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?

I posted this at reddit:

https://www.reddit.com/r/math/comments/45sri1/all_natural_numbers_are_finite_hence_set_of/

Sal:

Sal,

You’re confusing two distinct approaches here. If you apply that particular definition of “finite”, which is #1 in the Wikipedia article…

…then you don’t need the induction, because every natural number is already

definedas finite. The induction adds nothing.(Remember that in set-theoretic terms, a natural number n is

definedas the set {0, …, n-1}, with 0 being defined as the empty set.)If you

dowant to use induction, based on these premises……then the induction proceeds upwards, and you show that n+1 is finite only

aftershowing that n is finite. It’s redundant to turn around at that point, using the finiteness of n+1 to demonstrate the finiteness of n.What paradox?

Inclined to agree that proof is redundant, what do the professionals think?

Agree 0 is finite, but how do you define finite and demonstrate 0 is finite? Which wiki definition or any definition do you use to demonstrate finiteness?

Thank you for asking. If we define the natural numbers as Designed, then we don’t need any inference to design. Therefore God. QED

Sal,

My preferred definition is that a finite set cannot be put into a one-to-one correspondence with any of its proper subsets.

But again, the exact definition doesn’t matter, as long as your interlocutor accepts both premises:

If P(0) holds, and P(n) => P(n+1), then P(n) holds for all natural numbers, regardless of what P(n) actually means. It could mean “n is nonnegative”, or “n is not a cheeseburger”, or “n is finite”.

Sal,

Let’s ask Frankie.

*busts out laughing*

Yes. The induction is not actually needed.

What do

youthink, Frankie? Is every natural number finite, despite the fact that there are infinitely many of them?Mung,

How is

yourmathematical education coming along?Link, Link

I’m still wondering why all these numbers are allowed to prance around naked.

Thanks!

Whoa! Ok, thanks! Sounds right, but ouch, that’s a little hard.

It should be so, however it is a given that there will be many numbers too large to write out by hand in all the lifetimes in the world.

Also Hilbert’s Hotel is total nonsense…

keiths:

Sal:

Look at the opposite side of the coin: a set is infinite only if it has proper subsets that are as large as it is (i.e. have the same cardinality).

That’s a counterintuitive property that isn’t shared with finite sets. Virgil Frankenjoe is still struggling to understand it after years.

keiths:

Virgil:

It’s okay, Virgil. Just put the pencil down and tell me what you

thinkthe answer is — not what it “should be”.keiths,You are the one who is struggling, keiths. You can’t even grasp the ramifications of set subtraction which prove that some countably infinite sets have more elements than others.

BTW, I understand it. I understand that it is incorrect.

Your contributions would be more helpful if you would substitute explanation for insult once in a while.

Two counters, both representing countable infinite sets- one counts every second and the other counts every other second. The first counter will always have a higher count than the second- always and forever. That also means the first set has more elements than the second at any and every point in time.

keiths cannot deal with that one. And that is OK with me

You can’t handle explanations and all you do is insult. Evolutionism is void of explanative power. So perhaps you should focus on that

Frankie swimming in his own ignorance. And it’s a vast sea. No land in sight and never expected to be.

Frankie,

It’s good that you’ve progressed from choo-choo trains to counters, but you’re still wrong.

No need for me to reinvent the wheel. I’ve already explained this to you:

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.

How about that!

I’m rather disgusted with his condescending tone acting as if he’s an expert when he’s the one needing remedial training.

I asked KF a simple question. Estimate the entropy of:

a. functioning 747

b. broken 747

c. empty soda can

I provided my estimates.

Entropy( functioning 747) ~= 6.94 x 10^7 J/K

Entropy(broken 747 with wing ripped off) ~= 5.24 x 10^7 J/K

Entorpy (empty soda can) ~= 14.68 J/K

http://www.uncommondescent.com/computer-science/a-designed-objects-entropy-must-increase-for-its-design-complexity-to-increase-part-2/#comment-432383

You’d think if KF is a physicist he’d be able to the calculations I did! Instead he just did his usual word vomiting.

I then told him this after he just kept vomiting out obfuscations:

I gave up in disgust. Then I engaged him again and under threat of ban I left in disgust again.

I posed this rather simple math to Mr. Kairos “physicist” Focus:

http://theskepticalzone.com/wp/2lot-and-id-entropy-calculations-editorial-corrections-welcome/comment-page-1/#comment-59813

No credible rebuttal, just word vomiting. He couldn’t even do simple math and come to terms with the results.

He resorts to a lot of theater and obfuscation. I disagree with a lot of people here at TSZ, but at least they shoot straighter than KF.

Choo Choo math again! Are we rerunning the classics? Can we have “need stars to stop the earth falling into the sun”, “pyramid power” or “H2O2”?

I just read some of the post from 2013 by keiths trying to explaining to virgil, who was JoeG then, about infinite sets. I am now relieved from feeling that I need to write one word to Virgil about this. It really is astounding how some people can not know, and be so oblivious to their not-knowing.

Joe Gallien is in to “rescue” KF at UD!

Aleta,

Astounding — and entertaining.

This is hilarious. Sort of sad and hilarious at a time.

I tried to gently set JoeG straight here becaue I knew TSZ would be merciless. You can’t say I didn’t try to show the guy the light and keep him from skewering himself:

http://www.uncommondescent.com/intelligent-design/siding-with-mathgrrl-on-a-point-and-offering-an-alternative-to-csi-v2-0/#comment-455378

I even gave him a fine point on why the discussion was important:

http://www.uncommondescent.com/intelligent-design/siding-with-mathgrrl-on-a-point-and-offering-an-alternative-to-csi-v2-0/#comment-455256

Keiths picked up on the discussion and hammered him.

I actually didn’t look at what Keiths and others said in detail. I realized JoeG’s decimation would be too gruesome to look at and that he would be the butt of jokes forever after.

You can’t say I didn’t try to help him avert disaster. He baked his own cake, now he as to eat it.

Sal,

Along with its CSI.

Rather like you coming in to rescue keiths.

I never got the gentle treatment from Salvador. He just modified my posts or deleted them.

BWAAAHAHAHAHAHAHA,

That’s Edgar Postrado level of retardness

Can you give *one* example of that happening?

And I think that was a shitty thing to do (and have openly said so).

Mung:

Rich:

Yes. It’s shitty when Sal does it, and it’s shitty when Barry does it.

Don’t you agree, Mung? How’s your “integrity thingy” working today?

keiths,Nonsense. Obviously you do not grasp infinity.

Two counters, both representing countable infinite sets- one counts every second and the other counts every other second. The first counter will always have a higher count than the second- always and forever. That also means the first set has more elements than the second at any and every point in time.keiths cannot deal with that one. And that is OK with meJust cuz keiths can say it is wrong doesn’t make it. Again keiths doesn’t understand that infinity is a journey. That isn’t my fault.

keiths,One set will always have more elements than the other- always and forever. That you fail to grasp that is your problem, not mine.

keiths,Infinity is a journey and as such cannot be taken as a whole.

Here is another question not one of my detractors can’t answer:

If Cantor was wrong about countable infinite sets having the same cardinality, what would be affected or would there be no effect at all?

Can’t handle simple examples, eh, keiths?

All keiths has done is to repeat what I am refuting as if it refutes what I am saying. Clueless ’til the end…

keiths quoting keiths quoting keiths. I think keiths has a good grasp on the concept of infinity.

Why reinvent the wheel, when I’ve already answered Frankenjoe’s objections?

Frankie:

I explained this to you in 2013:

keiths,You were wrong in 2013:

Nope, keep fishing and attacking a strawman. And if a set never stops growing then how is it finite? You still cannot grasp that infinity is a journey.

keiths uses his already handy explanations forgetting that they have already been dispensed with.

In what sense is it a set if it keeps having members added to it?

These excerpts give a feel for how the conversation is going over at UD:

daveS:

daveS:

daveS:

daveS:

daveS:

Aleta:

Aleta:

Aleta:

hrun0815:

Aleta:

daveS:

daveS:

Collections can be added to, just not for infinity.