infinity is made up and all sets are finite, albeit with a really, really large LN that will grow until the universe ends.
At 2:51 PM, Joe G said…
Correction
two uncountable sets it depends on the sets. That's what my methodology says.
At 5:16 PM, socle said…
You can use Cantor's definition to compare cardinalities of any two sets.
Mine also. I just question the way Cantor "compares".
Ok, then how do the sets A and B, which I described above, compare?
But then again I say that infinity is made up and all sets are finite, albeit with a really, really large LN that will grow until the universe ends.
In that case, shouldn't you be able to find evidence for the existence of the LKN? Can you find any reference to it at all? If not, I would suggest the reason is that there is no such thing, and if that is the case, the set N is infinite.
At 5:50 PM, Joe G said…
Ok, then how do the sets A and B, which I described above, compare?
Is there a reason why you cannot compare them using my methodology?
You guys keep giving me work to do as if I am your slave and as if I really care about your games.
In that case, shouldn't you be able to find evidence for the existence of the LKN?
I can't find any evidence for the existence of infinity.
The existence of a largest number is based on the fact that there are numbers, a number line, a beginning and there will be an end to this universe.
Of course if you accept God exists and is eternal, as Cantor did, then infinity would be something to consider.
However, in a material world, numbers wouldn't even exist without us, ie clever organisms who can think of such things. Numbers and set theory will end with the extinction of we clever organisms.
A very finite existence indeed.
So which is it?
At 6:09 PM, Unknown said…
"The existence of a largest number is based on the fact that there are numbers, a number line, a beginning and there will be an end to this universe."
Who says numbers are limited to the life of the universe? Have you no imagination at all?
"Of course if you accept God exists and is eternal, as Cantor did, then infinity would be something to consider. "
WTF???
"However, in a material world, numbers wouldn't even exist without us, ie clever organisms who can think of such things. Numbers and set theory will end with the extinction of we clever organisms."
So, 1 + 1 = 2 might not be true? If numbers are relative that is.
At 6:09 PM, socle said…
Is there a reason why you cannot compare them using my methodology?
Yes. In your first post on Einstein's Train, you didn't say how to handle the situation where the "lead" in number of marks made shifts back and forth between the two trains. What's your rule for that case?
The existence of a largest number is based on the fact that there are numbers, a number line, a beginning and there will be an end to this universe.
Ok, I take it you didn't find any evidence of a program to track the LKN.
How do you know that a number line exists? How long is it?
At 6:23 PM, Joe G said…
In your first post on Einstein's Train, you didn't say how to handle the situation where the "lead" in number of marks made shifts back and forth between the two trains. What's your rule for that case?
Relativity it all depends on when you look. And it also depends on why it matters.
Ok, I take it you didn't find any evidence of a program to track the LKN.
I didn't even look.
How do you know that a number line exists?
I have used it.
How long is it?
Longer than I can see. But then again the center of this galaxy is beyond my sight yet it is a finite distance away.
At 6:25 PM, Joe G said…
Who says numbers are limited to the life of the universe?
Numbers are limited to agencies that can think of them.
So, 1 + 1 = 2 might not be true?
Ask a fish heck ask a chimp ask any organism besides a human.
At 8:22 PM, socle said…
In your first post on Einstein's Train, you didn't say how to handle the situation where the "lead" in number of marks made shifts back and forth between the two trains. What's your rule for that case?
Relativity it all depends on when you look. And it also depends on why it matters.
Let's say you look now, for example. Which set is larger, A or B? And what's the rationale for your decision?
At 9:03 PM, Joe G said…
LoL! The train starts down the number line and the counters start counting whenever a member of the set it is counting is passed.
If I look now I see the train waiting to go, with both counters set to zero. As the train passes 1 set A is greater. As the train passes 2 the sets are equal. As the train passes 3 set B is greater.
It ain't that difficult if you can actually follow along. It's a journey though so it's most likely difficult for you.
At 11:25 PM, socle said…
If I look now I see the train waiting to go, with both counters set to zero. As the train passes 1 set A is greater. As the train passes 2 the sets are equal. As the train passes 3 set B is greater.
So three different people can say that A < B, A = B, and A > B, all at the same time? Working with the same, unchanging sets A and B? That's not generally regarded as a good thing. This just says that ET is telling me something about the observer, not the sets.
Cantor's method is objective, and anyone who applies it will always get the same answer. Don't you agree that's better?
At 11:46 PM, socle said…
But then again I say that infinity is made up and all sets are finite, albeit with a really, really large LN that will grow until the universe ends.
The idea of working with finite sets only is interesting, but also very limiting. Of course the real numbers are (uncountably) infinite, so that's out. Do you really not accept that there are infinitely many real numbers in the interval [0, 1]? If not, how many are there?
Furthermore, why should the LKN number be increasing? I can assure you that there is no one keeping track of such a thing, because as you stated, as soon as you choose one, it immediately becomes obsolete. It would be very difficult to make a compelling case for such work on a grant application.
If no one is doing "research" in this area, doesn't that mean the "LKN" would stop growing? Does that mean no one can use numbers higher than the LKN until the researchers catch up? What if the Chinese have a LKN which is twice ours in North America? I'm really at a loss here.
At 7:27 AM, Joe G said…
So three different people can say that A < B, A = B, and A > B, all at the same time?
Nope. How is that even possible?
Cantor's method is objective,
No, it isn't.
and anyone who applies it will always get the same answer.
LoL! Anyone without the ability to think can use it!
At 7:30 AM, Joe G said…
The idea of working with finite sets only is interesting, but also very limiting.
The idea of working with infinite sets is also interesting, however no one can demonstrate that infinity exists.
Of course the real numbers are (uncountably) infinite, so that's out.
Infinity doesn't exist.
Do you really not accept that there are infinitely many real numbers in the interval [0, 1]? If not, how many are there?
Start counting them and tell me how far you have gotten just before you die.
Furthermore, why should the LKN number be increasing?
It will stop when we cease to exist. I take it that you are also unable to follow along.
At 8:46 AM, socle said…
So three different people can say that A < B, A = B, and A > B, all at the same time?
Nope. How is that even possible?
Say I contact 3 independent observers today and ask them to apply ET to the problem. They choose stopping points of 1, 2, and 3, as in your example, meaning that one says A < B, one says A = B, and one says A > B.
All three observers are telling the truth, correct?
If I then ask them to use Cantor's method, they will all get the same answer.
To put this in perspective, here's an illustration. Suppose I devise a very efficient means of testing for CSI. The only problem is that the result of the test depends on some arbitrary choices the observer must make. Due to "relativity", "it all depends on when you look" and so forth.
As a consequence, three observers can calculate three different values for the specified complexity of a certain artifact: 50 bits, 400 bits, and 1000 bits, let us say. So was the artifact definitely designed? The three observers would disagree.
Would that be acceptable to you, the way that your ET method is?
Furthermore, why should the LKN number be increasing?
It will stop when we cease to exist.
Ok, maybe so, but that's not my question. Why should it continue to increase when no one is working on the "problem"?
There are several people in ID with math backgrounds, Granville Sewell, Sal, and of course Dembski. Have you run your "LKN" concept by them?
At 9:10 AM, Joe G said…
Say I contact 3 independent observers today and ask them to apply ET to the problem. They choose stopping points of 1, 2, and 3, as in your example, meaning that one says A < B, one says A = B, and one says A > B.
Umm, that is not at the same time. Geez you should at least TRY to keep your story straight.
As for Cantor's "method" well if throwing your hands in the air and say "they're equal!" is a method, then you have something.
Due to "relativity", "it all depends on when you look" and so forth.
Example please.
Why should it continue to increase when no one is working on the "problem"?
1 It isn't a problem
2 Numbers march on until intelligent agencies cease to exist.
At 9:13 AM, Joe G said…
Granville Sewell says:
I think infinity is a concept that exists only in mathematics, I don’t believe an infinite amount of anything could actually exist.
At 9:30 AM, socle said…
Umm, that is not at the same time. Geez you should at least TRY to keep your story straight.
These are independent observers. AFAIK, Einstein's Train does not run according to a set timetable.
Furthermore, the answer to my cardinality question is an equation or inequality, not something like "A > B at time 1".
As for Cantor's "method" well if throwing your hands in the air and say "they're equal!" is a method, then you have something.
Let's compare:
ET method: Either A < B, A = B, or A > B.
Cantor: A = B
The ET actually doesn't tell us anything, because before doing any calculations, we already know that either A < B, A = B, or A > B is true.
Example please.
It's just hypothetical, I don't have one (although maybe Dembski's formula would be an example, since no one seems to be able to calculate it).
2 Numbers march on until intelligent agencies cease to exist.
I'm not so sure. I have no reason to believe that today's LKN is greater than last week's LKN. Are you concerned that this stagnation could lead to an "Idiocracy" scenario where other countries' LKNs overtake ours and leave us in the dust?
At 9:32 AM, Unknown said…
"As for Cantor's "method" well if throwing your hands in the air and say "they're equal!" is a method, then you have something."
You really don't understand Cantor's work. And you don't seem interested in finding out by doing some work and some reading.
At 9:34 AM, socle said…
I think infinity is a concept that exists only in mathematics, I don’t believe an infinite amount of anything could actually exist.
I agree. But we are doing maths here.
At 9:39 AM, Unknown said…
"I think infinity is a concept that exists only in mathematics, I don’t believe an infinite amount of anything could actually exist."
So? We're not talking about some physical pile of stuff. We're talking about mathematics.
Ask Dr Sewell about the lim of 1/x^2 as x approaches zero and see what answer he gives. He will say it tends to infinity.
At 10:08 AM, Joe G said…
These are independent observers. AFAIK, Einstein's Train does not run according to a set timetable.
YOU said "all at the same time" and yet you then changed to them looking at different times.
Furthermore, the answer to my cardinality question is an equation or inequality, not something like "A > B at time 1".
No, the answer to your question depends on when one is looking at the counters.
The ET actually doesn't tell us anything, because before doing any calculations, we already know that either A < B, A = B, or A > B is true.
Cantor doesn't do any calculations. He just throws up his hands and sez "they're equal!". No work, nothing juts "they're equal".
It's just hypothetical, I don't have one
Then it is meaningless.
I'm not so sure. I have no reason to believe that today's LKN is greater than last week's LKN.
What stopped it from marching on?
At 10:08 AM, Joe G said…
Jerad:
You really don't understand Cantor's work.
Fuck you you cowardly assface.
At 10:09 AM, Joe G said…
So? We're not talking about some physical pile of stuff.
keiths was.
At 10:15 AM, Joe G said…
I agree. But we are doing maths here.
Actually we are discussing if infinity exists. And obvioulsy it only "exists" in our minds.
At 10:25 AM, Joe G said…
As for understanding Cantor's "work", well I would love to see his work pertaining to countable and infinite sets.
At 10:47 AM, socle said…
YOU said "all at the same time" and yet you then changed to them looking at different times.
Yes, I contact three independent observers and ask them to report back to me at 3 PM with the answer to my question. They all give me different answers simultaneously (at the same time), even though they are looking at the same set.
No, the answer to your question depends on when one is looking at the counters.
Then I would suggest your definition of cardinality is flawed. If you, KF, and BA^77 all calculate the specified information in an artifact, is it ok if you say 50 bits, KF says 400 bits, and BA^77 says 1000 bits? Let's assume that the artifact was Lizzie's glacier image, and you each did your calculations on a different day last week, so that relativity may have played some role.
Cantor doesn't do any calculations. He just throws up his hands and sez "they're equal!". No work, nothing juts "they're equal".
No, under Cantor's method, you look for (and find) a onetoone correspondence between elements of A and B.
What stopped it from marching on?
The fact that no one is searching for such a thing. Suppose all searches for large primes are halted. Do you expect the largest known prime number to continue increasing?
Actually we are discussing if infinity exists. And obvioulsy it only "exists" in our minds.
I actually don't have a problem with that. I avoid talking about a generic "infinity" when possible, because the word has many different meanings, some less precise than others.
Numbers are abstract things. They don't exist in the physical world. But you had no problem saying you believed that the "number line" "exists", when that is clearly also an abstract entity that you cannot locate in space and time.
If you're going to accept that numbers and the number line "exist", then it's perfectly sensible to say that infinite sets such as N also exist in the same sense. After all, what is the number line, other than an infinite set of points?
At 11:43 AM, Joe G said…
Yes, I contact three independent observers and ask them to report back to me at 3 PM with the answer to my question. They all give me different answers simultaneously (at the same time), even though they are looking at the same set.
Fuck you. Do you really think that just because you can be a belligerent ass that my claim somehow is in trouble?
If all three people look at the sets at the same time they will all give you the same answer.
However, as I have said before, in the case of large sets you need to first determine a pattern. And you cannot do that just by looking at one or two elements from any set.
No, the answer to your question depends on when one is looking at the counters.
Then I would suggest your definition of cardinality is flawed.
That makes me happy as it pretty much proves my definition is correct.
No, under Cantor's method, you look for (and find) a onetoone correspondence between elements of A and B.
LoL! Yeah one "looks" for the ... at the end, puts one set on top of the other and draws a line from e1 to e1, e2 to e2, e3 to e3 regardless of what e1, e2 and e3 are. His "method" is that of a frustrated child.
What stopped it from marching on?
The fact that no one is searching for such a thing.
How would that stop it? Are people searching for infinity?
Suppose all searches for large primes are halted. Do you expect the largest known prime number to continue increasing?
OK I get it, known prime numbers don't stop and the largest number doesn't stop.
There is no way of knowing what every person on this planet knows wrt large numbers.
After all, what is the number line, other than an infinite set of points?
A very long but finite set of points. Infinity is not a number and doesn't belong on any number line.
BTW I still haven't had to roll more than 10 times to roll a 4.
Just sayin'...
At 12:26 PM, Joe G said…
socle,
If infinity is a journey, does it depend on when, into that journey, you look to determine where you are?
Or is it that you just cannot grasp the journey concept?
At 1:26 PM, socle said…
If all three people look at the sets at the same time they will all give you the same answer.
Why? Each of them would use their own ET to judge the sizes of the sets. As I said, there is no universal timetable for ET. If these observers are independent, they could all give me contradictory answers.
How would that stop it? Are people searching for infinity?
The current largest known prime number is 2^57,885,161 − 1. It has 17,425,170 digits. There was a 5year gap between its discovery and that of the previous largest known prime number. How will the next largest prime number be found if no one is looking for it? Will it just suddenly appear on wikipedia?
There is no way of knowing what every person on this planet knows wrt large numbers.
Then how can it make sense to speak of the LKN? You've refudiated the concept right there.
A very long but finite set of points. Infinity is not a number and doesn't belong on any number line.
No one said that infinity is on any number line.
BTW I still haven't had to roll more than 10 times to roll a 4.
Just sayin'...
How many trials did you do? I ran a simply computer simulation and roughly 16% of 10roll sequences had no 4s. That was with on the order of 1000 to 10000 trials.
If infinity is a journey, does it depend on when, into that journey, you look to determine where you are?
I didn't say infinity is a journey. This question doesn't even make sense as far as I can tell.
The set {1, 2, 3, ...}, as understood by mathematicians, is infinite. It doesn't matter "when" you look at it to "determine where you are". It doesn't change when you look at it. It's just a set.
At 1:42 PM, socle said…
Do you really not accept that there are infinitely many real numbers in the interval [0, 1]? If not, how many are there?
Start counting them and tell me how far you have gotten just before you die.
Ok, let's assume there are only finitely many real numbers in [0, 1]. If there are n of them, we can represent them as x1, x2, x3, ..., xn, where x1 < x2 < x3 < ... < xn. n is obvioulsy a large number, greater than 2.
That's supposedly the complete list. But now consider the real number x' = (x1 + x2)/2, the average of x1 and x2.
As you can check, x1 < x' < x2, so x' was not on our original list. That means our list was incomplete.
This argument works for any n > 2, so any finite list cannot include all real numbers in [0, 1]. The set must therefore be infinite.
At 1:46 PM, Joe G said…
Why? Each of them would use their own ET to judge the sizes of the sets.
Not according to my methodology. All is equal ONE TRAIN TWO COUNTERS.
Then how can it make sense to speak of the LKN?
That's why I have changed to the largest number. I see that you are still having trouble following along.
No one said that infinity is on any number line.
That is what we are talking about.
How many trials did you do?
Over 1000.
I ran a simply computer simulation and roughly 16% of 10roll sequences had no 4s.
LoL! I have already been over that. As I said you just cannot follow along.
I didn't say infinity is a journey.
olegt did. And what he says means more than what you say.
This question doesn't even make sense as far as I can tell.
So if you are on a journey you are in the same place every time you look?
The set {1, 2, 3, ...}, as understood by mathematicians, is infinite.
"Infinite" appears to be meaningless and nonexistent.
At 1:50 PM, Joe G said…
Ok, let's assume there are only finitely many real numbers in [0, 1]. If there are n of them, we can represent them as x1, x2, x3, ..., xn, where x1 < x2 < x3 < ... < xn. n is obvioulsy a large number, greater than 2.
That's supposedly the complete list. But now consider the real number x' = (x1 + x2)/2, the average of x1 and x2.
As you can check, x1 < x' < x2, so x' was not on our original list. That means our list was incomplete.
Just relabel your list, duh. IOW your labeling system sucks.
At 1:53 PM, Joe G said…
If there are n of them, we can represent them as x1, x2, x3, ..., xn, where x1 < x2 < x3 < ... < xn. n is obvioulsy a large number, greater than 2.
Maybe and maybe not. That method may work on natural numbers but that doesn't mean it is OK to use on all real numbers.
At 1:55 PM, Joe G said…
Joe’s main difficulty with the concept of infinity is a failure to realize that infinity is a journey, not a destination. olegt
At 3:02 PM, socle said…
Not according to my methodology. All is equal ONE TRAIN TWO COUNTERS.
That would make it very difficult for two independent observers to get consistent results. On the other hand, Cantor's method has worked fine for over 100 years, without worrying about nonsense such as syncing up trains.
That's why I have changed to the largest number. I see that you are still having trouble following along.
I'll probably regret asking, but what is the "largest number"?
How many trials did you do?
Over 1000.
:O
Is that 1000 rolls, or 1000 trials of 10 rolls each? With real dice?
Maybe and maybe not. That method may work on natural numbers but that doesn't mean it is OK to use on all real numbers.
What specifically is wrong with the argument? What did I do that might not be ok to use on all real numbers?
At 3:14 PM, Joe G said…
That would make it very difficult for two independent observers to get consistent results.
Only if the observers were evoTARDs.
On the other hand, Cantor's method has worked fine for over 100 years
Worked? Worked on what? How does it "work"?
I'll probably regret asking, but what is the "largest number"?
It is what it is the largest number.
Is that 1000 rolls, or 1000 trials of 10 rolls each? With real dice?
I have hit 4 over 1000 times. I have never rolled over ten times to do so.
I have been watching the game of thrones and rolling, and rolling, and rolling. I take breaks but at 5 seconds a roll (slow) I can get in quite a few rolls in 1 hour. And there are 5 discs for season 2 each with 2 episodes. I have one episode left.
Do the math.
What specifically is wrong with the argument?
Where do you put 1.5 in the set of integers? 1+2/2=1.5
At 3:52 PM, socle said…
Worked? Worked on what? How does it "work"?
By giving noncontradictory answers.
It is what it is the largest number.
Can you give me an orderofmagnitude estimate of this number?
I have hit 4 over 1000 times. I have never rolled over ten times to do so.
That's remarkable. If I understand you correctly, the probability of such an event is miniscule.
Where do you put 1.5 in the set of integers? 1+2/2=1.5
1.5 isn't an integer of course. 2 is not in the interval [0, 1], so my argument would never involve averaging 1 and 2. All the x1, x2, x3, up to xn are greater than or equal to 0 and less than or equal to 1.
At 4:03 PM, Joe G said…
Worked? Worked on what? How does it "work"?
By giving noncontradictory answers.
It depends on what you mean by "noncontradictory" of course. Mine doesn't give contradictory answers if you know how to use it and don't mind doing some work.
Cantor's "works" for lazy people, so have at it.
Can you give me an orderofmagnitude estimate of this number?
Does it matter? Infinity doesn't have an order of magnitude.
1.5 isn't an integer of course.
And of course it doesn't work with those real numbers then, does it?
BTW, your number, x'=(x1+x2)/2, is already covered in the set of real numbers between 1 and 2 that is if it is a real number.
At 4:04 PM, Joe G said…
Oops between 0 and 1
At 4:58 PM, socle said…
Can you give me an orderofmagnitude estimate of this number?
Does it matter? Infinity doesn't have an order of magnitude.
Yes, but the LN is not infinite, right? I want to know the size of the current LN.
1.5 isn't an integer of course.
And of course it doesn't work with those real numbers then, does it?
BTW, your number, x'=(x1+x2)/2, is already covered in the set of real numbers between 1 and 2 that is if it is a real number.
I think we have a failure to communicate here. You claim the set [0, 1] is finite. I said if that is the case, put them all in a list. Then I produced a real number in [0, 1] which was not on your list. Therefore you did not in fact have all the real numbers in [0, 1] in your list.
We can apply this argument to any finite list (with >= 2 elements). Therefore no finite list can include all the real numbers in [0, 1].
[0, 1] must therefore be infinite.
At 5:37 PM, Unknown said…
"And of course it doesn't work with those real numbers then, does it?"
Do you know what a real number is in a mathematical sense?
At 5:57 PM, Unknown said…
"Fuck you you cowardly assface."
I revel in your eloquence. Is this how you resolved disputes with your Calculus teacher? What did you tell them when they asked you to show that
½ + ¼ + ⅛ + 1/16 + . . . converged?
Did you say: fuck you asshole, you can't add up an infinite series of numbers?
Infinity comes up over and over again in Calculus courses. You say you took a Calculus course. So . . .did you just take all that as some worthless crap you didn't have to understand?
At 6:30 PM, Joe G said…
You claim the set [0, 1] is finite. I said if that is the case, put them all in a list.
It's your set, YOU put them all in a list start with the first element what is it?
At 6:58 PM, Joe G said…
Do you know what a real number is in a mathematical sense?
Yes.
At 7:02 PM, Joe G said…
Is this how you resolved disputes with your Calculus teacher?
What disputes?
What did you tell them when they asked you to show that
½ + ¼ + ⅛ + 1/16 + . . . converged?
Did they ask that? If so I most likely told them what they wanted to hear.
Infinity comes up over and over again in Calculus courses.
So what?
So . . .did you just take all that as some worthless crap you didn't have to understand?
What's there to understand about something that cannot be seen, studied and most likely doesn't even exist?
At 7:07 PM, Joe G said…
Hey socle,
Can you even have a set when you can't even write down what the first element is?
At 9:45 PM, socle said…
You claim the set [0, 1] is finite. I said if that is the case, put them all in a list.
It's your set, YOU put them all in a list start with the first element what is it?
lol. You're the one rewriting all the set theory textbooks. If you think the set [0, 1] is finite, just tell me what the smallest element is. Hint: I'd go with 0.
Can you even have a set when you can't even write down what the first element is?
Yes. The set (0, 1) has no "first" (i.e., smallest) element.
At 10:57 PM, Joe G said…
You're the one rewriting all the set theory textbooks.
Not really. I'm just pointing out the laziness of Cantor's accepted "method". And in the process I am demonstrating that the concept that countable and infinite sets having the same cardinality, is totally usless and meaningless.
I am also questioning the existence of infinity and it appears I was correct it is just a made up concept whose utility can be replaced by the UPB or some other very large number.
The set (0, 1) has no "first" (i.e., smallest) element.
How is it a set if a set is a collection of things and you can't even say what any of the things actually are?
And if a real number is a value that represents a quantity along a continuous line (wikipedia), wouldn't there have to be a smallest element in that set?
At 12:24 AM, socle said…
How is it a set if a set is a collection of things and you can't even say what any of the things actually are?
We can list lots of elements of the set (0, 1). For example, 1/2, 0.1234, sqrt(2)/2.
I am also questioning the existence of infinity and it appears I was correct it is just a made up concept whose utility can be replaced by the UPB or some other very large number.
Even if you could get by without "infinity" of some kind, many problems are simpler with it. I think the infinite series 1 + 1/2 + 1/4 + 1/8 + ... came up somewhere here. Using the standard definition of infinite series, it simply equals 2.
There is a very useful formula:
1 + a^1 + a^2 + ... = 1/(1  a)
that holds for all a < 1 For example,
1 + (1/3)^1 + (1/3)^2 + ... = 1/(1  1/3) = 3/2
If you truncate the series to make it have finite length, the formula is more complicated:
1 + a + a^2 + ... + a^n = (1  a^(n + 1))/(1  a)
If n is very large, the two sums are almost equal anyway, so why not use the one with the simpler formula?
And if a real number is a value that represents a quantity along a continuous line (wikipedia), wouldn't there have to be a smallest element in that set?
No, (0, 1) has no smallest and no largest element. If you choose any number in (0, 1), say x, then x/2 and (x + 1)/2 are both in the set, and x/2 < x < (x + 1)/2. This means that x can be neither the smallest nor the largest number in (0, 1).
At 1:21 AM, Unknown said…
" 'The set (0, 1) has no "first" (i.e., smallest) element.'
How is it a set if a set is a collection of things and you can't even say what any of the things actually are?
And if a real number is a value that represents a quantity along a continuous line (wikipedia), wouldn't there have to be a smallest element in that set?"
You specify infinitely many numbers in (0, 1) but you cannot say what the smallest number in that set is.
I cannot understand how someone who says they took a Set Theory course and a Calculus course can't grasp these concepts. Unless you just memorised the procedures and didn't understand how or why you were doing things.
At 9:14 AM, Joe G said…
You specify infinitely many numbers in (0, 1)...
Umm no one can specify infinitly many numbers that is impossible.
but you cannot say what the smallest number in that set is.
But then that continuous line cannot get going.
I cannot understand how someone who says they took a Set Theory course and a Calculus course can't grasp these concepts.
What concepts can't I grasp, Jerad? Please be specific.
At 9:19 AM, Joe G said…
We can list lots of elements of the set (0, 1). For example, 1/2, 0.1234, sqrt(2)/2.
Yet you cannot list the first element.
Even if you could get by without "infinity" of some kind, many problems are simpler with it.
Simpler doesn't make it right.
And if a real number is a value that represents a quantity along a continuous line (wikipedia), wouldn't there have to be a smallest element in that set?
No, (0, 1) has no smallest and no largest element.
That goes against the definition of a real number it has to be a quantity yet you cannot say what that is. That should be a problem.
If you choose any number in (0, 1),
There are numbers that can't even be choosen because they cannot be written, which means they cannot be quantified, which goes against the definition of a real number.
At 9:57 AM, socle said…
Yet you cannot list the first element.
Right, because there is no such thing.
Simpler doesn't make it right.
Sure, but if the formula is easier to use, and gives you an answer you know will be close enough, why not use it?
One other example that I remember from basic physics class: Suppose you need to calculate the electric field generated by a charged plate. It's actually easier mathematically if you assume the plate is infinite. The result you get will be accurate enough under certain circumstances. That's another example of how the concept if infinity is both useful and practical.
And if a real number is a value that represents a quantity along a continuous line (wikipedia), wouldn't there have to be a smallest element in that set?
I don't see why. Take the set {x in R : x > 5} That's all real numbers greater than 5. There is no smallest element.
No, (0, 1) has no smallest and no largest element.
That goes against the definition of a real number it has to be a quantity yet you cannot say what that is. That should be a problem.
What is "it"? If I asked you in what year the Texas Rangers won their first World Series, then you would simply say there is no such year.
There are numbers that can't even be choosen because they cannot be written, which means they cannot be quantified, which goes against the definition of a real number.
Are you talking about transcendental numbers? The number pi/4 is transcendental, and also lies in the interval (0, 1). So is 1/e. These can both be "chosen" in the argument I posted above. I don't see any conflict between these numbers and the properties of the real numbers.
At 11:51 AM, Unknown said…
"What concepts can't I grasp, Jerad? Please be specific."
Well, that the open set (0, 1) has no smallest element.
Or that you can specify an infintite number of elements of (0, 1). Like ½, ⅓, ¼, 1/5, . . . . . that's an infinite set and if you give me a number I can tell you whether it's in that set or not.
(The lest set which obviously has the same cardinality as the postitive integers but which does not diverge to infinity.)
Or that you can compare the 'sizes' of sets by attempting to put them in a onetoone correspondence.
And you say things like: "But then that continuous line cannot get going" which just shows how little you understand what the real numbers are.
And this is aside from the fact that you frequently end up name calling and using profanity as if that's a dabating tactic.
When I took a Set Theory class it was a 300 or 400 level mathematics course, something I took AFTER I had completed 3 semesters of Calculus, Differential Equations and some other courses. You say you took a Set Theory course but your understanding of basic mathematical concepts (like there is no smallest number in the open interval (0, 1) ) makes me doubt you ever did take such a course.
"But then that continuous line cannot get going." doesn't even make sense! The real numbers are not like the positive integers. They don't 'start' somewhere. Nor do the rational numbers.
Did you really take a 300 level Set Theory course? What textbook did you use?
At 8:15 PM, Joe G said…
Well, that the open set (0, 1) has no smallest element.
Questioning something that doesn't make sense is better than just blindly following it, Jerad.
Or that you can specify an infintite number of elements of (0, 1). Like ½, ⅓, ¼, 1/5, . . . . . that's an infinite set and if you give me a number I can tell you whether it's in that set or not.
Specify the first number after 0.
Or that you can compare the 'sizes' of sets by attempting to put them in a onetoone correspondence.
Yeah, look for the elipsis and then do a pancake stack. Cantor probably had the idea eating flapjacks.
The real numbers are not like the positive integers. They don't 'start' somewhere. Nor do the rational numbers.
They do in the open set (0,1) you ignorant fuck.
And as I said Jerad, I took calculus over 30 years ago and only used it when I took it again in 1992 as part of my computer science course.
However I do know that the open set (0,1) has a starting point. It ain't as if 1 is in the set.
What does the word "quantity" mean to you in a mathematical scenario?
At 8:34 PM, Joe G said…
One other example that I remember from basic physics class: Suppose you need to calculate the electric field generated by a charged plate. It's actually easier mathematically if you assume the plate is infinite. The result you get will be accurate enough under certain circumstances. That's another example of how the concept if infinity is both useful and practical.
True, but that doesn't mean infinity exists.
As you pointed out it also makes other equations simpler without changing their effect.
Useful concept that doesn't exist/ will ease to exist when beings aware of the concept cease to exist.
If I asked you in what year the Texas Rangers won their first World Series, then you would simply say there is no such year.
What? There is a smallest number after 0. You just don't know what it is.
JoeMath would have 10^150 as that point. And that would be the interval/ increment spacing on the number line. 10^150 quantified points between consecutive integers.
At 9:03 PM, socle said…
Useful concept that doesn't exist/ will ease to exist when beings aware of the concept cease to exist.
We might as well make use of it while we are here, then. And that infinite plate approximation is based on the assumption, or fact, depending on your point of view, that the set of real numbers is infinite.
What? There is a smallest number after 0. You just don't know what it is.
JoeMath would have 10^150 as that point. And that would be the interval/ increment spacing on the number line. 10^150 quantified points between consecutive integers.
Somehow I knew 10^150 would make an appearance. Why arbitrarily choose 10^150?
Let's see how well it works, anyway. What is the length of the hypotenuse of a right triangle with legs of length 1, in JoeMath?
At 9:14 PM, Joe G said…
And that infinite plate approximation is based on the assumption, or fact, depending on your point of view, that the set of real numbers is infinite.
But we know there aren't any infinite plates and I bet 10^150, ie a really large number, will do just fine.
What is the length of the hypotenuse of a right triangle with legs of length 1, in JoeMath?
As long as it has to be to connect the two points and form the triangle. It's going to be close to the square root of 2. And the square root of 2 out to 10^150 places would be good enough. I would round up that last digit just to make sure the points were fully connected and no infinitely small gap existed. :)
At 9:50 PM, socle said…
But we know there aren't any infinite plates and I bet 10^150, ie a really large number, will do just fine.
I agree that there aren't infinite plates, but if R is not infinite, then you would have to throw away the simple approximation formula. Actually I think people would still secretly use it.
As long as it has to be to connect the two points and form the triangle. It's going to be close to the square root of 2. And the square root of 2 out to 10^150 places would be good enough. I would round up that last digit just to make sure the points were fully connected and no infinitely small gap existed. :)
Actually 150 places I believe.
Fractions would certainly be fun in JoeMath. 1/3, 1/7, 1/9, and a whole lot of others are missing. In fact, 4/7 + 4/7 =/= 8/7, after converting to your system. Are you sure this is a good idea?
At 10:16 PM, socle said…
It's going to be close to the square root of 2.
Hmm, I guess I missed this. It seems that you do acknowledge that the actual square root of 2 exists, presumably along with the other algebraic numbers. Sure, you can find JoeMath approximations to all of them, but why bother when you can use the exact values?
At 2:25 AM, Unknown said…
"However I do know that the open set (0,1) has a starting point. It ain't as if 1 is in the set."
1 is not in the open inteval (0, 1) Nor is 0 or 1.
If you KNOW the open interval has a starting point then what is it?
If you can't be specific then you don't KNOW there is such a thing.
And if you won't be specific then your honesty shall be doubted.
"Quantity" can have several mathematical meanings. Usually, of course, it means some numeric value. But 'a quantity' can also mean a term or a variable or a variable expression. It not as specific as term as are other mathematical terms. Like square root, integral, Euler circuit, prime number, amicable numbers, inequality, least upper bound, greatest lower bound, countably infinite, pi, etc.
So, did you take a Set Theory course? Did your experience of Calculus include Taylor and Maclaurin series? Where you find infinite polynomials that are equal to sine and cosine? Where you talk about intervals of convergence? Where you talk about divergence?
At 9:00 AM, Joe G said…
In fact, 4/7 + 4/7 =/= 8/7, after converting to your system.
What? 4/7 + 4/7 = 8/7 under my system.
Sure, you can find JoeMath approximations to all of them, but why bother when you can use the exact values?
OK what is the EXACT value for the square root of 2?
At 9:04 AM, Joe G said…
1 is not in the open inteval (0, 1) Nor is 0 or 1.
that's what I said dumbass. It's as if you don't understand anything.
If you KNOW the open interval has a starting point then what is it?
Wahtever the next highest value, ie QUANTITY, is above 0 duh.
Ya see your methodology doesn't allow you to put a value on it. And that means there isn't a quantity nor a point to be traversed.
Infinity is a journey and the journey can't get started in your scenario....
At 9:08 AM, Joe G said…
real number:
In mathematics, a real number is a value that represents a quantity along a continuous line.
That means a specific place on the number line.
At 9:37 AM, Joe G said…
Fractions would certainly be fun in JoeMath. 1/3, 1/7, 1/9, and a whole lot of others are missing.
Why would fractions be missing? Why would rounding up the 150th decimal place affect the fraction?
Who even writes out the decimal equivalent of a fraction to 150 decimal places?
At 9:57 AM, socle said…
What? 4/7 + 4/7 = 8/7 under my system.
Try converting each side of that equation to JoeMath, then compare the last digits.
OK what is the EXACT value for the square root of 2?
The exact value is simply sqrt(2). This number does not exist in JoeMath.
The point is that when you said "It's going to be close to the square root of 2.", you implicitly acknowledged that it's convenient to use the (infinite) set of real numbers which has an exact square root of 2, even if your ultimate goal is to calculate in JoeMath.
I think this discussion has just about run its course. To get back to the original question, we've seen that (0, 1) is a nonempty set with no smallest element (just asserting that 10^150 is the smallest number in JoeMath doesn't help; the set of real numbers does not exist in JoeMath). You won't be able to reconcile this with your statement that all sets are finite.
At 10:00 AM, socle said…
Why would fractions be missing? Why would rounding up the 150th decimal place affect the fraction?
Just because rounding a number changes it.
Who even writes out the decimal equivalent of a fraction to 150 decimal places?
I don't know. I generally avoid that, preferring to use expressions such as 1/3 rather than 0.333...3 (out to 150 places).
At 10:29 AM, socle said…
Correction: Actually, if you always round up your decimals instead of following the usual rounding rules, 4/7 + 4/7 = 8/7 in your system. However, then 1/7 + 1/7 =/= 2/7 (after converting to JoeMath).
At 11:01 AM, Unknown said…
" 'If you KNOW the open interval has a starting point then what is it?'
Wahtever the next highest value, ie QUANTITY, is above 0 duh."
So, what is it? You say it exists but you can't tell me what it is? Are you sure it exists if you can't be specific?
"Ya see your methodology doesn't allow you to put a value on it. And that means there isn't a quantity nor a point to be traversed."
Nothing to do with Cantor here. Everyone except you has thought there is no smallest value in any open interval for hundreds of years. Go figure.
"Infinity is a journey and the journey can't get started in your scenario…."
Sure it can: Let's try Zeno's Paradox shall we?
So, you don't remember Taylor or Maclaurin series. You know a bit of Engineering though. You know about Fourier Transforms?
So . . . you didn't take a Set Theory course even though you claimed you did?
At 11:07 AM, Unknown said…
"OK what is the EXACT value for the square root of 2?"
It's the number, when squared gives you 2.
Or
It's the ratio of a diagonal of a unit square to one of it's sides.
Or
It's 1/arcsin(pi/4)
Take your pick. The point is when you do algebra with notation like root 2 you are not losing any places of accuracy whereas if you use a truncaation of the decimal expansion you lose at least one place of accuracy for every operation. Same for pi which is why I used that in one of my above examples instead of an approximation.
At 11:14 AM, Unknown said…
"Why would fractions be missing? Why would rounding up the 150th decimal place affect the fraction?"
Some fractions would be missing in your system because they have infinitely long decimal expansions.
Why would rounding up affect the fraction?
0.171717171717171717171717……….
Makes a difference which place you round to when you add it to something else.
At 11:15 AM, Joe G said…
socle,
Then according to your math 1/3 + 1/3 + 1/3 =/= 1
At 11:15 AM, Joe G said…
"OK what is the EXACT value for the square root of 2?"
It's the number, when squared gives you 2.
So you don't know
At 1:03 PM, Unknown said…
"So you don't know"
Sure I know. It's the number that when you square it you get 2.
You mean: can I exaclty specify root 2 as a decimal?
That's not possible, no one can do it as the decimal expansion is infinitely long and has no pattern.
So, let's see . . .
You didn't answer the Seth Theory course question. I'll assume you didn't take one then 'cause otherwise you'd say yes.
You apparently don't remember Taylor or Maclaurin series both of which deal with infinite sequences which converge and lead to Fourier analysis which is widely used in engineering.
And you can't seem to specify the smallest number in the open interval (0, 1). I guess it doesn't exist then.
Do you know Zeno's paradox? Hint: it's about a journey of infinite steps . . .
At 2:14 PM, Joe G said…
Some fractions would be missing in your system because they have infinitely long decimal expansions.
LoL! So the FRACTIONS would be OK just their decimal equivalents would change ever so slightly.
Why would rounding up affect the fraction?
It wouldn't. It would affect te decimal equivalent.
At 2:16 PM, Joe G said…
You didn't answer the Seth Theory course question.
Jerad, you aren't anyone to be asking me questions.
At 2:53 PM, socle said…
socle,
Then according to your math 1/3 + 1/3 + 1/3 =/= 1
No, only in JoeMath. By definition, 1/3 is the real number x such that 3x = 1. So 1/3 + 1/3 + 1/3 = 3*(1/3) = 1 in the real numbers. Apparently in JoeMath 1/3 + 1/3 + 1/3 > 1.
And going back to the value of the square root of 2, you're obvioulsy aware of the fact that not all real numbers have a finite decimal representation. That's ok. Nobody ever said they did. There's no need throw up our hands and round everything off to 150 decimal places.
At 3:13 PM, Unknown said…
"Jerad, you aren't anyone to be asking me questions."
You're always asking other people questions: can we prove this or that. How do we know certain things.
Anyway, I'll just take your nonresponse as a no then. Along with Fourier analysis and the smallest number in (0, 1) which you assert exists but you can't tell me what it is. And that really is your claim which you have to defend. It's not up to me to find something you say must be there.
And you don't want to talk about Zeno's paradox . . . too bad. it's a nice illustrative example. Oh well.
At 6:25 PM, Joe G said…
By definition, 1/3 is the real number x such that 3x = 1.
Good luck doing that with the decimal representation.
Just how would one go about multiplying a number that never ends?
At 6:27 PM, Joe G said…
Along with Fourier analysis and the smallest number in (0, 1) which you assert exists but you can't tell me what it is. And that really is your claim which you have to defend.
Umm my defense is the number line. There IS a point on that number line that is just after 0.
I thought you said you knew what the number line is?
At 6:29 PM, Joe G said…
The point is that when you said "It's going to be close to the square root of 2.", you implicitly acknowledged that it's convenient to use the (infinite) set of real numbers which has an exact square root of 2,...
There isn't any exact square root of 2. And the set of real numbers doesn't have to be infinite to include the square root of 2.
At 7:39 PM, socle said…
Good luck doing that with the decimal representation.
You don't have to. Just use fractions.
Just how would one go about multiplying a number that never ends?
Infinite series.
There isn't any exact square root of 2. And the set of real numbers doesn't have to be infinite to include the square root of 2.
*yawn*
Have you read any of Berlinski's books? I know he's about the worst writer on the planet, but at least his math is more or less accurate. For example:
"And the natural numbers, it must not be forgotten, are infinite."
"The natural numbers and the even numbers are similar; they can be put into onetoone correspondence; and they thus do share the same cardinal number. It follows that they are the same size.
So much for common sense. Whereupon common sense does what common sense always does, and retires flustered from the controversy."
lol
At 8:19 PM, socle said…
Those Berlinski books cover just about everything that has come up in these discussions: Cantor, infinite sets and cardinality, irrational numbers, etc. Why don't you read them, then do a post revisiting all those topics?
At 1:46 AM, Unknown said…
"Umm my defense is the number line. There IS a point on that number line that is just after 0."
Then tell us what it is? You say it exists but you can't tell us what it's value/quantity is. I'm not making that claim, Richie's not making that claim, socle's not maikng that claim. But you are. It's down to you to find and display that value.
"I thought you said you knew what the number line is?"
Absolutely. Which is why I'm not saying there is a number after zero. But, I will say there are a countably infinite number of rational numbers in (0, 1) but there are an uncoutably infinite number of real numbers.
"There isn't any exact square root of 2. And the set of real numbers doesn't have to be infinite to include the square root of 2."
Of course there is an exact root of 2, it just doesn't have a finite or repeating decimal representation.
If the reals weren't infinite then there would be a maximum number of decimal places necessary to exactly represent all real numbers. And then root 2 and pi and e and root 3 and a lot of other real numbers would only ever be represented approximately. In fact, your assumption that the number of reals doesn't have to be infinite contradicts itself immediately. Congratulations, you've just done your first proof by contradiction. Well done.
You should look up Fourier analysis sometime, fascination stuff. And Zeno's paradox. And if you really want something interesting look up Gabriel's horn . . . the mathematical one. Take the function 1/x from 1 to infinity and rotate about the xaxis. The object you get has finite volume but infinite surface area. Very cool.
At 7:18 AM, Joe G said…
You don't have to. Just use fractions.
Provide the fraction for the square root of 2.
At 7:20 AM, Joe G said…
"There isn't any exact square root of 2. And the set of real numbers doesn't have to be infinite to include the square root of 2."
Of course there is an exact root of 2, it just doesn't have a finite or repeating decimal representation.
You have no idea what "exact" means. You have no idea what "quantitiy" means.
And infinite doesn't exist...
At 7:23 AM, Joe G said…
socle
I have shown this onetoone corresponcence. Amd I have also shown that during the infinite journey the set natural numbers will always outnimber the set of even positive number 2to1. It is only if we do NOT treat infinity as a journey do the sets have the same cardinality.
At 7:25 AM, Joe G said…
If the reals weren't infinite then there would be a maximum number of decimal places necessary to exactly represent all real numbers.
A maximum could still be more numbers than we could write down in 1,000 lifetimes.
And then root 2 and pi and e and root 3 and a lot of other real numbers would only ever be represented approximately.
They are.
At 9:07 AM, socle said…
Provide the fraction for the square root of 2.
Of course you already know that most real numbers cannot be represented exactly as a fraction (which Berlinski also covers).
I have shown this onetoone corresponcence. Amd I have also shown that during the infinite journey the set natural numbers will always outnimber the set of even positive number 2to1. It is only if we do NOT treat infinity as a journey do the sets have the same cardinality.
You can google this passage by Berlinski as well:
"The number system is now dense, and not discrete, infinite in either direction, as the positive and negative integers go on and on, and infinite between the integers as well."
If you can't trust a Senior Fellow at the DI, who can you trust??
At 9:26 AM, Unknown said…
" 'Of course there is an exact root of 2, it just doesn't have a finite or repeating decimal representation. '
You have no idea what "exact" means. You have no idea what "quantitiy" means."
Oh don't be insulting. Exact does not mean you can write a finite or repeating decimal expansion of the number!! If we used a different number base then all our decimal expansions would change! In a root two based number system root 2 would be 10.
"And infinite doesn't exist…"
Tell that to the engineers who use Fourier transforms.
" 'If the reals weren't infinite then there would be a maximum number of decimal places necessary to exactly represent all real numbers. '
A maximum could still be more numbers than we could write down in 1,000 lifetimes."
That's still finite and therefore not an exact value for root 2.
" 'And then root 2 and pi and e and root 3 and a lot of other real numbers would only ever be represented approximately.'
They are."
Wrong. Suppose I take my expression root 2 and a decimal expansion for it. Perhaps a very, very accurate decimal expansion but any one will do. If I square both of those numbers I should get 2. Exactly. But that won't work with a decimal expansion. Squaring it will never give me 2 exactly. But that's the definition of root 2, the number that when I square it I get 2. It exists, I can write it as root 2, I can do arithmetic and algebra and calculus with it just like I can with 3 or 17 just by always writing down the symbols root 2 when I use it.
And, let's keep track here . . . you seemed to have given up on the smallest number in (0, 1) claim. Which I take as a tacit assumption that either it doesn't really exist or you haven't a clue of how to find it. Just 'cause you're trying to shift the argument doesn't mean you won the previous wave. And you can't claim you did: you avoided answering a challenge.
"I have shown this onetoone corresponcence. Amd I have also shown that during the infinite journey the set natural numbers will always outnimber the set of even positive number 2to1. It is only if we do NOT treat infinity as a journey do the sets have the same cardinality."
II don't get this infinity is a journey thing but I'll let it go. You did not provide a onetoone correspondence. A onetoone correspondence means lining up every element of one set and lining up every element of another set and seeing if by linking the first two elements and the second two elements and the third two elements, etc you will get every element of one set linked up with an element of the other set. Learn to use the terminology correctly.
At 9:50 AM, Joe G said…
socle,
Ask Berlinski, or any mathematician, what happens to the numbers once all intelligent agencies and the universe, are gone.
At 9:55 AM, Joe G said…
Exact does not mean you can write a finite...
Yes, it does. Otherwise it ain't exact.
Tell that to the engineers who use Fourier transforms.
I would ask those engineers for evidence of it's existence, which they could not provide.
And, let's keep track here . . . you seemed to have given up on the smallest number in (0, 1) claim.
Nope. It has to exist for the reasons provided. Strange that you ALWAYS ignore my reasons and prattle on anyway.
II don't get this infinity is a journey thing but I'll let it go.
Talk to olegt about it. Perhaps he can clue you in. Not my fault that you don't understand the terminology.
You did not provide a onetoone correspondence.
That's because there isn't one, duh.
At 10:43 AM, socle said…
Ask Berlinski, or any mathematician, what happens to the numbers once all intelligent agencies and the universe, are gone.
That's irrelevant to the present question. Anyway, he's your guy, how about you contact him?
At 11:52 AM, Unknown said…
"Ask Berlinski, or any mathematician, what happens to the numbers once all intelligent agencies and the universe, are gone."
You think the Pythagorean Theorem won't be true after all intelligent life in the universe is gone? Really?
" 'Exact does not mean you can write a finite... '
Yes, it does. Otherwise it ain't exact."
Uh huh. So the ratio of the circumference of a circle to its diameter is only ever an approximate value? What if you define Pi = 4*arctan(1)?
What do you think of imaginary numbers then? Which are used in engineering. You do know about imaginary numbers I trust. The square root of minus one. How do you write a decimal expansion of that? ("Invented" long before Cantor by the way.)
" 'Tell that to the engineers who use Fourier transforms. '
I would ask those engineers for evidence of it's existence, which they could not provide."
Whatever.
" 'And, let's keep track here . . . you seemed to have given up on the smallest number in (0, 1) claim.'
Nope. It has to exist for the reasons provided. Strange that you ALWAYS ignore my reasons and prattle on anyway."
If your reasoning is good then you should be able to produce the number in question. I didn't ignore your reasons. I disagree with them but I figured I could skip the arguing and just cut to the chase. Produce the number. Or find someone who can. Or find a real proof that it has to exist not just some vague argument based on the way you think things should work. Otherwise . . . you lose.
" 'II don't get this infinity is a journey thing but I'll let it go. '
Talk to olegt about it. Perhaps he can clue you in."
Yeah I thought about that. But I just don't care that much. It's all clear in my mind without the analogy.
" 'You did not provide a onetoone correspondence.'
That's because there isn't one, duh."
Well, you said: I have shown this onetoone correspondence. I don't know what you meant then.
Here's another one:
1 <> ½
2 <> ⅓
3 <> ¼
4 <> 1/5
etc.
n <> 1/(n+1)
Two countably infinite sets. One grows without bounds, it diverges. The other converges to zero. But never gets there. Good huh?
At 12:52 PM, Joe G said…
Ask Berlinski, or any mathematician, what happens to the numbers once all intelligent agencies and the universe, are gone.
That's irrelevant to the present question.
What question? My question is does infinity exist other than in our minds.
At 1:00 PM, Joe G said…
You think the Pythagorean Theorem won't be true after all intelligent life in the universe is gone?
Who will test it? Triangles will be gone, Jerad. And so will the therom.
So the ratio of the circumference of a circle to its diameter is only ever an approximate value?
Yes. If you had noticed, using Pi, the circle never closes.
What do you think of imaginary numbers then?
They're imaginary more so that the alleged real numbers.
If your reasoning is good then you should be able to produce the number in question.
I should? Please, make your case I dare you.
Well, you said: I have shown this onetoone correspondence. I don't know what you meant then.
Read this: How Cantor Looked for and Found a "onetoone correspondence..."
One grows without bounds, it diverges.
Nope. It will cease to grow once we are gone.
At 3:04 PM, socle said…
Joe, according to Berlinski, your challenge:
OK, so here is what my opponents have to do:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
has been met.
If you are still unconvinced, maybe you could do a chapterbychapter analysis/refutation of the relevant parts of one Berlinski's books?
At 4:46 PM, Unknown said…
" 'You think the Pythagorean Theorem won't be true after all intelligent life in the universe is gone? '
Who will test it? Triangles will be gone, Jerad. And so will the therom."
It already has been tested and proven. Over 2000 years ago. So why should it cease being true if it's true now? There won't be triangles then? Really?
You're really grasping at straws here. You think mathematics is some fleeting truth. Some passing fad. Some artificial construct.
How does E = mc^2 depend on someone there to measure it? Either it is true or it's not true.
Either the 2nd Law of Thermodynamics is true or it isn't. And, if it's true, then it always has been true and it alwyas will be true. No matter who is there to verify it.
" 'So the ratio of the circumference of a circle to its diameter is only ever an approximate value?'
Yes. If you had noticed, using Pi, the circle never closes. "
You really don't understand mathematics. Or you are just trying to continue an argument for the fun of it. Like Dr Berlinski. Just for the sport.
" 'What do you think of imaginary numbers then? '
They're imaginary more so that the alleged real numbers."
Alleged real numbers? WTF?
Mathematically you're falling back to the early Greeks you know. You're throwing away more than 2000 years of mathematics. This discussion was settled a long time ago you know. Or maybe you don't.
" 'If your reasoning is good then you should be able to produce the number in question.'
I should? Please, make your case I dare you."
You made a claim which you can't prove. You lose. Simple as that. Your position is not the default. You are the one saying everyone else is wrong. You have to establish your hypothesis. Which you seemingly can't. End of story. Just like someone who claims to have invented a perpetural motion machine. Okay, show me. You can't. You don't get to claim the high ground here. And if you think you can then enjoy being left behind.
" 'Well, you said: I have shown this onetoone correspondence. I don't know what you meant then.'
Read this: How Cantor Looked for and Found a "onetoone correspondence…" "
I did read it. Not accurate. Not funny. A badly written screed attempting to make fun of a person instead of addressing the mathematics which you cannot do.
" 'One grows without bounds, it diverges.'
Nope. It will cease to grow once we are gone."
Clearly you lack some fundamental mathematical reasoning ability. You cannot extrapolate beyond the perceiveable realm. Too bad. Have a nice life in the past. Please try and clean up after yourself at least.
At 5:08 PM, Unknown said…
For more than 2000 years people have known that if you take two lines exactly one unit long and place them end to end at exactly 90 degrees then the diagonal between the other ends is a number which when you square it has to come out to 2. Not approximately two but exactly two. And they realised they could not represent that with a finite decimal expansion, i.e. a rational number.
2000 years. Root 2 is anumber that you can prove exists but which you cannot represent as a finite decimal expansion.
If you really care you can go look it up and check it out.
But you won't. Your loss.
Just don't try and tell those of use who have learned how it all works that we're wrong or stupid or blind. Stop saying thay you know better and that we have to walk you through it all when we don't. It's up to you to learn and catch up like the rest of us have. You're failure at doing so is not a disproof of 2000 years of mathematics. It is, however, a statement about how willing you are to try and learn.
You can now swear at me and call me names if it makes you feel better. Thankfully, it won't change a thing and you'll still be stuck in the past. Left behind. Unable to prove that there is a number after zero. Have fun.
At 7:44 AM, Joe G said…
Joe, according to Berlinski, your challenge:
OK, so here is what my opponents have to do:
1 Provide a mathematically rigorous definition of infinity
2 Demonstrate infinity exists
has been met.
Bullshit. If you think so then you have provide the reference(s).
At 7:53 AM, Joe G said…
It already has been tested and proven. Over 2000 years ago. So why should it cease being true if it's true now? There won't be triangles then? Really?
The therom will ceasse to exist once we cease to exist REALLY.
Mathematics is an artificial construct we constructed it.
Yes. If you had noticed, using Pi, the circle never closes. "
You really don't understand mathematics.
Because a coircle never closes using Pi, I don't understand mathematics? Only a fuckhead would make such a "connection, and here you are.
You made a claim which you can't prove.
So have YOU, asshole.
Read this: How Cantor Looked for and Found a "onetoone correspondence…" "
I did read it. Not accurate.
Of course it's accurate.
Clearly you lack some fundamental mathematical reasoning ability.
Clerarly you cannot deal with REALITY.
At 7:54 AM, Joe G said…
For more than 2000 years people have known that if you take two lines exactly one unit long and place them end to end at exactly 90 degrees then the diagonal between the other ends is a number which when you square it has to come out to 2. Not approximately two but exactly two. And they realised they could not represent that with a finite decimal expansion, i.e. a rational number.
Strange that the LINE is FINITE.
At 8:54 AM, socle said…
Joe,
Now that you have received some feedback on your mathematical theories, are you going to actually pick up a maths book or two and compare your ideas to others'? Maybe even change your mind based on the arguments you find? Or are you going to dig in your heels and refuse to entertain the possibility that you're mistaken?
If you want to see what this discussion looks like from our point of view, here's an example:
http://forums.randi.org/showthread.php?t=141270
That's "MacM", an internet antirelativity crank of some note from the 2000's. I read that he has passed away already. Notice how he cannot escape from his misconceptions because he refuses/is unable to understand the arguments that the others in that thread present.
BTW, have you hit 2000 4's yet without getting 10 non4's yet? According to my numbers, if you do so, you will have crashed through the UPB.
At 9:01 AM, Unknown said…
"The therom will ceasse to exist once we cease to exist REALLY.
Mathematics is an artificial construct we constructed it."
You're entitled to your opinion.
"Yes. If you had noticed, using Pi, the circle never closes. "
I don't even know what that means: the circle never closes.
"Because a coircle never closes using Pi, I don't understand mathematics? Only a fuckhead would make such a "connection, and here you are. "
It's not my fault you say things that don't make sense.
" 'I did read it. Not accurate.'
Of course it's accurate. "
And you would know 'cause you've read what Cantor wrote.
"Strange that the LINE is FINITE."
The point is that root 2 has been known to be an irrational number for a very long time. That it cannot be written as a rational number. That a finite approximation is not the same as the actual value of root 2. That the real value of root 2 has an infinitely long decimal expansion with no pattern.
I think you're losing track of your own arguments, such as they are. You just keep throwing out things which aren't true and then you can't remember why you said them and can't be bothered to prove your claims. Like that (0, 1) has a smallest element. Did you think we'd forget? Or that you don't understand Fourier analysis which uses infinite sequences. Or that you didn't really take a Set Theory course. Or that two countably infinite sets have different cardinality.
I think, in actuality, you just enjoy getting attention and getting people angry. Maybe that's the way you like getting attention. I might stick around to see if you say anthing else that's not true but I'm thinking I actually have better things to do.
At 9:30 AM, Joe G said…
The point is that root 2 has been known to be an irrational number for a very long time.
That's because only irrational people would even try to find it.
At 9:35 AM, socle said…
Correction:
BTW, have you hit 2000 4's without getting 10 consecutive non4's yet? According to my numbers, if you do so, you will have crashed through the UPB.
At 9:35 AM, Joe G said…
Now that you have received some feedback on your mathematical theories, are you going to actually pick up a maths book or two and compare your ideas to others'?
Is there a math book that proves infinity exists? Is there a math book that proves there are as many nonnegative even integers as there are nonnegative integers even though the nonnegative integers are the sum of the nonnegative even and odd integers?
How can that be?
Let A = all even nonnegative integers. Let B = all odd positive integers and C = all nonnegative integers:
If A + B = C AND neither A, B, nor C = 0, then how can A = C and B = C?
At 9:39 AM, Joe G said…
BTW, have you hit 2000 4's yet without getting 10 non4's yet?
I have yet to go over 19 rolls to get a 4. And I have had well over 1000 rolls.
At 9:41 AM, Joe G said…
"The therom will ceasse to exist once we cease to exist REALLY.
Mathematics is an artificial construct we constructed it."
You're entitled to your opinion.
That ain't an opinion.
At 12:32 PM, Joe G said…
The point is that root 2 has been known to be an irrational number for a very long time. That it cannot be written as a rational number. That a finite approximation is not the same as the actual value of root 2. That the real value of root 2 has an infinitely long decimal expansion with no pattern.
2 is a prime number. Do you know what a prime number is?
I ask because if you knew what a prime number is then you wouldn't go looking for a root of a prime number. Sure, curiosity may sink in and you have to take a peak. Then you learn why they call it "irrational".
At 1:30 PM, socle said…
Is there a math book that proves infinity exists? Is there a math book that proves there are as many nonnegative even integers as there are nonnegative integers even though the nonnegative integers are the sum of the nonnegative even and odd integers?
Start with this.
Can you find any books or other references that support your crazyass assertions?
At 2:08 PM, Joe G said…
That's basically the same thing keiths sed.
It seems as if this is proof by repeated assertion. And it also seems that infinity will cease to exist as soon as agencies that can think of the concept have ceased to exist.
We know the universe had a beginning. And given what we know of the laws of physics, it will have an end.
Billions upon billions of years may seem like infinity to us, but it ain't.
At 2:12 PM, Joe G said…
And socle,
I would love to see how a mathematician responds to Einstein's train one train, two counters one counting all nonnegative integers and one counting only the positive even integers.
First I would have to know if the mathematician understood that infinity is a journey.
Maybe you or Jerad would like a crack at that one which counter will count more numbers, for infinity?
At 3:16 PM, socle said…
It seems as if this is proof by repeated assertion. And it also seems that infinity will cease to exist as soon as agencies that can think of the concept have ceased to exist.
In math, you start by accepting certain axioms (e.g., the Peano Axioms), some primitive terms, and rules of logic as your starting point. Then you can create defined terms and prove theorems using those rules of logic.
You seem to have accepted that the natural numbers "exist" (even stating so in this thread). Exactly what it means for them to exist is a philosophical question; they do not exist in the physical world in my view. But we are doing math and dealing with abstract entities. That comes with the territory.
The properties of the natural numbers are described by the Peano Axioms, so it would seem you accept those as reasonable as well.
That theorem in the book says that if N exists, an infinite set exists.
I'm not seeing any proof by repeated assertion there.
And the reference to the laws of physics, or the fact that the universe might have an end have no bearing here. What if I said that by your argument CSI will cease to exist at some point? Does that mean it doesn't "exist" now?
Regarding the ET, I mentioned before that it's related to density measures in N or Z (but not exactly the same). Apparently no one has found any reason to use it in connection with cardinality.
At 3:26 PM, Joe G said…
socle,
Yes, I agree that "infinity" exists as a concept, ie a mental construct.
What if I said that by your argument CSI will cease to exist at some point?
It will as soon as the design and the designer(s) cease to exist.
Regarding the ET, I mentioned before that it's related to density measures in N or Z (but not exactly the same).
No, it's related to taking a journey and counting mile markers or just miles. One counting all miles and one counting only the even numbered miles. Which counter will be counting a greater amount, for as long as the journey lasts?
Apparently no one has found any reason to use it in connection with cardinality.
And apparently no one has found any use for saying that all countable and infinite sets have the same cardinality.
So I am in good company.
At 3:35 PM, Joe G said…
And the reference to the laws of physics, or the fact that the universe might have an end have no bearing here. What if I said that by your argument CSI will cease to exist at some point? Does that mean it doesn't "exist" now?
So infinity exists now, but will cease to exist at some point. And that makes sense to you, how?
BTW, at the point it ceases to exist the set of nonnegative integers will have 2x the cardinality as the set of positive even integers.
At 3:38 PM, Joe G said…
Do we know when infinity came into existence? Or how? Or why? or Where?
At 5:32 PM, Unknown said…
" 'The point is that root 2 has been known to be an irrational number for a very long time.'
That's because only irrational people would even try to find it."
OH, right. So if you had a square field 1 mile x 1 mile then what woud you say it's diagonal was?
"Is there a math book that proves infinity exists? Is there a math book that proves there are as many nonnegative even integers as there are nonnegative integers even though the nonnegative integers are the sum of the nonnegative even and odd integers?"
Yup. Kaplansky's Set Theory. You should read it.
" ' "Mathematics is an artificial construct we constructed it."
You're entitled to your opinion.'
That ain't an opinion."
Ummm, yes, it is. And if you disagree then prove it.
"2 is a prime number. Do you know what a prime number is?"
Of course.
"I ask because if you knew what a prime number is then you wouldn't go looking for a root of a prime number. Sure, curiosity may sink in and you have to take a peak. Then you learn why they call it "irrational"."
Um . . . are you just making this shit up as you go along or is there some pont to it? Seriously.
You are falling way off the stupid end here.
You always accuse people of not staying on topic. Well, you've fallen off the deep end here.
Weird. And wrong.
"Maybe you or Jerad would like a crack at that one which counter will count more numbers, for infinity?"
As long as I can make a onetoone correspondence I'm good.
"And apparently no one has found any use for saying that all countable and infinite sets have the same cardinality."
No. Apparenly you haven't bothered to read the references we have provided. Not the same thing. Stop wasting our time. Do some work!!
It's like arguing with a 4year old. "i haven't seen it so I don't believe it." Go read. Do some research. Spend some time making at least an attempt trying to digest the work that's already been done.
But you won't. 'Cause you don't care. Or can't. It's all the same to me. But it would be adult if you would at least acknowledge your intentions. Which seem to be wasting others' time.
At 6:45 PM, Joe G said…
So if you had a square field 1 mile x 1 mile then what woud you say it's diagonal was?
It would be the length of a FINITE line connecting the two end points. 7467 feet 9/16 inches will do it.
And if you disagree then prove it.
Let's see, no record of math being around until humans came on the scene. No sign of mother nature producing math books.
Yeah, I would say it is a given that math is a manmade construct.
"Maybe you or Jerad would like a crack at that one which counter will count more numbers, for infinity?"
As long as I can make a onetoone correspondence I'm good.
You can't, not with the counters. So you are not good.
IOW you LOSE and you just refuse to deal with it.
"And apparently no one has found any use for saying that all countable and infinite sets have the same cardinality."
No.
No what? You have failed to tell me of any use. You just bluff your way along as if your bluffs mean something.
And fuck you, I have looked. I didn't find one word pertaining to what I am asking.
BTW, assface coward, I am not forcing you to post here nor respond to me in any way.
At 11:49 PM, socle said…
Yes, I agree that "infinity" exists as a concept, ie a mental construct.
Excellent.
And apparently no one has found any use for saying that all countable and infinite sets have the same cardinality.
Here's the situation: There are sets that are not finite, yet are not uncountable. I take it you agree with that. Cantor's method recognizes this category and gives the same cardinality for each.
You are the one who wants to take the truly useless extra step to further subdivide this class of sets. You want to say that {0, 1, 2, ...} is somehow larger than {1, 2, 3, ...}, {1, 2, 3, ...} is larger than {2, 4, 6, ...}, and so on.
Yet you cannot find a single example where this extra step has any utility. None. Not to mention your "methodology" gives contradictory results, which you attempt to mask with bogus references to "relativity".
Now you're saying that prime numbers don't have square roots. lol. If it's true that you "would love to see how a mathematician responds to Einstein's train", then write all this up and send it to Dembski or Granville Sewell and see what they say.
At 12:53 AM, Unknown said…
" 'So if you had a square field 1 mile x 1 mile then what woud you say it's diagonal was?'
It would be the length of a FINITE line connecting the two end points. 7467 feet 9/16 inches will do it.
And if you disagree then prove it."
No, I don't disagree with real life approximations, what else can you do? My question has to do with increasing the size. My next question would have been: what about a square field 10 miles on a side, ignoring curvature of the earth. The ppoint being that as you ramp up the size you have to use more and more decimal places in the expansion of root 2 so, if you ever say 'so many places is enough' you might find that it isn't enough for some application.
Practically, on a daily basis, we could easily get by with fairly few decimal places, I agree.
Anyway, it's still true that there is an exact value for root 2 but we cannot, in a finite period of time, write it down in a decimal expansion.
"Let's see, no record of math being around until humans came on the scene. No sign of mother nature producing math books.
Yeah, I would say it is a given that math is a manmade construct."
So, the second law of thermodynamics . . . that wasn't true before humans were around? Or the basic rules of arithmetic? Do you think we can't model natural processes before man with mathematics?
" 'As long as I can make a onetoone correspondence I'm good.'
You can't, not with the counters. So you are not good."
I can with the number sequences though and that's the point. Your counters are irrelevent. It's not the rate the elements in the sets are counted that's important.
"No what? You have failed to tell me of any use. You just bluff your way along as if your bluffs mean something.
And fuck you, I have looked. I didn't find one word pertaining to what I am asking. "
What did you look at? Try taking a Set Theory course, that would be the best way.
I'm not bluffing. It's like learning to ride a bicycle. If you want to know somethings you have to do them.
"BTW, assface coward, I am not forcing you to post here nor respond to me in any way."
Temper, temper! I never said you did. I just wondered about your motivation. What is your problem? Why can't you have a disagreement without swearing? Why can't people disgaree with you without being losers or cowards? You're never wrong (apparently) and most everyone else is a knave or a fool. Weird.
At 7:32 AM, Joe G said…
And apparently no one has found any use for saying that all countable and infinite sets have the same cardinality.
Here's the situation: There are sets that are not finite, yet are not uncountable. I take it you agree with that. Cantor's method recognizes this category and gives the same cardinality for each.
Nonsequitur.
You are the one who wants to take the truly useless extra step to further subdivide this class of sets.
How is it useless? Please be specific.
BTW you have forgotten to tell me how Cantor's method is useful. It's as if you and Jerad are knownotthing cowards.
You want to say that {0, 1, 2, ...} is somehow larger than {1, 2, 3, ...}, {1, 2, 3, ...} is larger than {2, 4, 6, ...}, and so on.
I have explained how IOW "somehow" proves that you are an asshole who refuses to follow along.
Yet you cannot find a single example where this extra step has any utility.
Cantor's method doesn't have any utility mine has the utility of following reality.
Not to mention your "methodology" gives contradictory results, ...
Nope.
Now you're saying that prime numbers don't have square roots.
They don't. If you think they do then please give the exact square root of 2,3,5 and 7.
If it's true that you "would love to see how a mathematician responds to Einstein's train", then write all this up and send it to Dembski or Granville Sewell and see what they say.
Well you abnd Jerad have choked on it. That is good enough for me.
At 7:39 AM, Joe G said…
I don't disagree with real life approximations, what else can you do?
What approximation?
Anyway, it's still true that there is an exact value for root 2 but we cannot, in a finite period of time, write it down in a decimal expansion.
No, there isn't an exact value for teh square root of 2. If there were you would be able to write it down.
I can with the number sequences though and that's the point. Your counters are irrelevent.
Spoken like a true coward, Jerad.
Cantor's method is irrelevant and you cannot demonstarte any utility for it.
It's not the rate the elements in the sets are counted that's important.
Right, it the number of elements that are importatnt. And my method captures that number whereas Cantor thrrows up his hands and sez "they're equal".
I'm not bluffing.
Yes, you are. Ya see Jerad, if you weren't bluffing you could tell me in your own words and produce a reference that is a passage of a book as opposed to telling me to go read a book.
And the reason I insult you Jerad is because you are being an asshole.
At 9:24 AM, socle said…
Now you're saying that prime numbers don't have square roots.
They don't. If you think they do then please give the exact square root of 2,3,5 and 7.
Lol. Berlinski discusses Dedekind cuts in A Tour of the Calculus. If you construct the real numbers using that method, sqrt(2) = {x in Q : x < 0 or x^2 < 2}. The square roots of 3, 5, and 7 are defined similarly.
Good thing you've been brushing up on set theory.
At 9:52 AM, Unknown said…
" 'I don't disagree with real life approximations, what else can you do?'
What approximation?"
Any finite decimal expansion of root 2 or root 3 or pi or e is an approximation.
" 'Anyway, it's still true that there is an exact value for root 2 but we cannot, in a finite period of time, write it down in a decimal expansion. '
No, there isn't an exact value for teh square root of 2. If there were you would be able to write it down."
You can't write down the square root of −1 but it's used very precisely. There are numbers that are not exactly representable in our number sheme but that doesn't mean they don't have an exact value.
" 'I can with the number sequences though and that's the point. Your counters are irrelevent.'
Spoken like a true coward, Jerad."
What? I'm just trying to answer your queries.
"Cantor's method is irrelevant and you cannot demonstarte any utility for it."
You will only accept things that you consider utilitarian. You won't accept that it one of the bases of Set Theory. But you say you took a course so I shouldn't have to point this out. AND, if you really wanted to know, you could go look it up.
" 'It's not the rate the elements in the sets are counted that's important.'
Right, it the number of elements that are importatnt. And my method captures that number whereas Cantor thrrows up his hands and sez "they're equal"."
Nope, you just don't grasp the onetoone correspondence. Which is fine, not everyone does. But don't say you're right and everyone else is wrong.
" 'I'm not bluffing.'
Yes, you are. Ya see Jerad, if you weren't bluffing you could tell me in your own words and produce a reference that is a passage of a book as opposed to telling me to go read a book."
I have been telling you in my own words. If you would really, seriously consider a quote from a book then I'll look for one. That would take time and work and, in my experience, your past pattern is to just claim it's all crap anyway.
"And the reason I insult you Jerad is because you are being an asshole."
In what way? You've got an attitude the size of a planet. I'm trying to answer your questions but you're just name calling and avoiding dealing with things YOU BROUGHT UP!! Like the smallest number in (0, 1). You say it exists, no one else does. It's up to you to show it exists. Do you? Nope. You ask me for a statement from a book to justify my view but do you provide one for your views? Nope. You say prime numbers don't have square roots and then say if you can't write down a full decimal expansion of a number then it doesn't exist. WTF?? Where did you get that? Root 2 is the number that, when squared, gives you two. I can write that down: a square root sign with a 2 inside it.
At 10:19 AM, Joe G said…
They don't. If you think they do then please give the exact square root of 2,3,5 and 7.
Berlinski discusses Dedekind cuts in A Tour of the Calculus. If you construct the real numbers using that method, sqrt(2) = {x in Q : x < 0 or x^2 < 2}. The square roots of 3, 5, and 7 are defined similarly.
LoL! Unfortunately for you that isn't an exact anything.
What's the exact number, for the square root of 2, socle?
At 10:33 AM, Joe G said…
Any finite decimal expansion of root 2 or root 3 or pi or e is an approximation.
The line connecting the two end points is not an approximation. It can be measured.
You can't write down the square root of −1 but it's used very precisely.
Nonsequitur.
There are numbers that are not exactly representable in our number sheme but that doesn't mean they don't have an exact value.
You mean we can represent them as some value that can be used, somehow.
What? I'm just trying to answer your queries.
Nope, you are avoiding the scenario, just as predicted.
You will only accept things that you consider utilitarian.
If they don't have any utility then they can be neither proven nor disproven ie it is meaningless.
You won't accept that it one of the bases of Set Theory.
It's a phony base.
But you say you took a course so I shouldn't have to point this out. AND, if you really wanted to know, you could go look it up.
Actually set theory was part of the math courses I took. And your continued bluff is duly noted.
Nope, you just don't grasp the onetoone correspondence.
Yes, I do. YOU just cannot grasp teh fact that all that does is realign the nember line such that it is no longer a reflection of a number line.
But don't say you're right and everyone else is wrong.
My challenge, which you cowardly avoid, demonstrates that i am right and you are wrong.
I have been telling you in my own words.
Liar You have NOT said one word pertaining to the utility of saying all countable and infinite sets have the same cardinality.
In what way are you an asshole?
1 For avoiding the Einstein Train challenge and cowardly declaring it irrelevant
2 For your total inability to tell me the utility of saying that all countable and infinite sets have the same cardinality
That's just for starters.
At 10:33 AM, socle said…
LoL! Unfortunately for you that isn't an exact anything.
What's the exact number, for the square root of 2, socle?
Exactly what I posted. If A = {x in Q : x < 0 or x^2 < 2}, then A^2 = 2. Therefore A is the (principal) square root of 2.
It's times like this that I really do suspect you are a deep cover sock. If so, carry on!
At 10:34 AM, Joe G said…
Is there a math book that proves there are as many nonnegative even integers as there are nonnegative integers even though the nonnegative integers are the sum of the nonnegative even and odd integers?
How can that be?
Let A = all even nonnegative integers.
Let B = all odd positive integers and C = all nonnegative integers:
If A + B = C AND neither A, B, nor C = 0, then how can A = C and B = C?
At 10:36 AM, Joe G said…
So infinity exists now, but will cease to exist at some point. And that makes sense to you, how?
BTW, at the point it ceases to exist the set of nonnegative integers will have 2x the cardinality as the set of positive even integers.
At 10:39 AM, Joe G said…
I would love to see how a mathematician responds to Einstein's train one train, two counters one counting all nonnegative integers and one counting only the positive even integers.
Jerad and socle choked.
First I would have to know if the mathematician understood that infinity is a journey.
And they don't seem to grasp that infinity is a journey.
At 10:41 AM, Joe G said…
Exactly what I posted. If A = {x in Q : x < 0 or x^2 < 2}, then A^2 = 2. Therefore A is the (principal) square root of 2.
Umm, that ain't an exact number, socle. That is a represenation of the number because the number cannot be produced.
Huge difference.
At 10:54 AM, socle said…
Umm, that ain't an exact number, socle. That is a represenation of the number because the number cannot be produced.
Huge difference.
You are aware that the symbol 2 is not a number either right? It's a numeral which represents the number 2, which is an abstract entity. You cannot write down numbers, only representations of numbers.
There is a famous book called Mathematical Cranks which is a compilation of letters written by crazy people who have claimed to square the circle, trisect general angles, and so on. It's as if you are typing stuff directly out of it here.
At 11:03 AM, Joe G said…
You are aware that the symbol 2 is not a number either right?
How do you know?
It's a numeral which represents the number 2, which is an abstract entity.
Absract entity, as in artificial, as in will cease to exist when intelligent agecies cease to exist just like infinity.
But anyway, there still isn't any exact number for the square root of two. Yes you can make an alphanumeric represenation, or use some other symbols to represent it, but you cannot write down the exact number.
It's as if you are typing stuff directly from your ass.
At 11:17 AM, socle said…
But anyway, there still isn't any exact number for the square root of two. Yes you can make an alphanumeric represenation, or use some other symbols to represent it, but you cannot write down the exact number.
You can't write down any number, strictly speaking. Not even the number represented by "2".
I guess no numbers exist in JoeMath.
At 11:22 AM, Joe G said…
You can't write down any number, strictly speaking.
And yet we do. You lose.
At 11:40 AM, Unknown said…
From a discussion on Quora:
"As to the [computer] halting problem, you often need to assume countablility in order to work with many mathematical constructs like induction (barring transfinite induction). If you set up an infinite process and you want to be assured that there is a next thing to do and that things can eventually end and be well ordered, then you have to work with countable sets. Uncountable sets don't provide you with a "next" element nor a notion of an end."
http://www.quora.com/Algorithms/WhatisthedifferencebetweencountableinfinityanduncountableinfinityWikipediadefinesbothbutitisnotclearhowthisdifferientationworkspracticallyasinanapplication
And from a The Straight Dope discussion
"Two direct applications of Cantor's diagonalization method are Goedel's Incompleteness Theorem and Turing's proof of the Halting Problem.
The latter (and its variations) is extremely important in CS. There will never be an automated method to check correctness of computer programs. The number of related uses is huge. E.g., there can never be a "perfect" computer virus detector. And on and on.
Method aside, there is still the following simple observation: There are a countably infinite number of programs (solutions) but an uncountably infinite number of problems. Ergo, there are a lot more problems that cannot be solved than can be solved. (But ut's nice to have simple to understand ones like the halting problem at hand.)
With Math, Goedel's is quite important (to say the least) as well. A lot of interesting problems can be phrased as relatively short Math Theorems. But there is not only no easy way, there is no way at all, to prove such Theorems in general. There is an inherent limitation on human knowledge."
http://boards.straightdope.com/sdmb/showthread.php?t=344139
A discussion of the Halting Problem:
http://en.wikipedia.org/wiki/Halting_problem
At 11:52 AM, Joe G said…
As to the [computer] halting problem, you often need to assume countablility in order to work with many mathematical constructs like induction (barring transfinite induction). If you set up an infinite process and you want to be assured that there is a next thing to do and that things can eventually end and be well ordered, then you have to work with countable sets. Uncountable sets don't provide you with a "next" element nor a notion of an end."
And what does that have to do with the question at hand:
What utility does it have to say that all countable and infinite sets have the same cardinality?
At 11:53 AM, Unknown said…
" 'Any finite decimal expansion of root 2 or root 3 or pi or e is an approximation.'
The line connecting the two end points is not an approximation. It can be measured."
Only to a certain degree of accuracy, i.e. it's an approximation of the real value.
" 'There are numbers that are not exactly representable in our number sheme but that doesn't mean they don't have an exact value.'
You mean we can represent them as some value that can be used, somehow.'
Yup, like pi or e or root 2. We can use them without writing out their entire decimal expansion.
"If they don't have any utility then they can be neither proven nor disproven ie it is meaningless."
You're not really an abstract type of guy are you?
" 'Nope, you just don't grasp the onetoone correspondence.'
Yes, I do. YOU just cannot grasp teh fact that all that does is realign the nember line such that it is no longer a reflection of a number line."
So? When you're comparing two sets who cares what order the elements are listed in? What matters is if every element of one set has a partner in the other set. And the positive integers and the positive even integers do:
n <> 2n
Simple.
"My challenge, which you cowardly avoid, demonstrates that i am right and you are wrong."
Uh . . . I don't think so.
"Liar You have NOT said one word pertaining to the utility of saying all countable and infinite sets have the same cardinality."
Whatever.
"1 For avoiding the Einstein Train challenge and cowardly declaring it irrelevant"
It is irrelevent. When you are comparing the sizes of two infinite sets the speed you count them or the order you count them is irrelevent. Rearrange them all you like. All that matters is that there is a onetoone correspondence. Anyway, disagreeing with you is not being an asshole or a coward.
"2 For your total inability to tell me the utility of saying that all countable and infinite sets have the same cardinality"
I will try to justify it to you if you try and find the smallest number in (0, 1)
Or are you exempt from your own rules? Let's see if you can meet your own standards . . . rather than just hoping we'll forget 'cause you're banging on about something you could look up if you were really interested.
At 11:58 AM, Unknown said…
From a discussion on CrossValidated:
"Another concept of "categorical" is that each outcome must be distinguishable from every other. This strongly suggests that any probability measure must be totally discrete: that is, all subsets are measurable, implying that each category will have its own welldefined probability. (This is not the case for continuous distributions.)
This would seem to indicate that the number of categories should be finite or at most countable, but that is not evident in the literature. For instance, an archetypal example of a categorical variable is a set of names. The set of all possible names on any finite alphabet is countable but not finite. It is therefore useful to allow countably infinite sets to be categorical. For example, if we are studying names given to babies, it is convenient to let the sample space consist of all possible names (rather than all names that we know of).
A slightly less realistic, but still conceivable, example of a categorical variable would be one that uses real numbers for names. In effect, such a variable would ignore all the usual mathematical structure on this set. I don't see any problem with such a usage, but it's worth observing that the axioms of probability imply that any probability distribution valid in this context would (a) assign a nonnegative value to each real number and (b) would assign a nonzero value to at most a countable infinity of the reals.
One application involving an uncountable sample space that supports categorical random variables of infinite, even uncountable, support is the study of random graphs. To understand the rate of growth of some property of graphs, we would want to contemplate graphs on 0, 1, ..., n, ... nodes, so it's convenient to allow graphs to have countably many nodes. Random variables defined on this set can have various types. For instance, the mean vertex degree (if finite) could be considered of ratio type; the total vertex degree could be considered of ordinal type (and, thereforeby forgetting the orderingis a nice example of a countable discrete variable). If we also allow a graph to have arbitrarily many edges and are interested in, say, its connected components, then we would have a naturally occurring category that is uncountable (because each connected component determines the subset of nodes it contains and there are uncountably many subsets of a countable set).
To summarize, it is reasonable to allow categorical values to attain an uncountable infinity of possible values, while recognizing that at most a countable number of them could ever have positive probabilities. This must be a discrete distribution, because all subsets are measurable, which is not the case for continuous distributions."
http://stats.stackexchange.com/questions/17029/cancategoricaldataonlytakefinitelyorcountablyinfinitelymanyvalues
At 12:01 PM, Unknown said…
From Topology and it's applications:
"If G is an Abelian group then G# is G with its maximal totally bounded group topology. We show that if G contains an infinite Boolean subgroup, then G# contains a countable infinite closed subset that is not a retract of G#. This answers a question posed by E.K. van Douwen."
http://www.sciencedirect.com/science/article/pii/0166864194000520
Do you know what a Abelian group is? Do you know what topology is? Hmmm??
At 12:04 PM, Unknown said…
From a discussion of Every Infinite Set has a Countably Infinite Subset on Wikipedia:
"What this in effect shows is that countably infinite sets are the smallest possible infinite sets."
http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset
At 12:08 PM, Unknown said…
"What utility does it have to say that all countable and infinite sets have the same cardinality?"
If two sets are shown to have the same cardinality and one is countably infinite then so is the other.
And countably infinite means being put into a onetoone correspondence with the postitive integers. So, if you're working on the halting problem and you have some looping structure that may generate intinite responses but they can be put into onetoone correspondence with the integers then they are countably infinite.
At 12:08 PM, Joe G said…
Only to a certain degree of accuracy, i.e. it's an approximation of the real value.
It is the real value.
Yup, like pi or e or root 2. We can use them without writing out their entire decimal expansion.
You cannot write their entire decimal expansion, that is the point.
When you're comparing two sets who cares what order the elements are listed in?
It matters if you want to compare reality or a realignment of reality.
What matters is if every element of one set has a partner in the other set.
And how they are aligned matters in if they have a onetoone correspondence.
And the positive integers and the positive even integers do:
Only if they are realigned, ie taken out of their natural state on the number line.
"My challenge, which you cowardly avoid, demonstrates that i am right and you are wrong."
Uh . . . I don't think so.
Of course it does.
Try it start at 0 and count every nonnegative integer with one counter and every positive even integer with another. The counter counting the nonnegative integers will always be at least 2x that as teh other counter, ie it will always have more elements ALWAYS as long as infinity exists and especially when infinity ceases to exist.
1 For avoiding the Einstein Train challenge and cowardly declaring it irrelevant"
It is irrelevent.
So counting is irrelevant when counting the number of elements? Who would have thought!?
When you are comparing the sizes of two infinite sets the speed you count them or the order you count them is irrelevent.
Umm we are counting both at the same speed as they appear on the number line.
Rearrange them all you like.
Umm, Cantor rearranges them. I keep them in their natural place on the number line. Please TRY to follow along.
All that matters is that there is a onetoone correspondence.
There is only a onetoone correspondence if you rearrange them. And that should tell you something but obvioulsy you are clueless.
2 For your total inability to tell me the utility of saying that all countable and infinite sets have the same cardinality"
I will try to justify it to you if you try and find the smallest number in (0, 1)
10^150.
BTW I did look it up and found NOTHING. And if there was some utility then YOU should be able to tell me, but you can't. So either there isn't any utility or you are ignorant of it.
At 12:12 PM, Joe G said…
"What utility does it have to say that all countable and infinite sets have the same cardinality?"
If two sets are shown to have the same cardinality and one is countably infinite then so is the other.
LoL! That's a utility? Really?
And countably infinite means being put into a onetoone correspondence with the postitive integers.
Only if you rearrange them, ie take them out of their natural place on the number line.
At 12:12 PM, Joe G said…
From a discussion of Every Infinite Set has a Countably Infinite Subset on Wikipedia:
"What this in effect shows is that countably infinite sets are the smallest possible infinite sets."
http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset
What does that have to do with anything I am posting?
Literature bluffer Jerad
At 12:14 PM, Joe G said…
From a discussion on CrossValidated:
What does that have to do with anything I have posted?
At 12:15 PM, Joe G said…
Jerad the gish gallop of irrelevant websites...
At 12:27 PM, socle said…
Joe, you are a great American.
At 12:41 PM, Joe G said…
Wow, that must mean something, coming from a sockpuppet and all...
At 2:18 AM, Unknown said…
" 'Only to a certain degree of accuracy, i.e. it's an approximation of the real value.'
It is the real value."
If you construct a square one unit on a side and measure the diagonal you will get an approximation to the square root of 2. If you square the measured value you will not get 2 exactly. It's an approximation. And the actual value must be the square root of two or the Pythagorean theorem is wrong. Along with Trigonometry.
"t matters if you want to compare reality or a realignment of reality."
What? What difference does it make which order you list the elements of a set as long as you don't miss any?
"Only if they are realigned, ie taken out of their natural state on the number line."
So, you do agree there is a onetoone correspondence! Whew!
"Try it start at 0 and count every nonnegative integer with one counter and every positive even integer with another. The counter counting the nonnegative integers will always be at least 2x that as teh other counter, ie it will always have more elements ALWAYS as long as infinity exists and especially when infinity ceases to exist."
Yup, that is obviously true. Well, not the part about infinity ceasing to exist. You are charmingly humancentric. BUT if you list the postitive integers and line that list up with a list of the positive even integers there is a onetoone correspondence between the lists. The speed that your trains are counting the elements of the sets has NOTHING to do with Einstein AND does not contradict the onetoone correspondence.
"So counting is irrelevant when counting the number of elements? Who would have thought!?"
The speed you are counting the elements is irrelevant. The total number of elements is relevant. To avoid the kind of confusion you are stuck in is why a onetoone correspondence is used.
"Umm we are counting both at the same speed as they appear on the number line."
It doesn't matter how dense they are on the number line!! The set {½, ⅓, ¼, 1/5, . . . } is also countably infinite and a train starting at one and going backwards to zero at some speed would be counting them faster than counting the positive integers when heading the other direction at the same speed. Does that mean there are more elements in that set? Please answer the question.
"Umm, Cantor rearranges them. I keep them in their natural place on the number line. Please TRY to follow along."
I am following along. I am disagreeing with you. For some reason you think disagreeing with you makes me an idiot.
"There is only a onetoone correspondence if you rearrange them. And that should tell you something but obvioulsy you are clueless."
At least you agree there is a onetoone correspondence. That's good.
" 'I will try to justify it to you if you try and find the smallest number in (0, 1)'
10^150."
I can think of infinitly many numbers in (0, 1) smaller than that:
10^151, 10^152, 10^153 . . . . .
Stop wasting time and really try.
" 'What this in effect shows is that countably infinite sets are the smallest possible infinite sets.'
What does that have to do with anything I am posting?"
It shows that putting a set into a onetoone correspondence with the postitive integers means it's countably infinite, can't be smaller and still infinite and, therefore, is numerable. Numberable means, in computer programming terms, that the halting issue has a resolution.
Try and keep up.
" 'From a discussion on CrossValidated'
What does that have to do with anything I have posted?"
I find discussions of applications of things being countably infinite and you scoff.
You asked. It's not my fault if you didn't understand the material I found.
At 7:36 AM, Joe G said…
If you construct a square one unit on a side and measure the diagonal you will get an approximation to the square root of 2.
I say there isn't any such thing as the square root of 2.
"t matters if you want to compare reality or a realignment of reality."
What? What difference does it make which order you list the elements of a set as long as you don't miss any?
Umm I told you in the sentence you responded to.
"Only if they are realigned, ie taken out of their natural state on the number line."
So, you do agree there is a onetoone correspondence!
Only with mental trickery.
"Try it start at 0 and count every nonnegative integer with one counter and every positive even integer with another. The counter counting the nonnegative integers will always be at least 2x that as teh other counter, ie it will always have more elements ALWAYS as long as infinity exists and especially when infinity ceases to exist."
Yup, that is obviously true.
Thank you! Finally you see the light!
Well, not the part about infinity ceasing to exist.
Seeing that infinity only exists in our minds it will end when we end.
BUT if you list the postitive integers and line that list up with a list of the positive even integers there is a onetoone correspondence between the lists.
Right, if you treat the numbers like arbitrary objects and play metal tricks, you can get a onetoone correspondence.
The speed you are counting the elements is irrelevant. The total number of elements is relevant.
YOU have already admitted that the total number of elements is different.
"Umm, Cantor rearranges them. I keep them in their natural place on the number line. Please TRY to follow along."
I am following along. I am disagreeing with you.
How can you disagree with a FACT?
"There is only a onetoone correspondence if you rearrange them. And that should tell you something but obvioulsy you are clueless."
At least you agree there is a onetoone correspondence.
There isn't. That only exists in the minds of the willfully ignorant.
" 'I will try to justify it to you if you try and find the smallest number in (0, 1)'
10^150."
I can think of infinitly many numbers in (0, 1) smaller than that:
10^151, 10^152, 10^153 . . . . .
Just because you can think of something that doesn't make it so.
What does that have to do with anything I am posting?"
It shows that putting a set into a onetoone correspondence with the postitive integers means it's countably infinite, can't be smaller and still infinite and, therefore, is numerable.
What does that have to do with anything I have posted?
I find discussions of applications of things being countably infinite and you scoff.
I did NOT ask about that you asshole.
Irt's as if Jerad is having two discussions one with me and one with someone else and he is responding to me when he should be responding to that other person.
At 8:56 AM, Unknown said…
"I say there isn't any such thing as the square root of 2."
I guess the Pythagorean theorem is wrong then. And Trigonometry is just made up crap.
" 'So, you do agree there is a onetoone correspondence!'
Only with mental trickery."
And if two sets have a onetoone correspondence then they are the same size.
"Right, if you treat the numbers like arbitrary objects and play metal tricks, you can get a onetoone correspondence."
In this case they are just elements in sets.
"YOU have already admitted that the total number of elements is different."
Nope, I said that the way you're counting them would always give you those answers FOR ANY FINITE PERIOD OF TIME.
"How can you disagree with a FACT?"
'Cause the way you count them is not relevent to the sizes of the sets.
" 'At least you agree there is a onetoone correspondence.'
There isn't. That only exists in the minds of the willfully ignorant."
Whatever.
" 'I can think of infinitly many numbers in (0, 1) smaller than that:
10^151, 10^152, 10^153 . . . . . '
Just because you can think of something that doesn't make it so."
So, the numbers I thought of AREN'T in (0, 1) and smaller than 10^150?
"Irt's as if Jerad is having two discussions one with me and one with someone else and he is responding to me when he should be responding to that other person."
Not my fault if you ask questions and then can't understand the answers.
Nor that you can't come up with the smallest number in (0, 1)
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