This Just In- The Stupidity Nevers Ends
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This just in:
The set of non-negative integers, {0,1,2,3,4,5,6,7,8,9,10,...} does NOT contain the set of non-negative even integers, {2,4,6,8,10,...} AND have the positive odd integers left unmatched.
For some reason it is NOT correct to use the naturally derived alignment, ie exact matching of numbers, and instead an artificially constructed, ie contrived, alignment, which transforms all members into generic elements, is preferred. Yet no one can why nor what good that does.
Are all mathematicians really that stupid to buy into that?
Really??
This just in:
The set of non-negative integers, {0,1,2,3,4,5,6,7,8,9,10,...} does NOT contain the set of non-negative even integers, {2,4,6,8,10,...} AND have the positive odd integers left unmatched.
For some reason it is NOT correct to use the naturally derived alignment, ie exact matching of numbers, and instead an artificially constructed, ie contrived, alignment, which transforms all members into generic elements, is preferred. Yet no one can why nor what good that does.
Are all mathematicians really that stupid to buy into that?
Really??
posted by Joe G @ 1:36 PM
64 Comments:
At 3:34 PM,
Unknown said…
Can't be bothered to cite where you gleaned this information? Or are you just misinterpreting what someone has seid in refutation of your claims? Hmmm??
At 5:00 PM,
Joe G said…
YOU, Jerad. YOU are the source of the OP.
And no, it isn't a misinterpretation.
At 5:18 PM,
Joe G said…
The naturally derived alignment should trump the artificial and contrived alignment.
That is doesn't is proof people have an agenda. And it isn't an agena based on honesty.
At 5:44 PM,
Unknown said…
"YOU, Jerad. YOU are the source of the OP."
Oh, well, glad to be of use.
"And no, it isn't a misinterpretation."
There are other opinions of course.
"The naturally derived alignment should trump the artificial and contrived alignment."
I think that it's quite natural when comparing the size of sets to line up the elements of both sets to see which is bigger.
"That is doesn't is proof people have an agenda. And it isn't an agena based on honesty."
Just because people disagree with you doesn't mean they're lying. You're starting to sound like GEM.
At 6:02 PM,
Joe G said…
Jerad,
Is {0,2,4,6,8,...} a proper subset of {0,1,2,3,4,...} because its 2 corresponds with the other set's 1, or because there is an exact match in the other set which is corresponds to?
We already have an alignment methodology that works. And it is naturally derived.
Cantor needs to stay consistent. That is all I am doing.
Strange that you are too stupid to understand that.
At 1:42 AM,
Unknown said…
"Cantor needs to stay consistent. That is all I am doing.
Strange that you are too stupid to understand that."
Cantor is very consistent. If you want to see if two sets have the same cardinality put then into a one-to-one correspondence. That's it. You just don't get it. You're all hung up on the way the set members are counted and their 'values'.
And you can't seem to ever deal with comparing the cardinalities of {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . }
You're method is in trouble 'cause you think the rate you count the elements matters and counting the second set is a problem for you. Not for Cantor's method. Go figure.
At 9:12 AM,
Joe G said…
Cantor is very consistent.
No, he isn't. He uses two different alignment techniques and sometimes he cares about the quality of a set's members then he doesn't.
If you want to see if two sets have the same cardinality put then into a one-to-one correspondence.
LoL! Magical transformers away!
BTW thank you for continuing to prove that you are a coward:
Is {0,2,4,6,8,...} a proper subset of {0,1,2,3,4,...} because its 2 corresponds with the other set's 1, or because there is an exact match in the other set which is corresponds to?
We already have an alignment methodology that works. And it is naturally derived.
At 9:28 AM,
socle said…
Joe,
When you extend your method so that you can compare any two sets, let us know.
As of now:
1) You can't answer Jerad's question
2) You have to duck the question (it depends on when you look, lol) if the lead between the marks collected by the trains shifts back and forth.
3) You can't deal with subsets of R that don't "go to infinity".
4) In fact, if the sets are not subsets of R, then forget it.
5) Did I mention that you can't answer Jerad's question?
At 9:34 AM,
Joe G said…
socle,
When you assholes can deal with te issue still on the table, let me know.
As of now:
1. You can't demonstrate any utility in Cantor's method for dealing with countably infinite sets
2. You can't demonstrate that Cantor's methods work
3. You cannot explain why this magical transformer technique is accepted
4. You cannot explain how one set can have all of the members of the other set AND have members the other set doesn't and still be of equal size
5. You chumps don't understand infinity
6. You assholes are also liars.
7. You don't grasp what "greater than or equal to" means
8. All you can do is spew false accusations
At 9:50 AM,
socle said…
Joe,
Use your method to compare A = {1, 2, 3, ...} and B = {1, 1/2, 1/3, ...}. Then tick the correct box:
□ |A| > |B|
□ |A| = |B|
□ |A| < |B|
At 11:59 AM,
Joe G said…
socle,
If your IQ was over 75 you would be able to see that my methodology tries to find a common denominator, so to speak- ie common ground between the two sets being compared.
That said, it is easy to see that A and B are reciprocals. Therefor in the magical world of infinities, even infinities between 2 finite points, the 2 sets have the same cardinality.
3) You can't deal with subsets of R that don't "go to infinity".
I say that I can. Do you have any evidence for your claim? Please present it.
At 12:33 PM,
Joe G said…
keiths sez:
1. Whether you realize it or not, you applied the mapping function F(n) = 1/n in order to establish a one-to-one correspondence between the two sets. You then concluded that their cardinalities are the same.
Just because you have been unable to grasp what I have been saying doesn't mean anything to me.
2. In other words, you applied Cantor’s method (finally!) and got the correct answer (finally!).
I applied my method and it is just a coincidence that Cantor and I came up with the same ansser.
3. On the other hand, JoeMath (and its patented choo-choo train method) were utterly unable to deal with Jerad’s problem.
I used JoeMath. Go figure.
Again keiths thinks his ignorance means something.
At 12:39 PM,
Joe G said…
keiths sez:
Of course we can justify that “alignment”. We choose the F(n) = 2n mapping because it produces a one-to-one correspondence between {1,2,3,…} and {2,4,6,…}.
The mapping shows the relative cardinality. And it also shows that you are using a magical trandsformation.
Why is a one-to-one correspondence important? Because one-to-one correspondences work when comparing the cardinality of two sets, whether finite or infinite.
The natural, ie dervied alignment works and demonstrates that this alleged one-to-one correspondence does not treat the sets as non-negative integrs vs non-negative even integers.
And again your cowardice is duly noted. Finite sets we can actually count the members.
The problem isn’t with the mathematicians, Joe.
It is if you, Neil Rickert and Jerad are mathematicians...
At 6:15 PM,
Joe G said…
keiths spoews more shit:
No, you used Cantor’s method with the mapping function F(n) = 1/n.
Nope, I used my method. As I said I can't help your ignorance and your ignorance means nothing.
You certainly didn’t apply JoeMath, because your patented choo-choo train method would have concluded that {1, 1/2, 1/3,…} is infinitely larger than {1,2,3,…}.
Yes I did apply Joemath and no it would not give the answer you stated.
Again your ignorance means nothing to me, keiths.
According to you, that means the cardinality of {1, 1/2, 1/3,…} is the reciprocal of the cardinality of {1,2,3,…}.
Nope. However it is hilarious that you think that your ignorance means something.
Can you explain why JoeMath gives contradictory answers to a single question?
If it ever does I will addfress it. However you don't get to say because you are an ignorant fuck.
At 1:52 AM,
Unknown said…
"If your IQ was over 75 you would be able to see that my methodology tries to find a common denominator, so to speak- ie common ground between the two sets being compared."
Which is not necessary when you're just trying to compare cardinalities.
"That said, it is easy to see that A and B are reciprocals. Therefor in the magical world of infinities, even infinities between 2 finite points, the 2 sets have the same cardinality."
In this case the two sets do have the same cardinality.
Now, what about the cardinality of the real numbers in (0, 1) ? Is it the same as {1, 2, 3, 4 . . . }
"I say that I can. Do you have any evidence for your claim? Please present it."
What is the cardinality of the reals in (0, 1)?
"I applied my method and it is just a coincidence that Cantor and I came up with the same ansser."
But you didn't apply your counting speed criteria.
"The natural, ie dervied alignment works and demonstrates that this alleged one-to-one correspondence does not treat the sets as non-negative integrs vs non-negative even integers."
It doesn't matter how it treats the elements, it only tries to find out whether the sets have the same cardinalities.
" 'No, you used Cantor’s method with the mapping function F(n) = 1/n.'
Nope, I used my method. As I said I can't help your ignorance and your ignorance means nothing."
No, you did not. You did not use your speed that the sets were being counted or your proper subset argument.
" 'You certainly didn’t apply JoeMath, because your patented choo-choo train method would have concluded that {1, 1/2, 1/3,…} is infinitely larger than {1,2,3,…}.'
Yes I did apply Joemath and no it would not give the answer you stated."
You can't even use your own method.
"If it ever does I will addfress it. However you don't get to say because you are an ignorant fuck."
Clueless.
What's the cardinality of the reals in (0, 1) ?
At 9:19 AM,
Joe G said…
Which is not necessary when you're just trying to compare cardinalities.
Just don't tell me that you are comparing the cardinality of the set of non-negative inetgers to the set of non-negative even integers. That would be a lie.
But you didn't apply your counting speed criteria.
So you are proud to be a moron.
The rate is the same you ignorant wanker.
It doesn't matter how it treats the elements, it only tries to find out whether the sets have the same cardinalities.
If it doesn't matter how you treat the elements then you ain't comparing sets of INTEGERS.
Nope, I used my method. As I said I can't help your ignorance and your ignorance means nothing.
No, you did not.
Yes, I did. YOUR ignorance is also meaningless.
You did not use your speed that the sets were being counted or your proper subset argument.
It's the same rate you fucking moron.
You can't even use your own method.
I can and have. Again just because you are an ignorant wanker, that doesn't mean anything to me.
And yes Jerad, you are clueless.
At 10:39 AM,
socle said…
That said, it is easy to see that A and B are reciprocals. Therefor in the magical world of infinities, even infinities between 2 finite points, the 2 sets have the same cardinality.
Presumably the same could be said of (0, 1] and [1, ∞). Every element of (0, 1] is the reciprocal of a unique element of [1, ∞), and vice versa. Therefore |(0, 1]| = |[1, ∞)| in JoeMath.
I'm just pointing this out because it's the first example that I recall seeing involving uncountable sets in JoeMath.
At 12:12 PM,
Joe G said…
No, socle, only in the magical world of infinities, even infinities between 2 finite points.
That said, for those of you who think that I am using Cantor's method, I offer this:
In Joemath, using the same method I used wrt {0,1,2,3,4,...} and {1,1/2,1/3,1/4,1/5,...}, The set of non-negative integers would have a cardinality that is greater than the set {1,1/2,1/4,1/8,1/16,...}.
If I was using Cantor's method the cardinalities would still be equal.
As I said it was only a coincidence that Cantor and I agree in the first scenario. And it is very telling that you morons are so stupid that you couldn't see that- even though I told you.
Nice job guys.
At 12:24 PM,
socle said…
No, socle, only in the magical world of infinities, even infinities between 2 finite points.
But you said that in JoeMath, |{1, 2, 3, ...}| = |{1, 1/2, 1/3, ...}|. Clearly no reference to Cantor there.
Why can't I also say "it is easy to see that (0, 1] and [1, ∞) are reciprocals", therefore |(0, 1]| = |[1, ∞)|, again, all in JoeMath?
At 2:23 PM,
Joe G said…
But you said that in JoeMath, |{1, 2, 3, ...}| = |{1, 1/2, 1/3, ...}|.
You have reading comprehension issues.
At 3:21 PM,
socle said…
But you said that in JoeMath, |{1, 2, 3, ...}| = |{1, 1/2, 1/3, ...}|.
You have reading comprehension issues.
So are you backing off the statements you made yesterday?
In particular, what did you mean when you said this:
I applied my method and it is just a coincidence that Cantor and I came up with the same ansser.
What was this "same ansser" that you and Cantor both came up with?
At 3:26 PM,
Joe G said…
That said, it is easy to see that A and B are reciprocals. Therefor in the magical world of infinities, even infinities between 2 finite points, the 2 sets have the same cardinality.
magical world of infinities.
I believe I have already made it clear that JoeMath doesn't like infinities, especially between two finite points. JoeMath is a TBD wrt both types of infinities.
So just back off, relax and try a little reading for comprehension.
At 4:00 PM,
socle said…
I believe I have already made it clear that JoeMath doesn't like infinities, especially between two finite points. JoeMath is a TBD wrt both types of infinities.
Nevermind about the issue of infinity, you said you applied your method and by coincidence got the same answer as Cantor, i.e., that A and B are equivalent.
If you want to stand down from this claim, which of course you have every right to do, then we have an answer to your challenge:
3) You can't deal with subsets of R that don't "go to infinity".
I say that I can. Do you have any evidence for your claim? Please present it.
At 4:03 PM,
Joe G said…
I stand by my statement.
At 4:06 PM,
Joe G said…
And the same answer that Cantor and I came up with is that the two sets have the same cardinality.
Cantor would also say that {1,1/2,1/3,1/4,...} has the same cardinality as {1,1/2,1/4,1/8,...}, whereas JoeMath would say the first set's cardinality is greater than the second set's cardinality.
At 4:16 PM,
socle said…
And the same answer that Cantor and I came up with is that the two sets have the same cardinality.
Cantor would also say that {1,1/2,1/3,1/4,...} has the same cardinality as {1,1/2,1/4,1/8,...}, whereas JoeMath would say the first set's cardinality is greater than the second set's cardinality.
And by exactly the same reasoning, (0, 1] and [1, ∞) have the same cardinality. The elements in one set are exactly the reciprocals of the elements in the other set.
It does have some strange implications, however. By your subset principle, |[2, ∞)| < |[1, ∞)|, so we conclude that |(0, 1]| > |[2, ∞)|.
At 4:46 PM,
Joe G said…
The elements in one set are exactly the reciprocals of the elements in the other set.
In the magical never land of infinity, even between two finite points.
The elements in one set are exactly the reciprocals of the elements in the other set.
Really? I never would have guessed that! Amazing. Reciprocals you say. Wonderfully interesting. Why didn't I think of that?
It does have some strange implications, however. By your subset principle, |[2, ∞)| < |[1, ∞)|, so we conclude that |(0, 1]| > |[2, ∞)|.
What's strange about that?
At 7:13 PM,
socle said…
What's strange about that?
Well, for every real number x in (0, 1], x + 2 is in [2, ∞). If x1 =/= x2, then x1 + 2 =/= x2 + 2. This means [2, ∞) should be at least as large as (0, 1].
At 10:04 PM,
Joe G said…
OK I have to back up because i wasn't paying attention:
And by exactly the same reasoning, (0, 1] and [1, ∞) have the same cardinality. The elements in one set are exactly the reciprocals of the elements in the other set.
You wouldn't use the reciprocals. They are already on the same terms.
They both start at 1 and start counting from there. And in magical infinity math neither would get anywhere.
JoeMath wouldn't use the train method. JoeMath would say [1, ∞)'s cardinality > (0, 1], and the reasoning would be that [1, ∞) has infinitely more integers and therefor infinitly more points between those integers.
At 2:45 AM,
Unknown said…
I've had enough of this. I'm fed up with your bullying and inability to understand basic, clear arguments and misunderstanding of your own techniques.
I think now that it's right not to argue with denialists, it just gives them the attention they crave and because they have ulterior motives for many of the positions they hold they won't back down or change their minds.
It's not about reason or trying to understand or find the truth. It's about something else.
I"ve been wrong many times in my life, just about every day I make a mistake of some kind. And I try to learn lessons from all those events. It's pointless to try and teach people who don't want to learn.
At 8:09 AM,
Joe G said…
Lol! You don't have any arguments, Jerad. And I do not misunderstand my own techniques. You are an imbecile.
And Jerad, you don't have anything to teach.
At 10:40 AM,
socle said…
You wouldn't use the reciprocals. They are already on the same terms.
My response would then be, you should be able to use the idea of reciprocals---why not, if it's valid in Jerad's example?
In fact, I could write the sets like this:
A = {x : 0 < x <= 1}
B = {1/x : 0 < x <= 1}
Then it's clear that every element of B is the reciprocal of an element in A, and also the other way around.
There are often several ways to solve a particular problem, and of course they all should give you the same result.
JoeMath wouldn't use the train method. JoeMath would say [1, ∞)'s cardinality > (0, 1], and the reasoning would be that [1, ∞) has infinitely more integers and therefor infinitly more points between those integers.
While I don't agree with your final conclusion, if you accept that [1, ∞) has infinitely more integers than (0, 1], then that certainly means that {1, 2, 3, ...} is an infinite set, and I would be happy to consider that point as settled.
At 9:47 PM,
Joe G said…
socle,
You only use the reciprocal if you need it to put the sets on common ground. That was needed in Jerad's example.
It was not needed in your example- we already had common ground.
And you weren't solving a problem. You were trying to create one.
And yes, it has been settled- infinity is a product of our imagination.
At 9:56 AM,
socle said…
And yes, it has been settled- infinity is a product of our imagination.
Just as the number represented by "3" is. And just as minus log base 2 and CSI are. I'm not disputing that numbers do not exist in the physical world.
The irony is that you keep using the concept of infinity yourself while at the same time asserting that JoeMath is an infinity-free zone. Go figure...
At 11:35 AM,
Joe G said…
Umm, CSI exists. Our computers, cars, homes, etc., none of that would exist without CSI.
So CSI is not a product of our imagination.
And I qualify my use of infinity- something I hacve done repeatedly in this thread alone- obvioulsy you are too stupid to grasp that.
Go figure, indeed...
At 11:39 AM,
socle said…
And I qualify my use of infinity- something I hacve done repeatedly in this thread alone- obvioulsy you are too stupid to grasp that.
What are these qualifications you place on your use of infinity? Please be specific.
At 11:59 AM,
Joe G said…
That said, it is easy to see that A and B are reciprocals. Therefor in the magical world of infinities, even infinities between 2 finite points, the 2 sets have the same cardinality.
magical world of infinities.
I believe I have already made it clear that JoeMath doesn't like infinities, especially between two finite points. JoeMath is a TBD wrt both types of infinities.
So just back off, relax and try a little reading for comprehension.
It's as if you are proud of your little pee-pee...
At 1:44 PM,
socle said…
My conclusion: You don't like infinity, for whatever reason, but you will use it when it's convenient. And no one else can, because it's JoeMath.
Just like the set {1, 1/2, 1/3, ...} is an infinite set of transformers, but (0, 1] isn't, because Joe declares it to be so.
At 1:52 PM,
Joe G said…
That is a big WHATEVER.
To me people invented infinities because they are convenient.
Just like the set {1, 1/2, 1/3, ...} is an infinite set of transformers, but (0, 1] isn't, because Joe declares it to be so.
Man you are dense. Neither would be infinite if I had my way.
As I said it's as if you are proud to have a little pee-pee.
At 2:08 PM,
socle said…
To me people invented infinities because they are convenient.
That's correct, at least in my view. Just like the game of chess is an invention. OTOH, I wouldn't be surprised if every reasonably advanced civilization had invented the real numbers; maybe not chess-like games though.
At 2:12 PM,
Joe G said…
LoL! Chess can be touched, observed and played. Infinity doesn't have any of those qualities.
At 2:46 PM,
socle said…
LoL! Chess can be touched, observed and played. Infinity doesn't have any of those qualities.
*facepalm*
I'd like to see someone "touching" chess then. Not touching a chess board or a chess piece, but _chess_ itself.
At 4:47 PM,
Joe G said…
Do you always wipe your ass with your bare hand before your facepalm?
At 5:15 PM,
socle said…
Have you ever heard of abstract thinking, Joe? According to wikipedia:
Thinking in abstractions is considered to be one of the key traits in modern human behaviour, which is believed to have developed between 50,000 and 100,000 years ago.
At 5:37 PM,
Joe G said…
Umm, I have a train on an infinite journey, going in both directions at the same time, counting integers along the way.
At 6:07 PM,
socle said…
Umm, I have a train on an infinite journey, going in both directions at the same time, counting integers along the way.
Yes, Joe, of course you do.
At 6:08 PM,
Joe G said…
Yes, socle, you never heard of abstract thinking.
At 6:27 PM,
socle said…
Lo! The turnabout accusation rhetorical attack!
At 7:37 PM,
Joe G said…
Grammatical judo...
At 11:42 AM,
DiEb said…
Perhaps a historical perspective will help: At first, Cantor was thinking about finite sets. How do we find out that finite sets have the same size? A shepherd may map his sheep to a bundle of tokens, making sure that each token represents exactly one particular sheep and vice versa. A mathematician may look at the sets {1,2,3} and {a,b,c} and see that there is a invertible one-to-one mapping between this sets: in fact, 3 out of the possible 27 mapping between these sets are injective and surjective (bijective). There is is no bijective mapping between - say - {apple, orange} and {1/2,1/3,1/4}. So, we may define that two finite sets have the same size if there is a bijective mapping between those two.
Cantor was a curious man. He just thought what happened when we take this definition to infinite sets. This lead to some statements which may appear paradoxical ("an infinite set is a set which has the same size as one of its proper subsets") and perhaps that is the reason that he used the expression "cardinality" instead of "size".
Nevertheless, it works - even if you don't like it. And the idea of "countable infinite" sets has proven very successful: you can't do analysis or probability theory without it (ever heard of a sigma-algebra?)
Bottom point: Cantor's ideas work for mathematicians. That's all we are caring about.
At 9:54 AM,
Joe G said…
Hi DiEB,
All one has to do is count the number of members in a finite set- one doesn't have to know anything about mapping- onr-to-one or not.
And what do you mean by "it works" wrt infinite sets?
No one even uses it for anything.
And how can one set contain all of the members of the other set PLUS have members the other set does not, meaning the first set obviously has more members/elements, and yet cantor sez they are the same size, ie have the same cardinality?
Good luck trying to explain your way out of that one...
At 5:02 PM,
DiEb said…
All one has to do is count the number of members in a finite set- one doesn't have to know anything about mapping- onr-to-one or not.
That's the beauty: you don't even have to know how to count when you wish to compare the size of to flocks of sheep - you only have to pair them up!
And what do you mean by "it works" wrt infinite sets?
We get a consistent theory.
No one even uses it for anything.
It is used the very moment you start calculus! Calculus is rooted in the idea of countable infinite sequences of numbers, and those are treated the way Cantor has shown!
Above, I gave another example: ever heard of a sigma-algebra?
At 5:19 PM,
Joe G said…
And how can one set contain all of the members of the other set PLUS have members the other set does not, meaning the first set obviously has more members/elements, and yet cantor sez they are the same size, ie have the same cardinality?
Strange that the cowards always ignore that fact.
That's the beauty: you don't even have to know how to count when you wish to compare the size of to flocks of sheep - you only have to pair them up!
You have to count to know they are paired.
The shepard should know how many are in his flock. To have to walk around to find another flock to pair them with sounds like your accepted methodology, but it is also pretty stupid.
And even then you wouldn't know how many you had!
We get a consistent theory.
In what way? As I have said it is inconsistent wrt comparing sets. We use one methodology for seeing if one set is a proper subset of another, and then use a totally different alignment for comparing cardinalities of infinite sets.
That is inconsistent. And you are being vague- another cowardly sign.
It is used the very moment you start calculus!
Stop being an asshole. Calculus does not need the premise that all countably infinite sets have the same cardinality.
No one needs it. It is not used for anything.
Get a grip.
Also deal with the bolded part. Failure to do so will just prove your cowardice. And cowardice is not a way to win an argument nor convince your opposition that you have a case.
And yes, I have heard of sigma-algebra. Do you have a point?
To me it looks like you jumped in here without a clue as to what is being debated.
Just sayin'...
At 5:50 PM,
DiEb said…
Stop being an asshole. I can't remember even to have started to be one!
But to answer your question: And how can one set contain all of the members of the other set PLUS have members the other set does not, meaning the first set obviously has more members/elements, and yet cantor sez they are the same size, ie have the same cardinality?
That's because they are infinite sets: it's a property of infinite sets to be of the same cardinality as some of their proper subsets! You don't have to like this, but that's how math is done.
At 10:49 PM,
Joe G said…
So infinite sets magically become equal even given what I said?
How does that help math?
At 4:00 AM,
DiEb said…
So infinite sets magically become equal even given what I said?
That has nothing to do with magic - it's about making some postulates and then thinking them through.
Take e.g. geometry: what happens when we don't say that there is always exactly one parallel through a point given a straight line, but many? We wouldn't expect it from our usual knowledge of the world, but it turns out that we can draw conclusions and get to the consistent theories of non-Euclidian geometries.
What happens when we declare the equation x^2-1=0 to be solvable and think of the properties of the solution? We get the imaginary numbers, again not obvious from daily life, but very useful.
The same for cardinalities: what happens when we declare that two sets have the same cardinality if there is a bijective mapping between these maps? It seems that we get very far!
Sequences play an important role in calculus. What happens to the limit of a convergent sequence if you change a finite number of it elements? What happens to the limit if you drop every second element? Nothing! Is it useful to think of the new sequence as having less elements than the original? Turns out, it isn't. That's why generations of undergraduates are confronted with the marvels of Hilbert's hotel: it is a strange place and it takes some time to understand its working, but it is fund to live there!
At 9:35 AM,
Joe G said…
That has nothing to do with magic
Yes it does.
it's about making some postulates and then thinking them through.
Obvioulsy no one thought it through.
As I said if one set contains all of the members of another set AND has members that other set does not have, it has to have more elements.
The same for cardinalities: what happens when we declare that two sets have the same cardinality if there is a bijective mapping between these maps? It seems that we get very far!
And yet you cannot say what use it is.
And Hilbert's hotel is total BS...
At 10:31 AM,
DiEb said…
This isn't about right or wrong, this is about one working concept vs. another working concept.
Obvioulsy no one thought it through.
Every student starting mathematics has to try to think it through. If he doesn't like it, fine, he can start with his own set of axioms and definitions and look where he is lead...
Your idea has some similarities to Nonstandard Analysis (which is much harder to understand than the usual way to think about numbers...)
The main drawback in your system: you cannot compare most of the sets which are usually discussed.
What's bigger (and why) in size: {0,1,2} or {1,2,3}
{0,1,2,3,...} or {1,2,3,4,...}
{0,1,2,3,...} or {-1,1,2,3,...}
{1,2,3,...} of {-1,-2,-3,...}
Cantor's system allows for a consistent way to compare the cardinalities of sets. Again, if you learn standard analysis or a little bit of measure theory and advanced analysis, topology, etc., it is used extensively.
At 10:47 AM,
Joe G said…
What's bigger (and why) in size: {0,1,2} or {1,2,3}
They are the same as both contain 3 elements.
And I have already been over the other sets. I am not going through this again. You need to read what I have already posted.
Cantor's system allows for a consistent way to compare the cardinalities of sets.
That is your opinion. And I have shown Cantor's error.
One more time-
The premise that all countable and infinite sets have the same cardinality is not used for anuything. It is useless. Nothing DiEB has posted has changed that...
At 10:49 AM,
Joe G said…
As I said if one set contains all of the members of another set AND has members that other set does not have, it has to have more elements.
At 2:52 AM,
DiEb said…
"The premise that all countable and infinite sets have the same cardinality is not used for anuything. It is useless."
Separable spaces spring to my mind...
At 9:11 AM,
Joe G said…
I am sure that there are many separable places in your mind...
At 9:05 PM,
socle said…
Joe, when discovering that JW's math book contradicts JOEC:
And what makes your math book correct?
Joe, less that 10 hours later:
Obviously this Ford guy and his text are wrong. *eyeroll*
lol
At 9:36 PM,
Joe G said…
LoL! Obviously all math textbooks that deal with set theory just blindly follow Cantor and the magical transformer hypothesis.
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