Veteran TSZers may recall an entertaining thread in which a bunch of us tried to explain the cardinality of infinite sets to Joe G:

**A lesson in cardinality for Joe G**

At UD, commenters daveS and kairosfocus are now engaged in a long discussion of the transfinite, spanning three threads:

**An infinite past can’t save Darwin?**

**An infinite past?**

**Durston and Craig on an infinite temporal past…**

The sticking point, which keeps arising in different forms, is that KF cannot wrap his head around this simple fact: *There are infinitely many integers, but each of them is finite.*

For example, KF writes:

DS, I note to you that if you wish to define “all” integers as finite -which then raises serious concerns on then claiming the cardinality of the set of integers is transfinite if such be applied…

The same confusion arises in the context of Hilbert’s Hotel:

KF:

Try, the manager inspects each room in turn, and has been doing so forever at a rate of one per second. When does he arrive at the front desk, 0?

daveS:

Re: your HH explanation: If the manager was in room number -100 one hundred seconds ago, he arrives at the desk now.

KF:

Yes a manager can span the finite in finite time. But the issue is to span the proposed transfinite with an inherently finite stepwise process. KF

daveS:

In the scenario I described above, the manager was in room -n n seconds ago, for each natural number n. Given any room in the hotel, I can tell you when he was there.

KF:

DS, being in room n, n seconds past does not bridge to reaching the front desk at 0 when we deal with the transfinitely remote rooms; when also the inspection process is a finite step by step process.

What KF doesn’t get is that **there are no ‘transfinitely remote rooms’.** Each room is only finitely remote. It’s just that there are infinitely many of them.

Any bets on when — or whether — KF will finally get it?

keiths:

Joe:

Now

that’smathematical rigor.Nice quote-mine. What does it matter where they went? I dare you to try to make a case

What’s especially funny is that we can use JoeMath to “prove” that {0,1,2,3…} and {0,17,34,51…} have the same cardinality.

Regarding his choo-choo trains, Joe writes:

Now consider the two sets

A = {0,1,2} and

B = {0,17,34}.

They have the same cardinality. Add 3 to set A and 51 to set B. Now you have

A = {0,1,2,3} and

B = {0,17,34,51}.

Still the same cardinality. Now add 4 to set A and 68 to set B:

A = {0,1,2,3,4} and

B = {0,17,34,51,68}

Still the same cardinality. Keep repeating this process and the two sets will have the same cardinality, “always and forever”. Thus, by JoeLogic, the cardinalities of these two sets are the same:

A = {0,1,2,3…} and

B = {0,17,34,51…}

That’s terrible reasoning, of course, but it

isJoeMath.JoeMath contradicts itself: {0,1,2,3…} has the same cardinality as {0,17,34,51…}, and it has a different cardinality.

Not being idiots, mathematically literate people reject JoeMath.

keiths,keiths is just making shit up again because he has proven that he doesn’t know anything about math or infinity. He thinks that you can remove all of the elements from an infinite set and he never supports what he says. he just repeats it.

With finite sets you can actually count the number of elements. You cannot do that for infinite sets. That means trying to use finite sets to show something about infinite sets is a losing proposition.

Nope, but I understand that you have to put words in your opponent’s mouth in order to “win”. Sad and pathetic

Joe,

Yet that’s exactly what you do in your choo-choo train exercise:

At any given time, the set of positive integers that have been encoutered by the train is roughly twice the size of the set of positive even integers that have been encountered. Those are finite sets, yet you foolishly conclude that what is true of the finite sets — that one has twice the cardinality of the other — must also be true of their infinite counterparts.

keiths,Nope. The train example goes on forever. The point is that every time you look one is always greater than the other, forever and ever. But thank you for continuing to misrepresent the argument. It is the only way you can “win” but it is still pathetic

Joe,

So does my example. You never stop adding elements to sets A and B.

And every time you look, sets A and B have the same cardinality.

The same (bad) reasoning that leads you to conclude that {0,1,2,3…} is twice as big as {0,2,4,6…} also “proves” that {0,1,2,3…} and {0,17,34,51…} have the same cardinality.

According to JoeMath,

a) {0,1,2,3…} and {0,17,34,51…} have the same cardinality, and

b) {0,1,2,3…} and {0,17,34,51…}

don’thave the same cardinality.Who would adopt such a useless system?

Set subtraction refutes your claim. We already have a way to align sets- by matching like elements from each set. You are changing that.

Wrong again. Obviously all you can do is misrepresent your opponent.

Still pathetic

Time to give JoeMath a decent burial next to “frequency = wavelength” theory.

You mean your strawman of it. And yes I agree it is time to bury your strawman version of JoeMath.

keiths, losing big time and punts. Life is good…

Is there a non-strawman version then? Where?

Most normal people would say “no, you are wrong because XYZ” .

If someone does not understand something you are saying chances are you have poorly explained it.

Why not actually try to explain why someone is wrong and provide a correction explaining why they are wrong?, instead of just saying they are wrong and deliberately so. If you don’t provide a detailed correction, you have no moral right to say someone is misrepresenting your position. All any onlooker has is your refusal to explain and your bald declaration you are in the right.

His reasoning seems sound. Why is it not?

Yes, OM, I am sure that you think you have something to ad. However it is obvious that you do not. All of your questions have been answered and I am not going over it again, especially with someone who couldn’t understand it.

It’s a little tough to extract the relevant math concepts from the hail of ad hominem attacks here, but I have an honest question. As a person who is completely unfamiliar with set theory (I’m still trying to get a grasp on how a ZF infinite can be a Dedekind Finite), I’m afraid that I might be oversimplifying the finite to infinite distinction.

a) {0,1,2,3,…,1*10^1000} has equal cardinality to {0,2,4,6,…,2*10^1000}

b) {0,1,2,3,…,Infinity} has equal cardinality to {0,2,4,6,…,2*Infinity}

c) 2*Infinity=Infinity

d) {0,1,2,3…} has equal cardinality to {0,2,4,6…}

Is that way off the mark??? Some of the commenters on here sound like very competent mathematicians and I’m trying to learn.

Also, does anyone use the term ‘transfinite’ anymore? I have only encountered it from Cantor and Dembski (by way of Kauffman). And obviously those two had very different reasons in mind.

Hi Roy. Others here are more knowledgeable than me, but here’s a reply.

a) is true. Each is a finite set with 10^1000 + 1 numbers. (Things would be a bit easier if we started counting at 1 in this case.)

b) These are usually written just {1,2,3 …}, without the “infinity” and {2,4,6 …} They both are infinite sets and they do have the same cardinality. Statement d is identical in meaning to b. Saying the second set goes to “2*infinity” is not a meaningful statement. Both sets can be extended indefinitely, and thus are said to have an infinite number of members.

The basic idea is that a set is infinite if it can be put in a 1:1 correspondence with a proper subset of itself. {2,4,6…] is a proper subset of {1,2,3,…}, and the elements can be put into a 1:1 correspondence by just extending the sets you mentioned in a indefinitely: 1 -> 2, 2 -> 4, 3 -> 6, … n -> 2n, …

Both {1,2,3 …} and {2,4,6, …} are infinite to the same degree, so to speak, and are said to have cardinality aleph null (a fancy symbol I can’t type here.)

Maybe you knew all that, but that’s an overview of what we’ve been discussing.

I admire your humility, but I have read several of your comments on this thread and I admire your perseverance in trying to break down these concepts for those unable or unwilling to get them.

Thanks for the clarification. I do indeed understand the discussion as you have just put it, but whenever I have doubts (and the interposition of sarcasm and vitriol in these comments is quite misleading at times) I prefer to go back to very basic principles.

I think that those who are completely unable to grasp this distinction are still stuck trying to treat an infinite like a finite. I would say that a weak analogy from physics is that of Heisenberg. Once you consider a set as having a discernible end (finite), it can no longer be considered in terms of infinity and vice versa.

KF is as confused as ever:

Endlessness is not something that is located “way out there” beyond any finite k. It’s a property of the sequence as a whole.

The endlessness of the natural numbers means that there’s no end at 1, no end at 2, no end at 3, and no end at any k in the set.

Agreed. In fact I find Aleta’s posting style charmingly familiar. 😉

“Familiar”??? 🙂 There is a possibility that Alan knows who I really am, from an interchange some years ago. I can’t remember the details, and may be wrong, but if so that is fine as I trust him with the information.

So if what I remember is correct, hi Alan.

At

Aeon:How thinking about infinity changes kids’ brains on mathkeiths, if only there were a study on how thinking about infinity changes adults’ brains:-)

That would have no effect on his brain, which is why he posted the link to juvenile brains.

At UD, daveS is still steering KF toward a realization regarding the finitude of every integer:

KF has been citing Craig as an authority throughout the thread. Will he accept that his authority is correct that every integer is finite?

Kudos to daveS for his persistence.

KF is as confused as ever:

Aleta:

KF:

What’s interesting is how many of his fellow IDers are also struggling to understand this stuff.

LoL. Go ahead keiths, resist the bait! I dare you.

Heh, then I’m compelled to ask you, what is the largest known integer?

A year later, and KF is still as confused as ever:

He’s gone full Gaulin.

I think he’s even managed to one-up Gary.

daveS comments, in response to the above bit from KF:

KF’s batshit response:

Amen. Anyone who corrects KF’s dumb mistakes regarding the transfinite is undermining civilization as we know it.

Does he know who ate the strawberries?

Curse those Radsecevomatscis!

Should we invite JoeG back to educate us on infinity?

Patrick,

If I thought he could actually come up with something new that was as entertaining as

his first time around, I’d be sorely tempted.But he can’t, so no.

daveS is still valiantly trying to pound it into KF’s thick skull:

Poor KF:

The same poor logic from one year ago:

An entire year, and zero progress.

Poor KF wouldn’t get it even if Jesus himself came down from heaven and explained it: