At his blog, Joe G. has worked himself into a lather over the cardinality (or roughly speaking, the size) of infinite sets (h/t Neil Rickert):
Of Sets, Supersets and Subsets
Oleg the Asshole, Still Choking on Sets
Of Sets and EvoTARDS
Subsets and Supersets, Revisted [sic]
In particular, Joe is convinced that the sets {0,1,2,3,…} and {1,2,3,4,…} have different cardinalities:
…the first set has at least one element that the second set does not, ie they are not equal.
So if two sets are not equal, then they cannot have the same cardinality.
As a public service, let me see if I can explain Joe’s errors to him, step by step.
First of all, Joe, sets do not have to be equal in order to have the same cardinality, as Neil showed succinctly:
I give you two sets. The first is a knife and fork. The second is a cup and saucer. Those two sets are not equal. Yet both have cardinality two.
Two sets have the same cardinality if their elements can be placed into a one-to-one correspondence. That’s the only requirement. Here is a one-to-one correspondence (or “bijection”) for Neil’s sets:
knife <–> cup
fork <–> saucer
Note that there is nothing special about that one-to-one correspondence. This one works just as well:
fork <–> cup
knife <–> saucer
It makes no difference if the elements are numbers. The same rule applies: two sets have the same cardinality if their elements can be placed into a one-to-one correspondence. Consider the sets {0,1,2} and {4,5,6}. They can be placed into the following correspondence:
0 <–> 4
1 <–> 5
2 <–> 6
As before, there is nothing special about that correspondence. This one also works:
1 <–> 4
0 <–> 5
2 <–> 6
There are six distinct one-to-one correspondences for these sets, and any one of them, by itself, is enough to demonstrate that the sets have the same cardinality.
Now take the sets {1,2,3} and {2,4,6}. They can be placed into this one-to-one correspondence:
2 <–> 2
1 <–> 4
3 <–> 6
However, note that it’s not required that the 2 in the first set be mapped to the 2 in the second set. This one-to-one correspondence also works:
1 <–> 2
2 <–> 4
3 <–> 6
As in the previous example, there are six distinct one-to-one correspondences, and any one of them is sufficient to demonstrate that the cardinalities are the same.
Now consider the sets that are befuddling you: the infinite sets {0,1,2,3,…} and {1,2,3,4,…}. They can be placed into a one-to-one correspondence:
0 <–> 1
1 <–> 2
2 <–> 3
3 <–> 4
…and so on.
A one-to-one correspondence exists. Therefore the cardinalities are the same.
Here’s what I think is confusing you, Joe: there is a different mapping from the first set to the second that looks like this:
0 doesn’t map to anything
1 <–> 1
2 <–> 2
3 <–> 3
… and so on.
You observe that all of the elements of the second set are “used up”, while the first set still has an “unused” element — 0. You comment:
They both go to infinity but the first one starts one number before the second. That means that the first one will always have one element more than the second which means they are NOT the same size, by set standards.
But that’s silly, because you could just as easily choose a different mapping, such as:
0 doesn’t map to anything
1 doesn’t map to anything
2 doesn’t map to anything
3 <–> 1
4 <–> 2
5 <–> 3
… and so on.
Now there are three unused elements in the first set. We can also arrange for the “unused” elements to be in the second set:
nothing maps to 1
nothing maps to 2
nothing maps to 3
0 <–> 4
1 <–> 5
2 <–> 6
… and so on.
There are now three unused elements in the second set. By your logic, that would mean that the second set has a greater cardinality. It clearly doesn’t, so your logic is wrong.
You can even arrange for an infinite number of unused elements:
0 <–> 2
1 <–> 4
2 <–> 6
3 <–> 8
… and so on.
Now the odd integers are unused in the second set. Since there are now infinitely many “unused” elements in the second set, then by your logic the second set is infinitely larger than the first. Your logic is wrong, and the mathematicians are right.
Two sets have the same cardinality if they can be placed into a one-to-one correspondence. The sets {0,1,2,3,…} and {1,2,3,4,…} can be placed into such a correspondence, so they have the same cardinality.
END (Hi, KF!)
Joe:
Joe, I want you to take several deep breaths and concentrate.
{1,2} is twice as big as {2}
{1,2,3,4} is twice as big as {2,4}
{1,2,3,4,…100} is twice as big as {2,4,6,8,…100}
{1,2,3,4,…,1,000,000,000,000} is twice as big as {2,4,6,8,…,1,000,000,000,000}
{1,2,3,4,…,1 googolplex} is twice as big as {2,4,6,8,…,1 googolplex}
In all of those cases we stopped. The sets are all finite. If you stop at some even number, no matter how large, you find that one set is twice the size of the other.
When comparing infinite sets, you don’t stop. If you stopped, the sets would be finite, not infinite. Infinity is not just a really, really big number. It goes on forever, by definition.
If you go on forever, you never run out of numbers. For every n in the set of non-negative integers, there is always a 2n in the set of of non-negative multiples of 2. The sets can be matched up one-to-one. They have the same cardinality.
Evidently even the Math Is Fun! site was beyond Joe’s abilities. Perhaps this explanation to a four-year-old, by the actor who played Sesame Street’s Count, is more compatible with Joe’s math skills.
Or he could just visit Mark Chu-Carroll’s blog for a proper spanking[*].
[*] Not that there’s anything wrong with that sort of thing among consensual participants.
Joe,
When you say that on the journey, one choo-choo train will “always have twice as many numbers” as the other choo-choo train, you are really saying that at every finite point in time, Choo-choo Train #1 will have (approximately) twice as many numbers as Choo-choo Train #2.
True enough, but you’re comparing finite subsets. Of course {1,2,3,4,5,6,7,8} has twice as many elements as {2,4,6,8}, and {1,2,3,…,1 gazillion} has twice as many elements as {2,4,6,…,1 gazillion}.
We’re asking a different question, which is does {1,2,3,…}, taken as a whole, have twice as many elements as {2,4,6,…}, taken as a whole? The answer is clearly no.
If {1,2,3,…} were twice as big as {2,4,6,…}, then the second set would run out of elements before the first one was exhausted. This doesn’t happen. For every n in {1,2,3,…}, there is a 2n in {2,4,6,…}. Neither set runs out of elements. The sets have the same cardinality. You can match their elements up one-to-one.
The equine’s corpse has been thoroughly flogged.
And when it has been flogged an infinite number of times, Joe will catch on.
keiths:
Joe’s “answer”:
Joe, no objects have been added or removed. Only the labels have been changed. Even you can see that the cardinality cannot change if you don’t add or remove any objects to/from the set. I think you’re just ashamed to admit it, because it means you’ve been spectacularly wrong throughout this entire discussion.
Suppose I have a set containing a rock, a Bible and a horseshoe. What is the cardinality of the set?
Now suppose I put labels on each of the objects. The rock gets a label with the number “1” on it. The Bible gets “2”, and the horseshoe gets “3”. What is the cardinality of the set?
Now I take the “1” off the rock and replace it with a “2”. The “2” comes off the Bible and I replace it with a “4”. The “3” label comes off the horseshoe and I replace it with a “6”. What is the cardinality of the set?
You don’t have to be a genius to see that the cardinality stays the same in all three scenarios. Why? Because we didn’t add or remove any objects to/from the set. All we did was change the labels.
The same holds true for any set, finite or infinite. If no elements are added or removed, the cardinality stays the same.
olegt found this comment somewhere on Joe’s blog:
That is so beautiful, it makes me want to cry.
Hey Joe, what happens when you add 1 to the LKN?
I’ve been doing so well avoiding the pure, straight from the source silliness and you wave something like that under my nose.
[ Edited to remove potentially offensive synonym for silliness. ]
It brings to my mind the sort of AI imagined by Douglas Adams.
“I know the largest known number” thought the computer with intense pride.
“Oh, wait a moment.”
Something whirred.
“Oh.”
The computer sank into a deep despair. If it had been given arms, it would have used them to pull the duvet over its head. Then it realised that it did now in fact know the Largest Known Number.
“I know the largest known number.” thought the computer with intense pride.
“Oh. Wait a moment.”
Something similar is happening with Joe, but the noise he makes is definitely not whirring.
Joe responds:
Thank you, Joe. 😀
Joe,
No, Joe, I’m thanking you for the entertainment. This is much better than Honey Boo Boo.
Really? What’s its current value, Joe? Do you suppose NIST tracks it?
Do you understand why everyone finds this hilarious?
Exactly. Finite sets and infinite sets behave differently. If only you could grasp that.
Apologies to Mark Frank and others for the length of this comment. Given the density of the intended audience, I think you’ll agree that it’s necessary.
Joe,
That statement shows exactly why you are hopelessly confused about infinity.
Please pay attention as I parse it:
As has been pointed out with much mirth here and elsewhere, the idea of an LKN is silly. Let’s replace it with something more sensible. In your thought experiment involving two trains rolling along the number line, there is at every point in time a largest number that has been encountered by the trains. Let’s call that the LNET, meaning “the largest number encountered so far by the trains.”
With that revision, your statement becomes:
First, you are confusing finite sets with infinite:
No. At every point in time, the finite set {1,2,3,…,LNET} will have twice the cardinality of the finite set {2,4,6,…,LNET} (assuming LNET is even). No matter when you look, the LNET is finite, not infinite, and the cardinality of the two sets is finite, not infinite.
But we’re not asking a question about finite sets. We’re asking whether the infinite sets {1,2,3,…} and {2,4,6,…} have the same cardinality. Answering the question for finite sets does not give us the answer for infinite sets.
No, Joe, they are not the same thing:
1. The LNET is a number. Infinity is not.
2. The LNET is a member of the set {1,2,3,…}. Infinity is not. That’s why we write {1,2,3,…} and not {1,2,3,…,∞}. Have you noticed that, Joe?
3. There is always a number greater than the LNET. This is not true of infinity.
As olegt and I have been telling you for weeks now, infinity is not just a really, really big number. It isn’t a number at all.
There is an important difference between finite and infinite sets that you either overlook or ignore. The cardinality of any finite set can be expressed as a number. Count the elements and you have the cardinality. The cardinality of an infinite set cannot be expressed as a number. You can’t count the elements.
Cantor recognized that even if you can’t count the elements in a set, you can still formulate a useful concept of cardinality based on one-to-one correspondences.
Imagine that you are a member of some isolated tribe that has no way of expressing numbers greater than two. You can’t count the fingers on your hand, because you don’t have a word or a symbol for the number five. Yet it is obvious to you that you have as many fingers on your right hand as you do on your left hand.
How do you know this without being able to count them? You bring your hands together with your fingertips touching. It is then clear that every finger on your right hand corresponds to a finger on your left hand. The set {right thumb, right index finger, right middle finger, right ring finger, right little finger} has the same cardinality as the set {left thumb, left index finger, left middle finger, left ring finger, left little finger}, and this is obvious even if you cannot count.
Cantor’s genius was to recognize that even when counting fails, the one-to-one correspondence method still works, and it works for both finite and infinite sets. That’s why mathematicians have embraced Cantor’s concept and why they reject yours, which only works for some finite sets.
Yet another difference between finite and infinite sets is that a finite set cannot be placed into a one-to-one correspondence with any of its proper subsets. Yet there are infinite sets that can be placed into an infinite number of one-to-one correspondences with their infinite proper subsets.
We already discussed one such case. {…,-3,-2,-1,0,1,2,3,…} can be placed into infinitely many one-to-one correspondences with {…,-6,-4,-2,0,2,4,6,…} using mapping functions of the form F(n) = ±2n ± 2c, where c is a constant integer.
Having seen all of these differences between finite and infinite sets, it’s amazing to me that you still believe that you can generalize from the behavior of finite sets to that of infinite sets. It’s ridiculous and obviously false.
A note to JoeG
I suggest that you take a look at Intuitionism and Mathematical Constructivism.
The intuitionists and constructivists have a different take on mathematics than most mathematicians. In particular, they talk of “potential infinities”, they tend to object to “completed infinities”. It’s not how I like to look at mathematics, but I give them credit for avoiding the kind of contradictions that come from your approach.
This is excellent. From what I’ve seen here (I won’t contribute to UD or Joe’s blog by sending them traffic), these seem to be the salient points:
Please correct me if I’ve misunderstood, but Joe seems not to realize that infinity is a mathematical concept, not an actual number. No one with that misconception is going to make any progress in learning about this topic.
Elebenty iz moar biggerer
Patrick,
Yep, that is the root of Joe’s confusion. He thinks that since the LKN/LNET increases without limit over time, the LKN/LNET is itself infinite:
And once Joe gets hold of a bad idea, he never lets go. As Richard once put it, he’s like a weasel with a rag dipped in rabbit juice.
Keiths, I’m touched! That and my Hempel’s paradox limerick…
I’ll bet Joe is, too.
I clicked on that link hoping for a video.
This is the best I can do.
Keep watching. It gets really funky around 0:45.
The latest:
“If not any of my opponents can:
1- Provide a mathematically rigorous definition of infinity
2- Demonstrate infinity exists
Then it is clear that infinity is arbitrary.”
Arbitrary, Could be 7.
Aleph-null bottles of beer on the wall, aleph-null bottles of beer. Take one down, pass it around, aleph-null bottles of beer on the wall. Aleph-null bottles of beer on the wall, aleph-null bottles of beer. Take one down, pass it around, aleph-null bottles of beer on the wall. Aleph-null bottles of beer on the wall, aleph-null bottles of beer. Take one down, pass it around, aleph-null bottles of beer on the wall….
I grew up hunting and fishing, and I had no idea weasels would take down rabbits.
I strongly recommend listening to either the Grateful Dead or Pink Floyd while watching that.
I have evidently risen in Joe’s estimation. I no longer am an asshole, I just argue like (and from) one:
keiths, argues from and like an asshole
Joe,
Infinity is logically coherent. The LKN isn’t. The moment you consider any candidate for the LKN, you already know of infinitely many larger numbers, starting with LKN + 1. The LKN is never the LKN. It’s self-contradictory.
What “alignment problem”? The “alignment problem” is only a problem for you and your “theory”. There is no “alignment problem” for Cantor. Any one-to-one correspondence works equally well in establishing that two sets have the same cardinality.
So does your “method”. An identity mapping is just as arbitrary as any other mapping when comparing the cardinalities of two sets. If I want to know whether {rock, paper, scissors} has the same cardinality as {scissors, rock, paper}, then the mapping
rock <–> scissors
paper <–> rock
scissors <–> paper
…is no better and no worse than…
rock <–> rock
scissors <–> scissors
paper <–> paper
The important thing is that they are both one-to-one correspondences, so either of them can be used to demonstrate that the cardinalities are the same.
In fact, your insistence on an identity mapping is a huge handicap, since it means that your method won’t work unless one set is a subset of the other. Cantor is much smarter than that.
Says the guy who argues that {1,2,3,…} has twice the cardinality of {2,4,6,…}, and thus depends on the (mathematical) existence of infinite sets. You’ve shot yourself in the foot, Joe.
In any case, it’s easy to meet your challenge:
1.Definition: Set A is infinite if every finite subset of A is also a proper subset of A.
2. Proof that infinite sets exist:
a. Assume that infinite sets do not exist. By the definition above, that means that every set has at least one finite subset that is not a proper subset.
b. Consider all finite subsets S1, S2, etc. of the set N of natural numbers. Every such subset contains a greatest element Ln (if there were any elements greater than Ln, then one of them would be the greatest element instead of Ln).
c. Ln is a finite natural number. Therefore Ln+1 exists and is also a finite natural number.
d. Ln + 1 is not a member of Sn, but it is a member of N. Therefore every Sn is a proper subset of N, thus contradicting a).
Assuming that infinite sets don’t exist leads to a contradiction. Therefore infinite sets do exist.
QED.
(I realize the chances are slim that Joe will be able to follow that proof, but that’s his problem.)
And if Newton was such a genius, then he must have been correct about alchemy. LoL!
Oh, really? Then what is the number q of elements in the set {1,2,3,…}? What is q+1?
Sure, you can count forever. That’s why Cantor coined the term “countable infinity”. But you never finish, which means you never get an answer. Counting won’t allow you to compare the cardinalities of infinite sets. Thus the need for one-to-one correspondences.
Yes, every finite set {1,2,3,…,LNET} will have twice the cardinality of the finite set {2,4,6,…,LNET}, and that will be true no matter how large the LNET gets. But you haven’t answered the real question, which is “does the infinite set {1,2,3,…} have twice the cardinality of the infinite set {2,4,6,…}?”
The answer to that question, as Cantor showed, is clearly “no”.
The LNET and infinity are not interchangeable, Joe. Infinity is not just a really, really big number.
How long before Joe notices you’re not making progress?
Maybe the problem is that he doesn’t know Hebrew.
Here you go, Joe:
BBC Horizon: Is Infinity a Number?
Oh, cool, I like that proof!
From the comments:
Infinity isn’t a number because infinity doesn’t exist. Well it exists in some people’s minds…
Sound like anyone you know?
🙂
As predicted, Joe is unable to follow my proof:
No, Joe, Ln is the largest element inside Sn. Ln + 1 is outside Sn.
No, Joe, Ln is the largest element inside Sn. Ln + 1 is outside Sn.
My “directions” don’t say anything about adding Ln + 1 to set Sn. Try again, Joe, and this time pay attention.
Joe is hopelessly confused:
Ln and n aren’t necessarily the same number, Joe. n is just an index. Ln is the largest element in the subset Sn.
Plus, you seem to have the bizarre idea that Sn must include all natural numbers from 0 to Ln. That’s not correct. Sn is just a subset of N. It might be {4,5,6}, or it might be {4, 219}.
What? That makes no sense, even in Joe Math.
The only improper subset of N is N itself, and N is infinite, not finite.
You’re flailing, Joe.
Slight correction to my proof:
…should read…
He’s added this:
Joe, N is the set of natural numbers. It isn’t finite. It doesn’t end at any Ln.
For Joe
http://youtu.be/FiMigmLwwTM?t=13m51s
Onlookers, marvel at the confusion.
Joe:
I’m not using it in my proof, Joe. I’m pointing it out to you because you don’t seem to get it.
I did.
That’s right! The cardinality of N is infinite, but it doesn’t contain any infinite elements. You learned something, Joe! Keep it up!
Awwwww. That was a short learning streak. Joe, there is no largest natural number. N = {1,2,3,…} does not have a greatest element.
Ummm, Joe — that’s a contradiction. Even if N had a greatest element, which it doesn’t, it would not be possible for Ln and Ln+1 both to be that greatest element.
You could define a set SLn that way, sure. But in my proof, Sn is just another finite subset of N. Read what I wrote:
{10} is one of those subsets. So is {72, 666}. So is {}, which is why I corrected my proof to refer to nonempty, finite subsets of N.
The point is that every one of those nonempty, finite subsets has a greatest element. In the case of {10}, the greatest element is 10. In the case of {72, 666}, the greatest element is 666.
But I can always form a number that is outside the subset by adding 1 to its greatest element. Therefore, the set is a proper subset of N, and my proof stands.
There is no largest natural number, Joe. Didn’t you learn your lesson from the “LKN” debacle? If you claim that x is the largest natural number, I can always prove you wrong by coming back with x+1, which is larger still.
Commentary would be superfluous:
Joe,
No matter how hard you try, it seems you just can’t let go of the idea that infinity is just a really big number:
Take that ridiculously large number. Square it. Square the result. Repeat this once for every particle in the universe. Then take the whole thing and raise it to the googolplex power. Do all of this a quintillion times a second for the rest of your life.
The stupendous number that results won’t account for 1%, or 0.01%, or even 0.000000000000000000001 % of infinity.
Mistaking that number for infinity is worse — infinitely worse — than mistaking a grain of sand for the entire universe.
Every finite number, no matter how staggeringly large, is infinitely small when compared to infinity.
Get it through your head, Joe — infinity is not just a Really Big Number. It’s a whole ‘nother ball game.
socle:
Joe:
There is no “highest integer” in the set of natural numbers, Joe.
Proof:
1. If n is a natural number, then so is n+1. (premise)
2. If n is a natural number, then n+1 is greater than n. (premise)
3. If p is the largest element in a set of natural numbers, then no element q exists for which q > p. (definition)
4. Assume that there exists an x that is the largest element in the set of natural numbers.
5. By #3, this means there is no natural number greater than x.
6. x+1 is a natural number (by #1), and it is greater than x (by #2).
7. Therefore, there is a natural number greater than x, contradicting #5.
8. The assumption in #4 led to a contradiction. Therefore the assumption is false.
9. There is no largest element in the set of natural numbers.
I think socle put his or her finger on the problem. You’re thinking of the natural numbers as a set that is growing over time, as people or computers or aliens in the Andromeda galaxy “discover” ever-larger natural numbers to add to the set.
That’s completely wrong. The set of natural numbers is static. All of the elements are already there, including numbers that no human, computer or alien has ever represented or contemplated.
socle:
Joe:
Numbers… the final frontier.
Joe,
You demanded a proof, thinking that no one would deliver and that you could get yourself off the hook that way.
I gave you a proof.
Now it’s your turn to deliver. Let’s hear your response to my labeling scenario:
And my follow-up point:
What’s really funny is that according to you, any pattern that holds for finite sets also holds for infinite ones. Since changing the labels doesn’t change the cardinality of the finite set described above, then according to you it shouldn’t change the cardinality of the infinite set that I described in my first scenario.
Not only are you “obvioulsy” wrong to claim that relabeling changes the cardinality of the infinite set — you’ve also contradicted yourself.
You’ve blown it, Joe, and you’ve done it in front of a large internet audience. Keep up the good work.
Joe:
Joe,
Count the elements of the following sets, and give us your answers:
{0,1,2,…}
{1,2,3,…}
{7,8,9,…}
{2,4,6,…}
{0,1,10,11,100,101,…}
{1,3,7,13,21,31, 43,…}
{0, 1/9, 2/9, 3/9, 4/9,…}
the set of all integers
the set of all rational numbers
the set of all real numbers
Joe,
What does your “methodology” say about the cardinality of the set of natural numbers versus the set of real numbers between 0 and 1, inclusive?
Joe:
keiths, still a total asshole
Joe, I realize that you’re angry and embarrassed about your performance, but lashing out isn’t going to solve anything. How about cracking open a math book or two?
No elements are added or removed when going from {1,2,3,…} to {2,4,6,…}. The existing elements are simply relabeled, as you just acknowledged. Yet according to you, the set loses half its elements during the relabeling process. Where do they go, Joe? Who removes them? You have no answers, do you?
How about that. You cut my list short and left out the very best ones, Joe:
Do those questions scare you? Let’s hear your answers.
If you think that {0,1,2,…} has one more element than {1,2,3,…}, then you think that ∞ is one greater than ∞ – 1. I guess you already forgot that infinity is not a number, huh?
Okay, so according to Joe Math:
{0,1,2,…} has a cardinality of ∞
{1,2,3,…} has a cardinality of ∞ – 1
{7,8,9,…} has a cardinality of ∞ – 7
{2,4,6,…} has a cardinality of ∞/2
{0, 10,000,000, 20,000,000,…} has a cardinality of ∞/10,000,000, according to Joe Math. That’s a really small infinity, isn’t it, Joe? 😀
Cantor wins, Joe loses. You see why we laugh?
And now the really good ones that you tried to ignore:
Joe Math gives different answers for {0,1,10,11,100,101,…}, depending on how you interpret the labels (and it’s true of the other examples as well). If you treat the labels as binary numbers, then the set has the same cardinality as decimal {0,1,2,…}. If you treat them as hexadecimal numbers, then the set has the same cardinality as decimal {0,1,16,17,256,257,…}.
Cantor gives the same answer regardless of how the labels are interpreted. Cantor wins, Joe loses.
In the set {1,3,7,13,21,31, 43,…}, the gap between the numbers increases as the elements get larger. Joe’s choo-choo train method can’t give an answer. Cantor can. Cantor wins, Joe loses.
Joe Math says that {0, 1/9, 2/9, 3/9, 4/9,…} has a cardinality of 9∞. Cantor knows that infinity is not a number. Cantor wins, Joe loses.
There are infinitely many rational numbers between each pair of successive natural numbers. According to Joe Math, the cardinality is therefore ∞ squared. What’s ∞ times ∞, Joe? Cantor gets the right answer: the rationals and the naturals have the same cardinality. Cantor wins, Joe loses.
I’ll let Joe show us how his wonderful method deals with the set of the integers and the set of the reals.
Impressive. Care to show us your derivation?
If you’ve refuted it, then you should be able to explain exactly where it is mistaken. Let’s hear it.
You’ve provided a lot of entertainment — I’ll give you that. However, I think that people deserve to be compensated for work that requires talent and/or diligence. Your bumbling requires neither, so I’m afraid you’ll have to keep on doing it for free.
kairosfocus won’t join us here or at AtBC because of how crude and offensive we are, but he welcomes Joe to the bosom of UD.
Patrick,
“My enemy’s enemy is my friend”
Yeah, I get that, but there’s also “With friends like these, who needs enemies?”
If only hypocrisy were physically painful….
Joe admits that you can relabel the elements of infinite sets:
…so I point out that…
…so Joe frantically backpedals and invents a bizarre rule:
Why not? If you’re worried about the set having two elements with identical labels during the relabeling process, then just move the existing label instead of creating a new one. C’mon, Joe — think.
1. You start out with an infinite set labeled {1,2,3,4,5,6…}.
2. Remove the “1” label from the first element. You now have {U,2,3,4,5,6,…}, where U signifies that the first element is unlabeled.
3. Move the “2” label from the second element to the first. You now have {2,U,3,4,5,6,…}.
4. Move the “4” label from the fourth element to the second. You now have {2,4,3,U,5,6,…}.
5. Remove the “3” label from the third element. You now have {2,4,U,U,5,6,…}.
6. Move the “6” label from the sixth element to the third. You now have {2,4,6,U,5,U,…}.
And so on. Using this method, you end up with {2,4,6,…}, and you never have two elements with the same label.
Joe,
You appear to believe that an infinite set is a finite set that keeps growing over time, forever. Not so.
An infinite set already has all of the members it will ever have. It is a static entity.
If an infinite set were the same thing as a growing finite set, then it would always be finite, never infinite.
And of course, your rule is inane anyway. It wouldn’t matter if there were duplicate labels during the relabeling process, given that you end up with no duplicates at the end in {2,4,6,…}.