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!)
I can’t tell if you have infinite (heh) patience or suffer from what my British friends call bloody mindedness. I’m finding it entertaining in either case, thanks!
keiths:
Joe:
No, the Joe G choo-choo train method fails utterly in that case, just as it does in so many others.
Choo-choo train #1 is counting the natural numbers {1,2,3,…}. Choo-choo train #2 is counting the members of the set {1,3,7,13,21,31, 43,…}.
After the 1 is encountered, train #1 and train #2 have the same count: 1. After the 2 is encountered, train #1 has a count of 2 and train #2 still has a count of 1.
Do this for a while, and you’ll see that (“obvioulsy”) the ratio of train #1’s count to train #2’s count is not constant.
Ratio of train #1’s count to train #2’s count after each successive natural number is encountered:
1, 2, 3/2, 2, 5/2, 3, 7/3, 8/3, 3, 10/3, 11/3, 4, 13/4, 7/2,…
Tell us, Joe — how much bigger is {1,2,3,…} than {1,3,7,13,21,31, 43,…}?
Cantor wins. Joe loses. Again.
I do it mainly for teh LULZ, but there is a nice side effect. I’m a firm believer in the adage “if you want to learn something, teach it.”
Joe may not have learned anything, but I understand infinity better for having tried to explain it to him.
For example, until I responded to Joe’s bogus demand for a proof that infinite sets exist, I had never realized that you could define an infinite set as one whose finite subsets are always proper.
Project much, Joe?
Joe:
That sounds a bit too Joe-like. Cantor was a smart man, so his thought process, boiled down, would have been more like this:
Cantor wins. Joe loses.
All this was presented in a wonderful book called One, Two, Three, Infinity by George Gamow. I read it 50 years ago in high school and have read it at least a couple of time since.
It’s still in print and rates five stars at Amazon.
Joe to Richard:
Joe, click here.
I’ve heard it’s good. Perhaps I’ll go pick it up at the library.
Maybe you should too, Joe.
Joe is taking a break from infinite sets today so that he can learn how to download PDF files.
(h/t Richardthughes)
Joe continues mangling set theory in a new post at his blog:
Set Theory: Counting is Irrelevant when Counting the Number of Elements?
It contains this memorable quote:
That’s the problem. Unlike Joe, mathematicians don’t treat numbers with respect.
Joe responds with an addendum:
In case you’re wondering what Joe’s on about — what horrible desecration my buddy Georg and I have perpetrated on the body of mathematics — it is this: we have asserted that the cardinality of a set does not depend on the identity of its elements — only on their numerosity. To Joe, that means that we are claiming that “numbers have no meaning.”
Substitute one number for another and the cardinality of a set doesn’t change: {1,2,3} has the same cardinality as {5,6,7}. A one-to-one correspondence can be established:
1 <–> 5
2 <–> 6
3 <–> 7
Yet according to Joe, everything changes if the sets are infinite. {1,2,3,…} cannot be placed into a one-to-one correspondence with {5,6,7,…}, because that would not be treating the numbers with respect.
It is now suddenly and inexplicably against the rules to continue the mapping we used above:
1 <–> 5
2 <–> 6
3 <–> 7
…cannot be extended to:
1 <–> 5
2 <–> 6
3 <–> 7
4 <–> 8
5 <–> 9
…and so on.
Why? You’ll have to ask Joe. JoeMath works in mysterious ways, and requires lots of ad hoc and contradictory rules.
So Mr. “I fix things- all kinds of things- mechanical, electrical, electronic and personal” was unable to connect to a website and download some PDFs without outside help?
Here’s what’s really funny. It has never dawned on Joe that the process of counting — which he wholeheartedly endorses, even for infinite sets — is exactly the sort of thing he says should never be done.
Why? Because it involves setting up a one-to-one correspondence between non-identical elements.
How do you count the elements in {5, 23, 41, 99, 666}? You mentally (or in Joe’s case, physically) point to the 5 and say “1” to yourself. You point to the 23 and say “2”, and so on.
In other words, you’ve established exactly the kind of one-to-one correspondence that Joe says should be outlawed:
1 <–> 5
2 <–> 23
3 <–> 41
4 <–> 99
5 <–> 666
Another own goal for Joe. Good job, Mr. Fixit.
If Joe had two beans and one had “15” pained on it, he’d have 16 beans by Joemath.
Joe should consider the cardinality of the set of people who use Joemath. I would love to see him on a crusade to share his system. Surely it is important to do so if his system is more useful? Joe posting on maths forums with the good news would be great fun.
“…as long as infinity exists and especially when infinity ceases to exist.”
Utter Joeberish.
Joe,
You are making the same mistakes again and again.
“Always” means “at every point in time”. Every point in time is finite. Infinity is not a point.
At every finite point in time, the Joe Train will have picked up roughly twice as many natural numbers as positive even integers. So what? You can’t generalize that result to infinity, because infinity is not a point. The train will never reach it.
You also have some very odd ideas about time and sets.
You seem to think that finite sets, if they are growing and will never stop growing, are already infinite. They are not. They are finite and therefore infinitely small compared to any infinite set. There is never a point in time when such a set “reaches” infinite size. It is always finite.
You also seem unable to think about infinite sets in non-temporal terms. The set of natural numbers is static, Joe. All of the numbers are already there. The set is not growing, and there is no “current largest element” in it.
And for FSM’s sake, set theory is a branch of mathematics, not physics. Cantor’s ignorance of relativity does not invalidate his work in the slightest, except in your fevered imagination.
The concept of infinity as anything other than some unimaginably large number is very difficult for some to grasp; as is the concept of something that cannot be added to to make it bigger
I’m afraid Joe will never do it.
Joe has now redefined the point as possessing dimension. Presumably lines – the geometric kind – possess breadth. Which is why they can never be infinite, because there isn’t enough lead in the universe to draw ’em. 🙂
Allan,
I think even Joe realizes that’s ridiculous. However, he is desperately searching for a way out of the corner that socle and Jerad have backed him into.
Joe:
socle:
Joe:
socle:
Joe:
Jerad:
Joe:
Joe:
socle, presumably rolling his or her eyes:
Jerad:
Joe:
Jerad:
Joe:
Joe can’t admit that he’s wrong, of course, so he has to alter the definition of a point. He needs to pretend that there is a least real number greater than 0, and the only way he can see to do that is to stipulate that points have a finite diameter, so that there really is a single point on the number line immediately to the right of 0.
So in the useless and ridiculous JoeMath, the intervals [0,1] and (0,1) contain only a finite quantity of real numbers, every real number has a successor, and the set of real numbers is countably infinite.
Hey, Joe — how do you know that points all have the same diameter? Could this be a fruitful area for ID research?
keiths:
Joe:
Joe,
In that metaphor, infinity is the entire journey, taken as an infinite whole. You are taking finite segments of that journey and arguing that whatever is true of a finite piece must also be true of the infinite whole, which is simply (and “obvioulsy”) false.
That is why we laughed when you wrote the following:
That’s as inane as claiming that 0 = 1 for sufficiently large values of 0.
I’m having a sense of déjà vu here.
Let S = 1 + 2 + 4 + 8 + 16 + 32 + …
Then 2S = 2 + 4 + 8 + 16 + 32 + 64 + …
But then we see that 2S is just S – 1. Subtract S from both sides and we get that S = -1.
But, from the first equation above, S goes to infinity.
Therefore, whenever we encounter infinity, then we can just replace infinity with -1 and keep going. Problem solved.
And what would you call the thing at their centres? Or the centre of that, or … it’s points all the way down!
Allan,
Somehow I doubt you’ll be surprised to hear that Joe has already fallen into that trap. Watch the confusion build in Joe’s mind as the following conversation progresses:
socle:
Joe:
socle:
Joe:
socle:
Joe:
socle:
Joe:
socle:
Joe:
Euclid is generally rather highly thought of! 🙂
In JoeMath, lines are either very thin rectangles, or touching circles; I can’t be sure which. Bounded by … er …
Are you archiving Joe’s comments, Keith? Even he is eventually going to realize how badly this reflects on him and delete it from his blog. It would be a tragedy on the scale of the burning of the Library of Alexandria if JoeMath (Does he really call it that himself?) were to be lost to the world.
I’m not so sure about that. I think he may actually go to his grave believing he has seen what that “frustrated child” Cantor could only dream of.
We’ve quoted the juiciest bits here and at AtBC. It make take centuries, but our best minds will someday reconstruct JoeMath in its monumental entirety, working from those precious fragments.
Hey Joe,
You insist that {1,2,3,…} is twice as large as {2,4,6,…}. If so, then it should be impossible to set up a one-to-one correspondence between them, because the smaller set should run out of elements before the larger one does.
Yet the mapping F(n) = 2n works just fine, with neither set running out of elements. For every n in {1,2,3,…} there is a 2n in {2,4,6,…}. No leftovers.
Cantor understood that this is because the two sets have the same cardinality. How does JoeMath explain it?
Or if you do think that {2,4,6,…} runs out of elements, show us how it happens, and show us which elements of {1,2,3,…} remain unmatched.
I want you to really think this through. It could help you understand why JoeMath is such a waste of time.
Joe to Jerad:
That’s my last irony meter turned to slag!
Joe spelled “obviously” correctly!
Joe,
That’s just silly, Joe. Consider the two sets {3,4,5} and {6,8,10}. Each has a cardinality of 3. The mapping function F(n) = 2n establishes a one-to-one correspondence between them. According to you, that means that one of them is twice as large as the other. That’s ridiculous, even by the standards of JoeMath.
Is {3,4,5} a thousand times as large as {3000,4000,5000}?
Joe, if one set is twice the size of the other then it should be impossible to create a one-to-one correspondence between them. The smaller set should run out of elements before the larger one does. Yet there are no “leftovers” when we map {1,2,3,…} to {2,4,6,…}. Why not?
Have you forgotten? Infinity is not a number. It doesn’t behave like a number. Infinity +1 is no larger than infinity itself. Add an element to an infinite set and it remains the same size.
Joe @ UD is resisting ‘correction’ by KF, but comes within a whisker of getting it.
Transformation neither adds nor removes members. If they have the same cardinality after, they must have had the same cardinality before.
Why not? {1,2,3,4 …} transforms to {2,4,6,8 …} by n->2n – precisely what you now accept.
I gave an example above in which I compared the sets {1,2,3} and {5,6,7}. They have the same cardinality, which can be demonstrated using the mapping function F(n) = n+4:
1 <–> 5
2 <–> 6
3 <–> 7
Add another element to each set, and they still have the same cardinality. |{1,2,3,4}| = |{5,6,7,8}|.
Keep adding elements in parallel to both sets, and the cardinalities remain equal.
Using Joe’s own emphatic logic and language:
JoeMath contradicts itself. The cardinality of {1,2,3,…} is both equal and not equal to the cardinality of {5,6,7,…}.
Poor Joe is still struggling:
Joe,
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,…}.
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.
For example, if you want to determine, without counting, whether {3,77,666} and {G,C,M,S,J} have the same cardinality, just look for a one-to-one correspondence between them. If you find one, then you can be certain that the sets have the same cardinality. If a one-to-one correspondence does not exist, then the sets have different cardinalities.
No matter how hard you try, you will never find a one-to-one correspondence between {3,77,666} and {G,C,M,S,J}. They do not have the same cardinality.
Your claim that the F(n) = 2n mapping is not a “naturally derived alignment” is just silly. It’s exactly the “alignment” we would use to count the elements in {2,4,6,…}! 2 is the first element, so 1 maps to it. 4 is the second element, so 2 maps to it. 6 is the third element, so 3 maps to it, and so on.
The problem isn’t with the mathematicians, Joe.
Joe is tantalizingly close to actually getting it.
socle asks him to solve the problem Jerad posed:
Joe responds:
Joe,
I want you to notice two things:
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.
2. In other words, you applied Cantor’s method (finally!) and got the correct answer (finally!).
3. On the other hand, JoeMath (and its patented choo-choo train method) were utterly unable to deal with Jerad’s problem.
Joe, to mathematicians: Try my method. It’s less capable, and it gives the wrong answer much of the time!
Mathematicians: No thanks. We’ll stick with Cantor.
Joe is embarrassed that he needed Cantor’s method to answer Jerad’s question:
No, you used Cantor’s method with the mapping function F(n) = 1/n.
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,…}.
Cantor wins. You lose.
What a train wreck. Joe contradicts himself in the very next comment:
Joe,
The mapping function for Jerad’s problem is 1/n. According to you, that means the cardinality of {1, 1/2, 1/3,…} is the reciprocal of the cardinality of {1,2,3,…}. Yet you already told us that the cardinalities are the same.
Can you explain why JoeMath gives contradictory answers to a single question?
Why would any sane mathematician use it?
Interesting that the set {1/n} for n = 1, 2, 3, … contains the terms in the harmonic series.
The alternating harmonic series is also interesting.
A = 1 – 1/2 + 1/3 + 1/4 – 1/5 + 1/6 – 1/7 + 1/8 – 1/9 + …
If this is added up in the order given, A = ln2.
However, added up in a different order gives any number we wish.
Take all the negative terms and put them in a series; and take all the positive terms and put them in another series. Both series add to infinity; one negative, the other positive.
Now take some numbers from the positive series and add them together to get lightly more than some number you would like the sum to equal. Then take a few from the negative series to make the sum slightly less than that number. Keep taking numbers from each series and successively go above and below your chosen number, getting closer and closer with each pick.
The point is that, since each pile of numbers adds up to infinity, one can do this for as long as one likes and converge on any sum you wish.
Therefore, ln2 equals any number you wish.
Young kids who understand this turn out to have considerable mathematical talent. If one presents this exercise to a student and the student gets hooked, you know you have a bright kid in front of you.
Joe’s latest response is a combination of “Nuh-uh!”, “Did so!”, and “I know you are, but what am I?”
How embarrassing for him. He couldn’t solve the problem using JoeMath, so he had to borrow a method from that “frustrated child” Cantor.
Turns out Cantor wasn’t so frustrated after all, eh, Joe?
Really. It’s not converting them into ‘generic’ elements, it’s mapping them to the set of integers, in which a member gives its own ordinal position. If a set has a member at position n, even if that member is not itself “n”, then it must map to the set of integers. Name an integer that is not in the set of integers. Give the ordinal position of any member of a countably infinite set that is not an integer.
The ordered set of positive integers has a member n at position n. All other countably infinite sets have something else at position n, but they still have a position n, precisely because you can count them but they are infinite. Even if the member at position n is actually n-1 (for example) – ie {0,1,2,3 …}, or 1/n (ie 1, 1/2, 1/3, 1/4 …}.
Joe, yesterday:
Joe, today:
Now that’s entertainment.
Entertainment or throwing rocks at the short bus? I know, I know, he’s doing it to himself. I’ve just been meditating on compassion lately.
Patrick,
Implying Joe is a special needs student is compassionate?
Patrick,
I think it’s possible to feel sorry for him and be entertained by him at the same time.
I truly hope that he learns something from the exchange, both mathematically and interpersonally. However, if he insists on being an ass, I’m not going to feel guilty for enjoying the show.
Joe to Jerad:
Joe,
That is true for finite sets, but not for infinite ones. The reason? The cardinality of finite sets is a number. The cardinality of infinite sets is not.
You still implicitly think that infinity is a number and that it behaves just like a number when it is added to or subtracted from.
For example, you think that the cardinality of {4,5,6,…} has to be less than the cardinality of {1,2,3,…}, because “one set contains all of the members of the other set AND has members the other set does not”.
That is tantamount to saying that
|{1,2,3,…}| = ∞
|{4,5,6,…}| = ∞ – 3;
therefore
|{1,2,3,…}| – |{4,5,6,…}| = 3
But to do that is to treat infinity just like a number. Infinity is not a number, and treating it like one leads to absurdities, as Mike’s example shows:
Not in the slightest.
My experience with Joe, when I used to read Uncommon Descent and before he made himself unwelcome here at TSZ, has been uniformly negative. He epitomizes the willfully ignorant intelligent design creationists who threaten science education in this country by unthinkingly block voting however directed by their religious leaders. As far as I’m concerned, mockery is a legitimate weapon against such people.
I said I was meditating, not that I was enlightened.
Re-reading what I wrote, I see that it could have been taken as a criticism. I apologize for that confusion. I’m right there with you in finding JoeMath amusing, I just had an odd compassionate instinct related to Joe’s inability to admit any error (and the related insight into his self-image) and felt the urge to share it.
There’s plenty to criticize about Joe’s presentation and positions. While I agree with you that it is entertaining, it’s also a hint sad.
Again, my apologizes if you found my comment critical of you personally.
Patrick,
No need to apologize. I think it’s an important question, well worth asking.
I’m a compatibilist in that I think we have a meaningful kind of free will: the freedom to desire, consider, evaluate and choose according to our own natures. I don’t think we are ultimately responsible for choosing our natures, however.
Joe didn’t arrange for his intelligence and temperament to be what they are, but he has to live with the consequences. In absolute seriousness, I think he deserves our sympathy and compassion for that unfortunate fate.
Not having chosen those things in any ultimate sense, Joe doesn’t deserve our mockery as a form of punishment. However, mockery is still an appropriate response to bad ideas, obnoxiously presented. It encourages the mockee (and others) to think more carefully, and entertainment is a side benefit.
I don’t see mockery as a punishment, but as a tool. We should never forget that this is a political conflict, not a scientific dispute. It’s worth continuing to point out the scientific errors made by the intelligent design creationists, to deny IDC the unearned respect its proponents so desperately crave, but that’s not the arena in which the battle will be won or lost.
Focusing solely on intelligent design creationism’s lack of scientific merit cedes the real field to those who would destroy science education, eliminate promising research avenues, and ultimately impose their sectarian views across society. We need to leverage their scientific failings to deny them credibility in the political realm. They would be perfectly happy to continually lose the scientific debate so long as they could spin it as being taken seriously by real scientists.
That’s why mockery is useful. If people are writing papers refuting you and angrily denouncing your views in public, an outside observer might think you have something meaningful to say. If a group of intelligent, well-educated people are pointing and laughing at you, that same outsider is less likely to take you seriously.
An exchange at Joe’s blog:
Joe:
socle:
Joe:
socle:
Joe:
socle:
Joe:
socle:
Joe: