# KF tackles the transfinite

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:

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?

## 387 thoughts on “KF tackles the transfinite”

1. keiths:

Where did all of those elements go, Joe?

Joe:

Who cares?

Now that’s mathematical rigor.

2. keiths:
keiths:

Joe:

Now that’s mathematical rigor.

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

3. 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:

One train with two (or more) counters will always count more positive integers than it will positive even integers- always and forever. So why would Cantor say the two sets are equal? It ain’t as if the counters are lying…

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 is JoeMath.

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.

4. 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.

Thus, by JoeLogic, the cardinalities of these two sets are the same:

A = {0,1,2,3…} and
B = {0,17,34,51…}

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

5. Joe,

That means trying to use finite sets to show something about infinite sets is a losing proposition.

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

One train with two (or more) counters will always count more positive integers than it will positive even integers- always and forever. So why would Cantor say the two sets are equal? It ain’t as if the counters are lying…

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.

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

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

7. Joe,

Nope. The train example goes on forever.

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

The point is that every time you look one is always greater than the other, forever and ever.

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’t have the same cardinality.

Who would adopt such a useless system?

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

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’t have the same cardinality.

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

Still pathetic

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

10. keiths:
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.

11. Frankie: And yes I agree it is time to bury your strawman version of JoeMath.

Is there a non-strawman version then? Where?

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

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.

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’t have the same cardinality.

His reasoning seems sound. Why is it not?

13. 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.

14. 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.

15. 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.

16. aleta: Others here are more knowledgeable than me, but here’s a reply.

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.

17. KF is as confused as ever:

DS, I spoke to something much more restricted, that endlessness is beyond any finite k no matter how large.

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.

18. RoyLT: 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.

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

19. “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.

20. keiths, if only there were a study on how thinking about infinity changes adults’ brains:-)

21. RoyLT: keiths, 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.

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

KF,

Some additional food for thought, from WLC’s website. Dr Craig is responding to a Muslim who is arguing for the possibility of an infinite past:

(Actually, as I think about it, it occurs to me that your argument is fallacious even if there was an infinite series of temporal points prior to creation! For from any point in the infinite past at which God chose to create there is only a finite distance to the present, just as any negative number you pick is only finitely distant from 0, even though there are an infinite number of negative numbers. From the fact that at any point in an infinite past God is able to create a world, it does not follow that God was able to create a world with an infinite past.)

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.

23. KF is as confused as ever:

Aleta:

To Dionisio, Origenes, HeKS, Silver Asiatic, and anyone else coming late to this party: to help explain what has kept this thread going so long, let me ask each of the following two questions:

Consider the integers on a standard number line, with the positive counting numbers 1, 2, 3, … to the right of zero and their negatives to the left.

1. True or False: there are an infinite number of integers

2. True or False: Every integer is a finite number.

What are your answers to questions 1 and 2.

Note: this is a purely mathematical question. It is not about time, or ladders, or tapes – just math.

KF:

Aleta, you and I are both aware the ellipsis cannot be exhausted. All numbers we can reach will be finite but endlessness remains onwards; as the pink vs blue punch tape examples highlight. And yes this points to limitations on what ordinary mathematical induction can reach by chaining, again only to the potentially infinite. And this is echoed in the problem of a claimed infinite succession of finite values at +1 increments from 0. The succession cannot be completed by any successive process and were it there is an incoherence involved as in effect the successor set to the last achieved is a copy of the sequence so far. If completed to actual endlessness, there would be endless and non finite members. The answer is, the endless cannot be traversed in successive +1 steps from 0. KF

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

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

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

Aleta: 1. True or False: there are an infinite number of integers
I am going to go with false. If they cannot be numbered, there is not an infinite number of them. Plus, infinity is not a number, so there cannot be an infinite number of anything.

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

Predictably, we will see implications of actually completed infinite successive, stepwise causal chains to date, backed up by distractions such as that at every point we identify a past infinity would already exist and attempts to deny or dismiss that implying such an infinite actual past requires that there be real world times that are endlessly separated and yet succeeded from the one to the other in finite stage successive steps, and more. When, it is readily seen that no such finite stage process can actually complete an endless span as beyond any kth step, there is a k+1, k+2, etc. onward, i.e. we can just as well call k = 0 and begin again, never able to span endlessness in steps.

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

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

Oddly enough, this very issue was what I had in mind in my first post in this thread. I simply believe that your argument involving the tapes is unsound. That’s it. No further agenda or motive.

KF’s batshit response:

DS, with all due respect evasive and inviting of dismissive projections. There is a kulturkampf in progress, it is now leading to riotous assemblies in the streets. The worldview engine that drives what is happening is the utterly destructive and pervasive, entrenched, powerful influence of radically secularist evolutionary materialist scientism and its fellow traveller ideologies and agendas [–> note this emphasis on a key distinction] that feed on its self-referential incoherence, its amorality and radical relativism, leading to a long rising tide of nihilisms that operate on the premise that might and manipulation make ‘truth,’ ‘knowledge,’ ‘right,’ ‘rights,’ ‘news,’ ‘education,’ ‘justice,’ ‘law,’ ‘history,’ etc etc. The impacts are so direct, so dominant and so utterly ruinous — starting with the ongoing holocaust of posterity of 800+ millions in 40+ years, mounting up at a million more per week — that we no longer have the luxury of ignoring or imagining we are neutral bystanders etc: s/he who does not stand up to fight against the onrushing destructive tide is now a part of the problem, regardless of what s/he may wish. Just as in the 1930’s and 40’s with Nazism. This pattern drives much of what happens in and around UD. KF

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

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

Predictably, we will see implications of actually completed infinite successive, stepwise causal chains to date, backed up by distractions such as that at every point we identify a past infinity would already exist and attempts to deny or dismiss that implying such an infinite actual past requires that there be real world times that are endlessly separated and yet succeeded from the one to the other in finite stage successive steps, and more. When, it is readily seen that no such finite stage process can actually complete an endless span as beyond any kth step, there is a k+1, k+2, etc. onward, i.e. we can just as well call k = 0 and begin again, never able to span endlessness in steps.

Should we invite JoeG back to educate us on infinity?

28. Patrick,

Should we invite JoeG back to educate us on infinity?

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.

29. Notice that Latemarch immediately understood that in your tape example, all cells have finite distance from one another. It’s “obvious”. All cells must be reachable (in a finite number of steps) from the end. If not, the unreachable cells could not be connected to the reachable portion, since no unreachable cell can be adjacent to a reachable cell. So your tape would consist of two parts that cannot touch. That’s not how tapes (even infinite thought-experiment tapes) behave.

30. Poor KF:

In this context to claim an infinitely remote past requires that there were stages that were once the present but are now transfinitely remote behind a chain of onward, causally connected finite step stages.

The same poor logic from one year ago:

Likewise, point to the copy of the sequence so far successor counting set principle. An endless repetition of successive defined counting sets would end up with one or more that are in themselves endless.

An entire year, and zero progress.

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

DS, On fair comment I have repeatedly shown why I concluded as above. You obviously disagree but instead of explaining why and showing what an infinite actual past means other than that there was at least one [and in fact indefinitely many] once present moments that are now after successive stages transfinitely remote, you have tried to suggest that I have failed to show why I have concluded as above. What part of durations in time are between specific stages is hard to understand? What part of, if stages X and Y have only finitely many stages between them, the duration is necessarily finite, is hard to acknowledge? What part of, if EVERY past time we can identify or even symbolise as having occurred will be finitely remote, then there are no such identifiable transfinite durations, is so hard to see? What part of, just proceeding with ever more finitely remote cases and suggesting ever more FINITELY remote cases can be added does not get us to an infinite duration, is so hard to then recognise? Going on, what part of, if even so there were an actually transfinitely remote stage I, then its step by step successors I+1, I+2 etc will face the impossibility of traversing an endless transfinite span in finite-stage steps, is so hard to acknowledge? KF

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