Set Theory: Counting is Irrelevant when Counting the Number of Elements?

The cardinality of a set refers to the number of elements it contains. A number that is arrived at by counting the elements in the set.
However when confronted with a set that is infinite, ie only ends when we* do, you cannot count them all. So, knowing that infinity is a journey, I said that one can take countably infinite sets and compare them by counting a finite representation of the set, establish a pattern that will also go on for infinity, and then compare those patterns.
For example:
Let set A = {1,2,3,4,...} and set B = {2,4,6,8,...}. Set A has a 2to1 advantage over set B in the finite represenation to 10. And that advantage will never change, ie it will always remain, always and forever, for infinity. Therefor set A's cardinality > set B's cardinality.
The difference is in the way I compare sets with the way Cantor does. He treats everything in a set as an arbitrary object, with no value. For Cantor, {1,2,3,4,5,6,7,8,9,10} and {2,4,6,8,10} line up:
{1,2,3,4, 5, 6,7,8,9,10}
{2,4,6,8,10}, with the first 5 elements "matching" and set A's last 5 left unmatched.
In JoeMath those sets would line up:
{1,2,3,4,5,6,7,8,9,10}
{ 2, 4, 6, 8, 10}, with the match being the actual matching number. That is because JoeMath treats the numbers with respect and allows them their place on the number line. Not only that JoeMath says that we actually have to count the members of the sets to figure out the cardinality.
No, Richie, one does not have to count every member. All one has to do is establish a pattern in order to get the relative cardinality that is one set's cardinality relative to another set's.
One train with two (or more) counters will always count more positive integers than it will positive even integers always and forever. So why would Cantor say the two sets are equal? It ain't as if the counters are lying...
Assface keiths chimes in:
So 8x7 can be anything? 15 can be a positive number? That is what can happen when you don't treat numbers with the respect I said, ie act like keiths and Cantor numbers have no meaning.
And on another note, no one can download a PDF from a website that they cannot connect to. Fucking morons...
The cardinality of a set refers to the number of elements it contains. A number that is arrived at by counting the elements in the set.
However when confronted with a set that is infinite, ie only ends when we* do, you cannot count them all. So, knowing that infinity is a journey, I said that one can take countably infinite sets and compare them by counting a finite representation of the set, establish a pattern that will also go on for infinity, and then compare those patterns.
For example:
Let set A = {1,2,3,4,...} and set B = {2,4,6,8,...}. Set A has a 2to1 advantage over set B in the finite represenation to 10. And that advantage will never change, ie it will always remain, always and forever, for infinity. Therefor set A's cardinality > set B's cardinality.
The difference is in the way I compare sets with the way Cantor does. He treats everything in a set as an arbitrary object, with no value. For Cantor, {1,2,3,4,5,6,7,8,9,10} and {2,4,6,8,10} line up:
{1,2,3,4, 5, 6,7,8,9,10}
{2,4,6,8,10}, with the first 5 elements "matching" and set A's last 5 left unmatched.
In JoeMath those sets would line up:
{1,2,3,4,5,6,7,8,9,10}
{ 2, 4, 6, 8, 10}, with the match being the actual matching number. That is because JoeMath treats the numbers with respect and allows them their place on the number line. Not only that JoeMath says that we actually have to count the members of the sets to figure out the cardinality.
No, Richie, one does not have to count every member. All one has to do is establish a pattern in order to get the relative cardinality that is one set's cardinality relative to another set's.
One train with two (or more) counters will always count more positive integers than it will positive even integers always and forever. So why would Cantor say the two sets are equal? It ain't as if the counters are lying...
Assface keiths chimes in:
That’s the problem. Unlike Joe, mathematicians don’t treat numbers with respect.
So 8x7 can be anything? 15 can be a positive number? That is what can happen when you don't treat numbers with the respect I said, ie act like keiths and Cantor numbers have no meaning.
And on another note, no one can download a PDF from a website that they cannot connect to. Fucking morons...
121 Comments:
At 2:14 PM, socle said…
Except when "it depends on when you look" of course, then you're screwed.
How do the cardinalities of {..., 6, 4, 2, 2, 4, 6, ...} and {1, 2, 3, ...} compare?
At 2:26 PM, Joe G said…
Except when "it depends on when you look" of course, then you're screwed.
Example please.
How do the cardinalities of {..., 6, 4, 2, 2, 4, 6, ...} and {1, 2, 3, ...} compare?
The second set's cardinality is always greater than or equal to the first set's.
It's not that difficult, socle.
At 2:37 PM, socle said…
Example please.
The one where you invoked "relativity".
How do the cardinalities of {..., 6, 4, 2, 2, 4, 6, ...} and {1, 2, 3, ...} compare?
The second set's cardinality is always greater than or equal to the first set's.
Based on statements you've already made, {..., 6, 4, 2, 2, 4, 6, ...} = 2{2, 4, 6, ...} and {1, 2, 3, ...} = 2{2, 4, 6, ...}, so {..., 6, 4, 2, 2, 4, 6, ...} should equal {1, 2, 3, ...} if your method is to be consistent.
At 2:53 PM, Joe G said…
The one where you invoked "relativity".
They all invoke relativity, even the example you gave.
Based on statements you've already made, {..., 6, 4, 2, 2, 4, 6, ...} = 2{2, 4, 6, ...} and {1, 2, 3, ...} = 2{2, 4, 6, ...}, so {..., 6, 4, 2, 2, 4, 6, ...} should equal {1, 2, 3, ...} if your method is to be consistent.
Relativity it all depends on when you look.
They start out equal no elements. Then the first set picks up a count, the set with the #1 is greater than teh otehr. The the first set picks up another count and at the same time the other counter gets two counts, now they are equal and so on for infinity.
Again, it ain't difficult.
You guys say I don't have any imagination and yet you guys cannot imagine a train with counters, counting and traveling forever down the number line.
Strange...
At 4:03 PM, socle said…
Let's set up some notation:
{1, 2, 3, ...} = X
{2, 4, 6, ...} = Y
{..., 6, 4, 2, 2, 4, 6, ...} = Z
Using my principle {1,2,3,4,...} will always have twice the cardinality as {2,4,6,8,...}.
So X = 2Y
Under your version of set theory, how does the cardinality of {1, 2, 3, ...} compare to {1, 2, 3, ...}?
They would be equal
Presumably this implies that {2, 4, 6, ...} = {2, 4, 6, ...} as well. Then the union is twice as large as either one, so:
{..., 6, 4, 2, 2, 4, 6, ...} = 2{2, 4, 6, ...}.
IOW Z = 2Y.
Since X = 2Y and Z = 2Y, X = Z.
This translates back to {1, 2, 3, ...} = {..., 6, 4, 2, 2, 4, 6, ...}.
This directly contradicts your statement that "the second set will always be larger than the first".
At 5:11 PM, Joe G said…
Umm, asswipe, I said greater than or equal to. It all depends on when you look.
My statement is:
X's cardinality is always greater than or equal to Z's.
And then I explained why.
At 5:55 PM, socle said…
So 8x7 can be anything? 15 can be a positive number?
It all depends on when you look...
At 6:04 PM, Joe G said…
I'm sure it does, to you guys.
Now I understand why you accept Cantor> you're just too stupid to think of anything else.
At 6:32 PM, socle said…
They start out equal no elements. Then the first set picks up a count, the set with the #1 is greater than teh otehr. The the first set picks up another count and at the same time the other counter gets two counts, now they are equal and so on for infinity.
Going back to your analysis of {..., 6, 4, 2, 2, 4, 6, ...} vs. {1, 2, 3, ...}, when are they equal after time 0? Using one train, two counts of course, as per protocol.
That doesn't ever happen. In fact, {1, 2, 3, ...} should be twice the size of {..., 6, 4, 2, 2, 4, 6, ...}, while it is also twice the size of {2, 4, 6, ...}.
Even though {..., 6, 4, 2, 2, 4, 6, ...} is twice the size of {2, 4, 6, ...}.
So we have X = 2Y, X = 2Z, and Y = 2Z. Depending on when you look, of course.
At 9:06 PM, Joe G said…
Going back to your analysis of {..., 6, 4, 2, 2, 4, 6, ...} vs. {1, 2, 3, ...}, when are they equal after time 0? Using one train, two counts of course, as per protocol.
I told you exactly when they are equal.
That doesn't ever happen.
It does if you can count. However you cannot and that is why you prefer Cantor's method it's brainless.
In fact, {1, 2, 3, ...} should be twice the size of {..., 6, 4, 2, 2, 4, 6, ...}, while it is also twice the size of {2, 4, 6, ...}.
Only if you cannot count. Geez I explained it for you.
One more time:
They start out equal no elements. Then the first set picks up a count, the set with the #1 is greater than teh otehr. The the first set picks up another count and at the same time the other counter gets two counts, now they are equal.
1+1=2 and 0+2=2. Every time the first counter counts it counts a single digit. Every time the second counter counts it counts 2 digits, ie the positive and negative counterparts.
So it starts 0=0, then 1>0, then 2=2, then 3>2, then 4=4, then 5>4, then 6=6, and so on, for infinity.
At 9:38 PM, socle said…
What the hell? How is the second counter counting two numbers at a time? That's not how the ET works. The train is in only one position at at time, so each counter can count at most one number at each point, ATAF.
At 10:09 PM, Joe G said…
How is the second counter counting two numbers at a time?
I told you how. Are you ignorant?
That's not how the ET works.
It works exactly how I have described it.
The train is in only one position at at time, so each counter can count at most one number at each point, ATAF.
No, not for a line extending in both directions. Either you absolute the negative side for morons like you that can't follow along in both directions at once or you follow along in both directions at once yup, one train heading in both directions at the same time, just like the number line.
At 10:35 PM, socle said…
yup, one train heading in both directions at the same time, just like the number line.
Woah, dude, check it out: The Cartesian Plane. F*** it, let's work in R^n. Along each ray starting at the origin, there's a train moving toward infinity. Only they're all the same train. #420
At 10:37 PM, Rich Hughes said…
"one train heading in both directions at the same time, just like the number line."
Bwahhahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahahahaahah
At 2:25 AM, Unknown said…
What about comparing
X = {1, 2, 3, 4 . . . . }
W = {1, ½, ⅓, ¼ . . . }
with two trains, both starting at one, heading in opposite directions at the same speed.
The train counting the elements in set W would count infinitily more elements before the other train got to 2 in X. In fact, the 'speed' the elements in W were being counted would increase.
Does that mean W has greater cardinality?
At 7:19 AM, Joe G said…
Yes Richie, you are a moron who doesn't have any imagination.
At 7:23 AM, Joe G said…
What about comparing
X = {1, 2, 3, 4 . . . . }
W = {1, ½, ⅓, ¼ . . . }
with two trains, both starting at one, heading in opposite directions at the same speed.
Why would you? That is like comparing apples to oranges.
At 8:48 AM, Unknown said…
"Why would you? That is like comparing apples to oranges."
They're two infinite sets of numbers. What are their cardinalities?
At 9:04 AM, Joe G said…
Please prove that both are infinite.
At 9:21 AM, Unknown said…
"Please prove that both are infinite."
You are serious?
Does {1, 2, 3, 4 . . . } have a largest element? No.
Does {1, ½, ⅓, ¼, . . . } have a smallest element? No.
They're both infinite sets. AND they have the same cardinality.
How does your counting technique handle them?
At 9:39 AM, Joe G said…
That is your "proof"? LoL!
Good luck showing that the one set doesn't have a largest number and teh other doesn't have a smallest number.
At 12:14 PM, Unknown said…
"Good luck showing that the one set doesn't have a largest number and teh other doesn't have a smallest number."
Let N be the largest number in the set {1, 2, 3, 4 . . . } the set of all positive integers.
N is an integer, therefore N + 1 is an integer and N + ! is greater than N therefore N was not the large positive integer.
Proof by contradiction.
Similarly for the set {1, ½, ⅓, ¼ . . } except that you assume some value is the smallest number in the set and then find one that is smaller that's in the set.
At 1:04 PM, Joe G said…
Let N be the largest number in the set {1, 2, 3, 4 . . . } the set of all positive integers.
N is an integer, therefore N + 1 is an integer and N + ! is greater than N therefore N was not the large positive integer.
N+1 becomes N. Just because the largest number changes, doesn't mean it doesn't exist. Selection pressures change and they still exist.
Refutation by reality.
At 5:45 PM, Unknown said…
"N+1 becomes N. Just because the largest number changes, doesn't mean it doesn't exist. Selection pressures change and they still exist.
Refutation by reality."
If any proposed lagest number is easily superceded then clearly the idea of a lagest number is nonsensical.
Then the idea of a largest number has no meaning. Anything you propose I can beat.
An infinite sequence of N + 1s means the whole concept is pointless.
There is no largest element of {1, 2, 3, 4 . . . .}
And you can't just keep saying "you haven't proved it". That's not an argument. What you have to do is FIND the largest element of that set. Which you, so far, have not been able to do.
Ball is in your court. What are you going to do?
At 5:57 PM, Joe G said…
If any proposed lagest number is easily superceded then clearly the idea of a lagest number is nonsensical.
Why is that? And how do you define easily?
Then the idea of a largest number has no meaning.
Why is that? Does the largest denomination of money have no meaning?
Anything you propose I can beat.
Here's the scenario I start writing a number that takes me many years to write down and just before you die I hand it to you. You don't even have time to read it and you die.
You lose, you didn't beat my number.
An infinite sequence of N + 1s means the whole concept is pointless.
What infinite sequence? Infinity exists only in our minds and will cease to exist when agencies capable of thinking of it are gone.
What part of that don't you understand?
At 6:36 PM, socle said…
Joe,
Whatever number system it is you are talking about, it's not N, the natural numbers. If you look at the 6th Peano Axiom from the wikipedia page:
For every natural number n, S(n) is a natural number.
S stands for successor, and in the usual interpretation of these axioms, the successor of n is denoted n + 1.
So if 10^150 is in N, 10^150 + 1 is in N. The axiom doesn't say five seconds later, 10^150 + 1 is in N.
You're reading things into the axioms that aren't there.
At 1:16 AM, Unknown said…
" 'If any proposed lagest number is easily superceded then clearly the idea of a lagest number is nonsensical.'
Why is that? And how do you define easily?"
Easily because all you have to do is add one to any proposed largest number.
" 'Then the idea of a largest number has no meaning.'
Why is that? Does the largest denomination of money have no meaning?"
What? There are not infinitely many ascending denominations of currency.
" 'Anything you propose I can beat.'
Here's the scenario I start writing a number that takes me many years to write down and just before you die I hand it to you. You don't even have time to read it and you die."
Then someone else will add one. Do I have to rephrase everything just so you can't nit pick?
"You lose, you didn't beat my number. "
But someone else will. Please stop with stupid objections. Your proposed number is beatable. Easily.
"What infinite sequence? Infinity exists only in our minds and will cease to exist when agencies capable of thinking of it are gone."
Why don't you prove that? I prefer to live in a not so limited intellectual world.
"What part of that don't you understand?"
Oh, here we go again. I disagree with you and your methods and there must be something wrong with me.
You don't get to be right just because you think everyone else is stupid. Your mathematical notion were abandoned, some centuries ago, because they couldn't produce the goods. They gave inconsistent and incorrect results. They couldn't handle some situations.
Root 2 exists. It's a number. The Greeks new that. They wanted to think every value could be written as a ration of two whole numbers. Then they found root 2. And so the family of numbers had to be enlarged.
{1, 2, 3, 4 . . . . } is an infinite set. It will be an infinite set always, no matter who is there to contemplate it. So is {1, ½, ⅓, ¼ . . . } AND they have the same cardinality.
Your counting technique is faulty. Count the positive even numbers. The first one is 2, the second one is 4, the third one is 6, the fourth one is 8, etc. Do you see it? That's a onetoone correspondence between the positive integers and the positive even integers. Just counting something is setting up the onetoone correspondence!! And if you can count the elements in a set and the set is inifinite then it's a COUNTABLY infinite set.
There is no smallest element of (0, 1). Nor is there a largest positive integer.
At 7:27 AM, Joe G said…
Whatever number system it is you are talking about, it's not N, the natural numbers.
Prove it.
S stands for successor, and in the usual interpretation of these axioms, the successor of n is denoted n + 1.
If N is teh largest number then when N+1 occurs it becomes the largest number.
And I am defining N as the largest number, not as the natural numbers.
At 7:37 AM, Joe G said…
Jerad,
Go fuck yourself. YOU don't know how to count. YOU cannot demonstrate infinity exists outside of our minds.
And YOU cannot tell me what the exact number for the square root of 2 is.
At 8:03 AM, socle said…
Whatever number system it is you are talking about, it's not N, the natural numbers.
Prove it.
I already posted that theorem which states if N exists, then an infinite set exists. Since no infinite sets exist in JoeMath, N doesn't either.
You can use whatever system of maths that you want, but JoeMath and normal math are two different things. You could also create your own variant on chess, let's say JoeChess, where the knights move 1 space in any direction instead of moving the normal way. Just because something can or cannot happen in JoeChess does not mean it can or cannot happen in normal chess.
And I am defining N as the largest number, not as the natural numbers.
I thought the LKN, erm, LN was the largest number? Anyway I'll use a boldface N for the natural numbers, which is standard.
At 8:28 AM, Joe G said…
I already posted that theorem which states if N exists, then an infinite set exists.
It's a theorem, not a law.
Since no infinite sets exist in JoeMath, N doesn't either.
That doesn't follow.
You can use whatever system of maths that you want, but JoeMath and normal math are two different things.
Only to the ignorant.
And I am defining N as the largest number, not as the natural numbers.
I thought the LKN, erm, LN was the largest number?
Look, you have to follow along if you are going to stick your nose into other people's discussions.
At 9:06 AM, Unknown said…
"Go fuck yourself. YOU don't know how to count. YOU cannot demonstrate infinity exists outside of our minds."
Uh huh.
"And YOU cannot tell me what the exact number for the square root of 2 is."
What do you think it is then?
At 9:08 AM, Joe G said…
So Jerad has nothing to say about infinity and he also cannot follow along Jerad, there isn't any exact number for teh square root of 2. 2 is a prime number which means it is divisble only by 1 and itself.
At 9:33 AM, Unknown said…
What do you think the length of the diagonal of a square with sides of length one is?
Does your result match with the Pythagorean theorem and Trigonometry?
root 2 = the secant of 45 degrees.
At 9:34 AM, Joe G said…
Jerad, there isn't any exact number for the square root of 2. 2 is a prime number which means it is divisble only by 1 and itself.
At 9:38 AM, Unknown said…
So, what the length of the diagonal of a unit square?
At 9:40 AM, Joe G said…
Measure it you stupid fuck.
At 4:34 PM, Unknown said…
" 'So, what the length of the diagonal of a unit square?'
Measure it you stupid fuck."
I have done, mathematically. What did you get?
And please, try and actually answer a mathematical question. The more you swear and bluster the less anyone is going to believe you.
If you can't answer that question, or the other ones YOU have said have answers (what is the smallest element in (0, 1) for example) you're just going to look foolish.
At 4:36 PM, Joe G said…
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 the other counter, ie it will always have more elements ALWAYS as long as infinity exists and especially when infinity ceases to exist.
And no one can demonstrate otherwise. All Jerad can do is act like the little whiny baby that he is.
At 4:58 PM, Unknown said…
What is the length of the diagonal of a unit square?
What is the smallest element of (0, 1)?
What is the cardinality of {1, ½, ⅓, ¼ . . . }
Simple questions. Answers which the mathematical community have agreed upon.
But what do you say? Your forum. Deliver the goods.
At 5:01 PM, Joe G said…
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 the other counter, ie it will always have more elements ALWAYS as long as infinity exists and especially when infinity ceases to exist.
And no one can demonstrate otherwise. All Jerad can do is act like the little whiny baby that he is.
At 5:27 PM, Unknown said…
Oh dear, Joe is staking his whole approach on one single flawed question. Despite the fact that his query/challenge has been addressed many times. And, even if he disagrees with the responses, you'd think he'd be more sensible that to fall back upon this one issue. Unless, perish the thought, that's all he's got.
What I'd really like to know is:
What is the smallest element of (0, 1)? Joe says it exists but he can't produce it.
What is the cardinality of {1, ½, ⅓, ¼ . . } Joe says his method is superiour to Cantor's but he can't answer this question.
What is the length of the diagonal of a unit square? Joe has refused to even guess at this.
At 5:45 PM, Joe G said…
Jerad,
You are too stupid to understand that the rate of count and teh number of elements are directly correlated, ie they have a onetoone correspondence.
And I have told you what the diagonal is. And if you want answers to your other questions then pay me. As I said you are too stupid to even grasp the fact that the rate and number of elements are the same thing.
At 6:05 PM, Unknown said…
"You are too stupid to understand that the rate of count and teh number of elements are directly correlated, ie they have a onetoone correspondence.
And I have told you what the diagonal is. And if you want answers to your other questions then pay me. As I said you are too stupid to even grasp the fact that the rate and number of elements are the same thing."
The rate you are counting and the number of elements are equivalent? Really?
So, how many elements are in {1, ½, ⅓, ¼ . . . ]
At 10:52 PM, Joe G said…
The rate you are counting and the number of elements are equivalent? Really?
Yes, there is a onetoone correspondence.
So, how many elements are in {1, ½, ⅓, ¼ . . . ]
More than the number of elements in the set {1, 1/2, 1/4, 1/8,...}
However I am sure you won't be able to grasp that either.
At 1:02 AM, Unknown said…
"Yes, there is a onetoone correspondence."
You don't understand what onetoone correspondence means.
" 'So, how many elements are in {1, ½, ⅓, ¼ . . . ]'
More than the number of elements in the set {1, 1/2, 1/4, 1/8,…}"
Nice try but you still didn't answer the question. And now you've got two infinite series whose cardinality you can't find.
"However I am sure you won't be able to grasp that either."
I know your contention is wrong. And since you can't tell me what the cardinalities are then your method is pretty useless isn't it? It can't deliver the goods. It doesn't work. AND you end up with lots and lots of different infinities which you cannot arrange in order of size.
At 9:19 AM, Joe G said…
You don't understand what onetoone correspondence means.
No, Jerad, YOU have no clue what a onetoone correspondence means.
What does Cantor say is the cardinality of {1,1/2,1/3,1/4,...}.
He cannot say "infinity" because infinity is NOT a number and cardinality refers to a number. So Cantor loses.
And now you've got two infinite series
No such thing exists in the real world, Jerad.
AND you end up with lots and lots of different infinities which you cannot arrange in order of size.
There aren't any infinities, Jerad. Not in the real world, anyway.
At 9:36 AM, Unknown said…
"What does Cantor say is the cardinality of {1,1/2,1/3,1/4,…}."
That set has cardinality aleph0, it's countably infinite because you can easily see the onetoone mapping or correspondence with the postitve integers: n <> 1/n. Easy.
"He cannot say "infinity" because infinity is NOT a number and cardinality refers to a number. So Cantor loses."
You really, really, REALLY don't understand cardinal numbers. Try shouting and stamping your feet, maybe that will help.
"No such thing exists in the real world, Jerad."
They exist in mathematics. Oh, except in JoeMath. My bad.
"There aren't any infinities, Jerad. Not in the real world, anyway."
Can't handle 'em eh? Too bad, so sad. Try not to get in the way of people who can then.
At 9:47 AM, Joe G said…
aleph0? LoL! So Cantor just makes up a new term.
And infinity will cease to end in mathematics as soon as we cease to exist.
IOW YOU can't handle infinity Jerad. You think it's some sort of magical equalizer.
At 9:51 AM, Unknown said…
"leph0? LoL! So Cantor just makes up a new term."
Looks like JoeMath never did take any set theory.
"And infinity will cease to end in mathematics as soon as we cease to exist."
JoeMath is looking a bit scared now.
"IOW YOU can't handle infinity Jerad. You think it's some sort of magical equalizer."
And . . . JoeMath chokes again!! It's just not his day.
At 12:21 PM, Unknown said…
Just in case it's escaped you (which it shouldn't have done IF you'd taken any set theory) is that aleph is the first letter of the Hebrew alphabet. Aleph0 or Alephnought or Alephnaught is the 'smallest' cardinal number associated with sets that are countably infinite.
Like {1, 2, 3, 4 . . . . } and {1, ½, ⅓, ¼ . . . . } and {1, 1, 2, 3, 5, 8 . . . . ] and {1, 1, 1, 1, 1, 1, 1, 1, . . } and {{ }, {{ }}, {{{ }}} . . . } and {1, −2, 3, −4, 5, −6 . . . . } (how would your train count that last set I wonder?
Better get them JoeMath engines warmed up!! The train is pulling out on an infinite journey, you don't want to be left behind now do you? Back in the Iron Age. When there was no infinity and people fell off the edge of the earth which was the center of the universe. And all those stupid 'elements'? Why some of them have only existed in a lab for picoseconds. How stupid is that. Much easier when there were only 4 elements. That makes sense. You can see that, all around you. I think they had beer back then which is good 'cause that'd be safer to drink than the water.
At 4:20 PM, Joe G said…
Jerad,
Shut the fuck up as you can't even count.
"Oh the rate is not the number!"
But if someone counts faster than another person they are counting more. And if that count continues for infinity it will ALWAYS count more!
"The rate is not the number!"
Except when it is you moron.
And I know what those terms are you asswipe. As I said Cantor had to invent new terms for his diatribe.
At 5:22 PM, Unknown said…
"Shut the fuck up as you can't even count."
Oh dear. JoeMath has to resort to profanity. How does this look in his performance statistics?
"But if someone counts faster than another person they are counting more. And if that count continues for infinity it will ALWAYS count more!"
Sad that JoeMath still doesn't grasp that techniques applicable to finite sets don't work with infinite sets.
"Except when it is you moron."
But then JoeMath can always resort to name calling. It doesn't solve or prove anything but it makes him fell better.
"And I know what those terms are you asswipe. As I said Cantor had to invent new terms for his diatribe."
Gosh, is it possible thet JoeMath has a better understanding of the work and the years that Cantor put into his ideas than we do? Well, if JoeMaths thinks that irrational numbers don't exists and yet people had figured out that they do exist about 2500 years ago then I'd say . . . Nope. JoeMaths probably hasn't got a clue about Cantor's work.
And, let's be honest, JoeMaths still can't answer his own assertion: what is the smallest number in (0, 1) ? JoeMath says it exists but JoeMaths can't find it.
Nor can JoeMaths decide which of these two sets {1, ½, ⅓, ¼ . . . } and {1, 2, 3, 4 . . . } has a greater cardinality. Funny that. It's almost like JoeMaths is actually pretty useless.
At 5:44 PM, Unknown said…
And, let's be clear, JoeMaths still has not found a number he says exists.
Can't keep the fans waiting forever. But if you don't produce they might shift their allegience to another team.
At 11:17 PM, socle said…
But if someone counts faster than another person they are counting more. And if that count continues for infinity it will ALWAYS count more!
Joe,
By this argument, if someone counts the set {1, 2, 3, ...} twice as fast as someone else, "continuing for infinity", then {1, 2, 3, ...} is twice as big as {1, 2, 3, ...}.
In a sense that is true, since 2*aleph_null = aleph_null.
At 2:16 AM, Unknown said…
"Shut the fuck up as you can't even count."
At least I can handle infinities and irrational numbers. JoeMaths can't. JoeMaths can't deliver the goods.
"But if someone counts faster than another person they are counting more. And if that count continues for infinity it will ALWAYS count more!"
Does it make you feel good to repeat the same incorrect notion over and over again even though your counting technique can't deliver the goods? Why are you clinging to a technique that can't handle some situations? Why can't you compare the cardinalities of
{1, 2, 3, 4 . . . . } and {1, ½, ⅓, ¼ . . . . }
Perhaps you should figure out how to handle things like that instead of just whining.
"And I know what those terms are you asswipe. As I said Cantor had to invent new terms for his diatribe."
Keep repeating that to yourself if it makes you feel better. The rest of us will get on with the 21st century. You've got a lot of catching up to do.
At 1:19 PM, Joe G said…
Dear Jerad,
Seeing that you are too stupid to understand that the faster the rate of count means that more elements will be counted and that more elements means a greater cardinality, perhaps mathematics isn’t your thing and you just should shut up.
At 1:20 PM, Joe G said…
By this argument, if someone counts the set {1, 2, 3, ...} twice as fast as someone else, "continuing for infinity", then {1, 2, 3, ...} is twice as big as {1, 2, 3, ...}.
socle, I get it, you are too fucking moronic to follow along.
At 3:12 PM, Unknown said…
"Seeing that you are too stupid to understand that the faster the rate of count means that more elements will be counted and that more elements means a greater cardinality, perhaps mathematics isn’t your thing and you just should shut up."
Casual onlookers, it appears Joe has been replaced with a mindless automation which just farts out the same response when challenged. Perhaps he's away on holiday. Or perhaps he's brain dead. It's so hard to tell. But let us not remember that he never did answer some questions:
What is the smallest element in (0, 1) ?
What is the length of the diagonal of a unit square?
How do the cardinalities of {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . } compare?
What is the circumference of the unit circle . . . exactly?
What is the cardinality of {1, −2, 3, −4, 5, −6 . . . . } ? (It would have to be a really bizarre train to count those values eh?)
And so many, many, many more questions.
Sigh.
Dearly beloved, we are gathered here today to note the passing of JoeMath. It wasn't with us long but, truth be told, it gave us great joy and happiness in its brief life. Since JoeMaths didn't believe in infinity it can't be with us today since, apparently, everything ends after death, but I think we should take at least the breath of one time point to mourn our loss.
Okay, done? Is there a pub nearby?
At 3:25 PM, Joe G said…
What is the smallest element in (0, 1) ?
Alpha 0
What is the length of the diagonal of a unit square?
Between 1.41 and 1.42 units
However , seeing that you are too stupid to understand that the faster the rate of count means that more elements will be counted and that more elements means a greater cardinality, perhaps mathematics isn’t your thing and you just should shut up.
At 5:08 PM, Unknown said…
" 'What is the smallest element in (0, 1) ?'
Alpha 0"
I say, we shoot JoeMaths now and put it out of our misery. Although we would lose the entertainment.
" 'What is the length of the diagonal of a unit square?'
Between 1.41 and 1.42 units"
Correct!! But not an exact answer. You did say there should be an exact answer didn't you? And your answer, when squared, has to equal 2 or Trigonometry is fucked. And the Pythagorean theorem as well. Gosh, sounds like we're heading back to the time when people thoght everything was made up of earth, air, fire and water. Guess the periodic table has to go then. Fair enough, who thought the symbol for tungsten should be W anyway?
"However , seeing that you are too stupid to understand that the faster the rate of count means that more elements will be counted and that more elements means a greater cardinality, perhaps mathematics isn’t your thing and you just should shut up."
Sorry for the reemergence of the JoeMath automated answering system. We are working on the problem and hope to have the situation resolved shortly.
Meanshile, we recommend you go back and read through Euclid's elements just in case you missed something. And, if you have the time, Newton's work on infitesimals. We know Liebnitz got it a bit better but still, at least Newton's work is in English.
At 8:20 PM, Joe G said…
'What is the smallest element in (0, 1) ?'
Alpha 0
I say, we shoot JoeMaths now and put it out of our misery.
Shoot yourself and do society a fovor.
And dickface, if you want to know the exact measurement of the diagonal of a unit square, make one and measure it!
What the fuck is wrong with you? It's as if you are proud to be a total pussy.
Then you take that length, duplicate it, make a square out of those 2 and measure the distance between the two end points. If you did it right you will have 2 units.
It works every time I have done it.
It's as if you are proud to be a fucking wanker.
At 8:47 PM, socle said…
Why isn't half of Alpha 0 also in (0, 1)?
At 9:15 PM, Joe G said…
Because it doesn't exist.
Why isn't half of an amino acid still an amino acid? Why isn't half of a nucleotide still a nucleotide?
Why don't half electrons orbit protons?
BTW have you figured out that the sequence determines the rate?
At 9:39 PM, socle said…
Because it doesn't exist.
Not true. The set of real numbers is closed under multiplication. Since 1/2 and Alpha 0 are real numbers, so is their product.
1/2 * Alpha 0 is nonzero because in R, if x and y are nonzero, so is x*y.
1/2 * Alpha 0 is nonnegative because both 1/2 and Alpha 0 are nonnegative.
1/2 * Alpha 0 is also less than 1 because Alpha 0 is less than 1.
This all means that half of Alpha 0 is in the interval (0, 1).
At 10:30 PM, Joe G said…
No 1/2 electrons. 1/2 Alpha 0 is indistinguishable from the coordinate 0,0. The movement is undetectable by any known instrument.
It's like saying that you have moved your rook because you shifted it's position within its residing square.
1/2 1Alpha 0 = Alpha 0
At 10:53 PM, socle said…
No 1/2 electrons. 1/2 Alpha 0 is indistinguishable from the coordinate 0,0. The movement is undetectable by any known instrument.
It's like saying that you have moved your rook because you shifted it's position within its residing square.
1/2 1Alpha 0 = Alpha 0
Then you're not doing math, Joe. Numbers are not electrons. Look up the axioms (or construction) for the real numbers. The set of real numbers is a field and has no zero divisors.
At 12:10 AM, Unknown said…
"And dickface, if you want to know the exact measurement of the diagonal of a unit square, make one and measure it!"
There you go focusing on dicks again. Is that some kind of obssession of yours?
I don't have to measure it, I know what the length of a diagonal of a unit square is. I can figure it out using real mathematics.
"Then you take that length, duplicate it, make a square out of those 2 and measure the distance between the two end points. If you did it right you will have 2 units.
It works every time I have done it."
Really? You can measure and reproduce root 2 exactly? What about the thickness of the lines you draw? Did you account for those?
I certainly agree with the second squares diagonal being 2. I can do the math without have to measure the things!! Pythgoras was a pretty smart guy!!
At 7:29 AM, Joe G said…
Whatebver socle. It's obvious that Cantor isn't doing math but that doesn't seem to bother any but me.
Numbers have a place on the number line, socle. And if one can fit an infinite number of points in a finite space, then that is totally useless mathematically.
BTW your set of real numbers contains numbers that cannot be added, subtracted, divided nor multiplied.
At 7:41 AM, Joe G said…
I don't have to measure it, I know what the length of a diagonal of a unit square is.
OK, give me an exact number then. Or shut the fuck up.
At 7:45 AM, Joe G said…
Let A = all nonnegative even integers
Let B = all positive odd integers
Let C = all nonnegative integers
A + B = C where neither A nor B = 0
So how can A=B=C? It's impossible unless you are a moron, or a Cantor follower.
At 8:37 AM, socle said…
Numbers have a place on the number line, socle. And if one can fit an infinite number of points in a finite space, then that is totally useless mathematically.
No, that's a basic property of the real numbers. And it's consistent with the fact that R is totally ordered. This way you also don't run out of midpoints of line segments.
BTW your set of real numbers contains numbers that cannot be added, subtracted, divided nor multiplied.
Which ones are those? You certainly can't divide by zero, but maybe you could give examples of real numbers that can't be added or multiplied.
Let A = all nonnegative even integers
Let B = all positive odd integers
Let C = all nonnegative integers
A + B = C where neither A nor B = 0
So how can A=B=C? It's impossible unless you are a moron, or a Cantor follower.
Apparently you are saying that A ∪ B = C, where A and B are nonempty, and are asking how it is possible that A = B = C.
What specifically would that violate? Infinite cardinal numbers are not real numbers, and there is no cancellation law for them that says x + x = x implies x = 0.
At 8:51 AM, Unknown said…
"Whatebver socle. It's obvious that Cantor isn't doing math but that doesn't seem to bother any but me."
You just can't follow his method.
"Numbers have a place on the number line, socle. And if one can fit an infinite number of points in a finite space, then that is totally useless mathematically."
Are you saying the set {1, ½, ⅓, ¼ . . . } isn't infinite then?
"BTW your set of real numbers contains numbers that cannot be added, subtracted, divided nor multiplied."
Such as?
" 'I don't have to measure it, I know what the length of a diagonal of a unit square is.'
OK, give me an exact number then. Or shut the fuck up."
The exact number is the square root of 2. You just can't write it down exactly as a finite decimal.
"Let A = all nonnegative even integers
Let B = all positive odd integers
Let C = all nonnegative integers
A + B = C where neither A nor B = 0"
Well, the union of sets A and B gives you C.
"So how can A=B=C? It's impossible unless you are a moron, or a Cantor follower."
The sets aren't equal but their cardinalities are.
At 8:57 AM, Joe G said…
In what way is R odered if you can't even start the count of R?
And irrational numbers cannot be added, subtracted, multiplied nor divided. that is because they do not have a last digit.
And if A+B=C then neither A nor B can = C.
As I said C has all of the elements of A PLUS elements A does not have. That alone tells us that C has greater cardinality. The same goes for C and B. C contains all of the elements of B PLUS has elements B does not.
At 9:01 AM, Joe G said…
You just can't follow his method.
What method? Sure I can throw my hands up and declare all countably infinite sets are equal. But that ain't a methodology.
The sets aren't equal but their cardinalities are.
That is impossible given the equatin A+B=C. And you are allegedly a mathematician? LoL!
The exact number is the square root of 2. You just can't write it down exactly as a finite decimal.
Then there isn't any exact number you moron.
At 9:11 AM, Joe G said…
If cardinality refers to the number of elements in a set, and I add elements to a set, how does that not change the cardinality of the set?
At 9:36 AM, socle said…
In what way is R odered if you can't even start the count of R?
If x and y are real numbers, then exactly one of x < y, x = y, or x > y is true. That implies R is totally ordered.
You don't need to be able to "start the count" at a specific number. Nobody said that was a requirement. In fact, because R is uncountable, this is impossible.
And irrational numbers cannot be added, subtracted, multiplied nor divided. that is because they do not have a last digit.
If that's a concern, remember that in JoeMath, you can't even add 1/7 + 1/7 and get an exact answer.
Anyway, in normal math, just because two numbers don't have a finite decimal representation doesn't mean you can't add them; their sum still exists. You also can sometimes simplify the sum if you have series representations of each number.
As I said C has all of the elements of A PLUS elements A does not have. That alone tells us that C has greater cardinality.
Only in JoeMath. This is not a problem in real math.
If cardinality refers to the number of elements in a set, and I add elements to a set, how does that not change the cardinality of the set?
It's amazing that these basic questions keep coming up weeks into the discussion. With all the effort you have expended on this internet debate, you could have actually read a book on the subject(!)
At 9:40 AM, Joe G said…
As I said C has all of the elements of A PLUS elements A does not have. That alone tells us that C has greater cardinality.
Only in JoeMath. This is not a problem in real math.
Then this alleged "real" math is total bullshit.
If cardinality refers to the number of elements in a set, and I add elements to a set, how does that not change the cardinality of the set?
What is amazing is the lengths you asswipes go through to avoid the obvious.
At 9:43 AM, Unknown said…
" 'You just can't follow his method.'
What method? Sure I can throw my hands up and declare all countably infinite sets are equal. But that ain't a methodology."
Not my problem. It's yours.
" 'The sets aren't equal but their cardinalities are.'
That is impossible given the equatin A+B=C. And you are allegedly a mathematician? LoL!"
You can't even write down set union correctly so don't lecture me.
" 'The exact number is the square root of 2. You just can't write it down exactly as a finite decimal.'
Then there isn't any exact number you moron."
JoeMaths, a infinity and irrational number free zone.
At 9:45 AM, Joe G said…
I am not talking about sets, Jerad. I am talking about the integers on the number line.
If cardinality refers to the number of elements in a set, and I add elements to a set, how does that not change the cardinality of the set?
At 9:48 AM, Joe G said…
If that's a concern, remember that in JoeMath, you can't even add 1/7 + 1/7 and get an exact answer.
Fuck you. Perhaps YOU can't add them but I can.
At 10:09 AM, socle said…
Fuck you. Perhaps YOU can't add them but I can.
Only by going outside of JoeMath and using properties of the rational numbers.
Sort of like when you used the Pythagorean Theorem.
But hey, that's the sensible thing to do. It just shows that it's convenient to have real math around, even if you prefer to work in JoeMath most of the time.
At 10:21 AM, Joe G said…
Only by going outside of JoeMath and using properties of the rational numbers.
Nope. I can add FRACTIONS. NO ONE can add the fraction's attempted decimal represenation.
Sort of like when you used the Pythagorean Theorem.
No, the square root of 2 would have a place in JoeMath, it just isn't a point.
Joemath sez dimensionless points are pointless.
And as I said, in the real world the diagonal of a unit square has an actual FINITE measurement.
At 10:31 AM, socle said…
Only by going outside of JoeMath and using properties of the rational numbers.
Nope. I can add FRACTIONS. NO ONE can add the fraction's attempted decimal represenation.
That's exactly what I said. There is no number x such that 7x = 1 in JoeMath, so 1/7 doesn't exist for you. If you disagree, show me the "attempted decimal representation" of 1/7 + 1/7 = 2/7 in JoeMath.
And as I said, in the real world the diagonal of a unit square has an actual FINITE measurement.
Your notions of what should happen in the real world aren't properties of R, however.
At 10:36 AM, Joe G said…
NO ONE can add the fraction's attempted decimal represenation.
That's exactly what I said. There is no number x such that 7x = 1 in JoeMath...
In any math.
If you disagree, show me the "attempted decimal representation" of 1/7 + 1/7 = 2/7 in your math. Please start adding the correct way, starting with the first number on the right.
Good luck with that.
Your notions of what should happen in the real world aren't properties of R, however.
loL! I am talking about what DOES HAPPEN in the real world. And perhaps R needs to be changed. HINT that is one of the purposes of JoeMath.
At 10:39 AM, Joe G said…
Back to the topic at hand
Given 2 sets A, B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's.
And the predicted unsupported and cowardly response of "Joe doesn't understand infinity", is duly noted.
At 10:44 AM, socle said…
If you disagree, show me the "attempted decimal representation" of 1/7 + 1/7 = 2/7 in your math. Please start adding the correct way, starting with the first number on the right.
Good luck with that.
I don't have to. Real math includes fractions such as 1/7. JoeMath only includes numbers that have a finite decimal representation.
And perhaps R needs to be changed.
Right, go ahead and get on that. You could start by writing down the axioms that your number system should satisfy. I think that deserves its own OP. lol.
At 10:57 AM, Joe G said…
Real math includes fractions such as 1/7.
So does JoeMath. Oh wait, I have told you that already.
Do you really think that your willful ignorance means something?
At 11:10 AM, socle said…
Now compare this with what you said a bit earlier:
The exact number is the square root of 2. You just can't write it down exactly as a finite decimal.
Then there isn't any exact number you moron.
By your reasoning, this means that there is no exact number 1/7, or even 1/3 correct?
At 11:19 AM, Joe G said…
By your reasoning, this means that there is no exact number 1/7, or even 1/3 correct?
Post the exact number ie decimal representation.
At 11:32 AM, socle said…
There is no finite, exact, decimal representation of 1/3 or 1/7. I guess those numbers don't exist?
At 12:25 PM, Joe G said…
There is no finite, exact, decimal representation of 1/3 or 1/7.
LoL! So it ain't just by my reasoning then.
I guess those numbers don't exist?
They work as fractions. Leave them be.
They are nice playthings for pattern finding and anomalies. But that is about it.
At 12:35 PM, socle said…
They work as fractions. Leave them be.
Hmmm, I wonder why they work as fractions?
Anyway, if you are willing to work in base 7, 1/7 = 0.1, exactly. I guess that's not an option in JoeMath.
At 12:51 PM, Joe G said…
Well if you are going to work in binary then there ain't no 1/2 of 1.
At 12:52 PM, socle said…
Well if you are going to work in binary then there ain't no 1/2 of 1.
You mean 0.1 (base 2)?
At 12:53 PM, Joe G said…
Hmmm, I wonder why they work as fractions?
I don't.
Anyway, if you are willing to work in base 7, 1/7 = 0.1, exactly.
JoeMath is OK with that.
I guess that's not an option in JoeMath.
You guessed incorrectly.
At 1:02 PM, Joe G said…
Well if you are going to work in binary then there ain't no 1/2 of 1.
You mean 0.1 (base 2)?
No, I mean there isn't any 2 in binary so there isn't any 1/2
At 7:09 PM, socle said…
You guessed incorrectly.
Well, it's good that these numbers are no longer just "nice playthings" in JoeMath!
No, I mean there isn't any 2 in binary so there isn't any 1/2
Yes, but that is easily fixed by using correct notation.
At 7:37 PM, Joe G said…
And guess what? There isn't any 7 in base 7. So I guess JoeMath would NOT be OK with:
Anyway, if you are willing to work in base 7, 1/7 = 0.1, exactly.
At 7:40 PM, socle said…
And guess what? There isn't any 7 in base 7. So I guess JoeMath would NOT be OK with:
Anyway, if you are willing to work in base 7, 1/7 = 0.1, exactly.
Yes, I should have written that 1/7 in base 10 equals 0.1 in base 7.
At 8:42 PM, Joe G said…
Yes, 1/10 = 0.1 in every base. JoeMath can handle that.
At 1:43 AM, Unknown said…
"I am not talking about sets, Jerad. I am talking about the integers on the number line."
You are terminally confused. And you can't do the math.
Is {1, ½, ⅓, ¼ . . . } and infinite set? Yes or no? What is its cardinality?
"If cardinality refers to the number of elements in a set, and I add elements to a set, how does that not change the cardinality of the set"
JoeMaths doesn't get infinity. Clearly. JoeMaths, doing things the very, very, very, very old fashioned way. With sticks, in the dirt.
At 9:52 AM, Joe G said…
Jerad the choke artist:
Given 2 sets, A and B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's.
And the predicted unsupported and cowardly response of "Joe doesn't understand infinity", is duly noted.
Let the flailing begin...
At 5:16 PM, Unknown said…
Please note al that Joe has once again failed to even acknowledge a perfectly reasonable question: how do the cardinalities of the positive integers and the set {1, ½, ⅓, ¼ . . . } compare? He's avoiding the issue because he can't answer the question and he hopes that he can swear and bully and bluff his wat around it. Naughty, naughty, bad Joe.
"Given 2 sets, A and B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's."
True for finite sets, not necessarily true for infinite sets. You really just can't grasp infinite sets.
"And the predicted unsupported and cowardly response of "Joe doesn't understand infinity", is duly noted."
It is true. You don't get it.
"Let the flailing begin…"
What ever you want to do in your own home with consenting adults is fine with me.
At 9:29 PM, Joe G said…
"Given 2 sets, A and B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's."
True for finite sets, not necessarily true for infinite sets.
It's true for infinite sets also. Nothing changes. It's all just more of the same.
You're just a dumbass, Jerad. You think something magical happens just because it's infinity.
At 1:30 AM, Unknown said…
"Given 2 sets, A and B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's."
Yawn.
"It's true for infinite sets also. Nothing changes. It's all just more of the same."
Yawn. Repeating it over and over doesn't make it true.
"You're just a dumbass, Jerad. You think something magical happens just because it's infinity."
Uh huh. If you're so good at this then compare the cardinalities of {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . }
I know you can't actually compare the cardinalities of those sets 'cause your method doesn't work. Too bad you're not mature enough to admit it. I just want to make sure that everyone knows you just keep avoiding the question.
At 9:27 AM, Joe G said…
Jerad,
What I said is true and you cannot demonstrate otherwise.
Given 2 sets, A and B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's."
"It's true for infinite sets also. Nothing changes. It's all just more of the same."
You can say that is wrong all you want. Saying that does not make it so.
OTOH my claim is easily proven. All of the even numbers in the set of nonnegative integers has an exact corresponding member in the set of nonnegative even integers. And all of the odd numbers in the set of nonnegative integers are left without a corresponding match.
At 9:28 AM, Joe G said…
And Jerad, seeing taht you are too stupid to understand that, you aren't in any position to ask anything of JoeMath.
At 9:40 AM, Unknown said…
"Given 2 sets, A and B, if A contains all of the members of B AND has members B does not, A's cardinality has to be greater than B's.
It's true for infinite sets also. Nothing changes. It's all just more of the same."
You stil don't get how to compare the number of elements in two infinite sets.
"OTOH my claim is easily proven. All of the even numbers in the set of nonnegative integers has an exact corresponding member in the set of nonnegative even integers. And all of the odd numbers in the set of nonnegative integers are left without a corresponding match."
Line up the positive integers. Next to that line line up the positive even integers. Stand between the lines. Point at the first element of one set with your right hand and the first element of the second set with your left hand and shout ONE! Move to the next two elements and shout TWO. That's your finite pattern. Continue on indefinitely. Nothing changes. At any given moment you are created two sets of equal size. And that continues on . . .
"And Jerad, seeing taht you are too stupid to understand that, you aren't in any position to ask anything of JoeMath."
Joe can't compare the cardinality of {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . } He knows it so he's running away from dealing with it. Sad.
At 1:14 PM, Joe G said…
You stil don't get how to compare the number of elements in two infinite sets.
My methodology is natural, ie derived from the number line and relies on exact matching. Cantor's methodology is artificial, ie contrived, and relies on transforming members of one set into members of the other OR transforming the members of all sets into genric elements.
"OTOH my claim is easily proven. All of the even numbers in the set of nonnegative integers has an exact corresponding member in the set of nonnegative even integers. And all of the odd numbers in the set of nonnegative integers are left without a corresponding match."
Line up the positive integers.
Cowardly choke.
Next to that line line up the positive even integers.
Followed by a gag.
Stand between the lines. Point at the first element of one set with your right hand and the first element of the second set with your left hand and shout ONE!
Why do that when I can point to EXACT matches, ie a true onetoone correspondence?
IOW Jerad refuses to deal with reality:
OTOH my claim is easily proven. All of the even numbers in the set of nonnegative integers has an exact corresponding member in the set of nonnegative even integers. And all of the odd numbers in the set of nonnegative integers are left without a corresponding match.
At 1:33 PM, Unknown said…
"My methodology is natural, ie derived from the number line and relies on exact matching. Cantor's methodology is artificial, ie contrived, and relies on transforming members of one set into members of the other OR transforming the members of all sets into genric elements."
There's no 'transformation'. You don't get onetoone correspondence. It's just a matter of showing that a set is countable. And the positive even integers are countable. The first positive even integer is 2. The second positive even integer is 4. The third positive even integer is 6. For first, second, third substitute 1, 2 and 3 and you've got the onetoone correspondence. But I forget. You're determined that it's not true that the positive integers have the same cardinality as the positive even integers so you'll just deny anything that suggests they do have the same cardinality.
" 'Line up the positive integers.'
Cowardly choke."
Nice that you've decided a head of time. No wonder you're not a mathematician or a scientist.
"Followed by a gag."
Joe can't understand the argument so, like a good playground bully, he casts aspersions.
"Why do that when I can point to EXACT matches, ie a true onetoone correspondence?"
Because, you dumbass, it's not the values that are the point. Did you pass your Calculus course? Really?
"OTOH my claim is easily proven. All of the even numbers in the set of nonnegative integers has an exact corresponding member in the set of nonnegative even integers. And all of the odd numbers in the set of nonnegative integers are left without a corresponding match."
JoeMaths, it's wrong but it has convictions.
You can't even behave properly Joe. You expect people to present arguments and evidence but when they do you swear and say they're wrong.
You need to grow up. Badly.
At 2:01 PM, Joe G said…
LoL! There is a transformation, Jerad. Your lies mean nothing anymore.
You are a fool and that is that.
At 2:28 PM, Unknown said…
"LoL! There is a transformation, Jerad. Your lies mean nothing anymore."
You continually mistake the number of elements in a set with some 'weight' or 'value' they have. It's only one of the ways you don't get infinities.
You are a fool and that is that."
And you still haven't compared the cardinalities of {1, 2, 3, 4 . . .} and {1, ½, ⅓, ¼ . . . .} And you can't even admit you haven't dealt with it. Truth.
And, you're such a coward, that you won't even copyandpaste this part of the discussion into your reply. Unless I challenge you on it. So . . . now, what do you do? If you mention it you lose because you haven't dealt with it. If you don't mention it then you lose because you're avoiding a challenge. Sounds like you lose either way. I guess it's time to swear and bluster and try to intimidate people.
Funny though that that doesn't actually work with people who actually know what they're talking about. Or people who disagree with you. You can't even handle disagreement can you?
What was your dad like? Was he abusive? Were you beaten? Did you 'learn' that intimidation and swearing were the ways to 'carry the day'? If so, then I am sorry. NO child should have to grow up like that. But my sympathy does not extend to agreeing that you're right about mathematics. And, if you're really interested, there are lots and lots of online resources which explain things.
At 4:03 PM, Unknown said…
"LoL! There is a transformation, Jerad. Your lies mean nothing anymore.
You are a fool and that is that."
According to you. And who agrees with you? Let's see some hands . . . gosh . . . that's not very many people really.
AND, just in case you thought you could ignore the issue: compare the cardinalities of {1, 2, 3, 4 . . . .} and {1,1/2, ⅓, ¼ . . . } and prove you're ideas and methods are sound.
Go on.
We're waiting.
We've been waiting.
Can't handle it? Well then at least be mature enough to admit it.
Or are you wrong AND juvenile? Your response will tell . . .
At 5:10 PM, Joe G said…
You continually mistake the number of elements in a set with some 'weight' or 'value' they have.
And more evidence that you think infinity is some sort of magical transforming equalizer.
As I have said FROM THE BEGINNING numbers have a specific place on the number line.
I guess your dad you to fuck you up the ass and dump his load in your mouth...
At 6:10 PM, Unknown said…
"As I have said FROM THE BEGINNING numbers have a specific place on the number line."
Great. These two sets have places on the number line: {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . }. So you should be able to compare their cardinalities. Yes?
"I guess your dad you to fuck you up the ass and dump his load in your mouth…"
Umm . . that's not a sentence. Or is it sentance? If you need someone to talk ablut things like that then I suggest you seek out a good therapist.
At 8:27 PM, Joe G said…
These two sets have places on the number line: {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . }.
Great, then you should be able to tell me where each member is on that line. Yet you can't.
Nor can you explain why Cantor cares about the quality of a set's members in one case and arbitrarily throws out the quality for another.
At 1:58 AM, Unknown said…
" 'hese two sets have places on the number line: {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . }.'
Great, then you should be able to tell me where each member is on that line. Yet you can't."
Pick an element and I'll tell you where it is.
"Nor can you explain why Cantor cares about the quality of a set's members in one case and arbitrarily throws out the quality for another."
Quality? What the hell is that?
I cannot believe after all this time that you still cannot grasp one simple concept. Cantor's method NEVER cares about the values of the sets when comparing their cardinalities. Only if you can put the sets into a onetoone correspodence.
Since YOU'RE so hung up on the 'qualities' whatever that is why don't you try and compare the cardinalities of these two infinite sets:
{1, 2, 3, 4 . . . } and {a, b, aa, bb, aaa, bbb . . . }
Go on, I'll wait.
Or how about these two sets:
{1, 2, 3, 4 . . . } and {1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1 . . . }
There is a pattern to the second set.
At 9:24 AM, Joe G said…
Pick an element and I'll tell you where it is.
The last element.
"Nor can you explain why Cantor cares about the quality of a set's members in one case and arbitrarily throws out the quality for another."
Quality? What the hell is that?
YOU brought it up in another thread, dumbass. The quality is its actual value.
I cannot believe after all this time that you still cannot grasp one simple concept. Cantor's method NEVER cares about the values of the sets when comparing their cardinalities.
You stupid fuck, THAT is my point. He cares about that quality in some cases but not in others. He is inconsistent.
At 2:08 AM, Unknown said…
" 'Pick an element and I'll tell you where it is. '
The last element."
There isn't one, try again. Or find the last element.
" 'I cannot believe after all this time that you still cannot grasp one simple concept. Cantor's method NEVER cares about the values of the sets when comparing their cardinalities.'
You stupid fuck, THAT is my point. He cares about that quality in some cases but not in others. He is inconsistent."
No, YOU care about the values.
At 9:07 AM, Joe G said…
Jerad,
If no one f no one except me cares about the values then it is obvious that they are NOT comparing nonnegative integers with nonnegative even integers.
Integers have values.
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