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:

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?

387 thoughts on “KF tackles the transfinite

  1. I posted this at reddit:

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

    If anyone can clean up my proof or provide a better proof, I welcome it….

    Let n = 1, then n is finite since there exist a number which is greater than n, namely n+1.

    By induction we show all natural numbers are finite since there exists for every n, a natural number greater than n, namely n + 1.

    For there to be a natural number n+1 for every n in the naturals implies the set of naturals is infinite.

    Thanks in advance for any feedback.

  2. Sal:

    Let S be the set of natural numbers less than or equal to a a natural number n. The set of those natural numbers can be placed into a one-to-one correspondence with the set of those natural numbers less than the natural number n+1. Hence n is finite.

    We show this for 1, and then 1+1, by induction we show this for all natural numbers, hence all natural numbers are finite, and this is possible because the set of natural numbers is infinite…

    Agreed, we start with n = 1.

    We show n+1 is finite because it is bounded from above by ( ( n+1) + 1).

    Rinse and repeat.

    Sal,

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

    1. S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.

    …then you don’t need the induction, because every natural number is already defined as finite. The induction adds nothing.

    (Remember that in set-theoretic terms, a natural number n is defined as the set {0, …, n-1}, with 0 being defined as the empty set.)

    If you do want to use induction, based on these premises…

    0 is finite.

    If n is finite, then n+1 is finite.

    …then the induction proceeds upwards, and you show that n+1 is finite only after showing 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.

    Paradox resolved.

    What paradox?

  3. …then you don’t need the induction, because every natural number is already defined as finite.

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

  4. 0 is finite.

    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?

  5. stcordova: …what do the professionals think?

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

  6. Sal,

    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?

    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:

    0 is finite.

    If n is finite, then n+1 is finite.

    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”.

  7. What do you think, Frankie? Is every natural number finite, despite the fact that there are infinitely many of them?

  8. Keiths:

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

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

  9. keiths:
    What do you think, Frankie?Is every natural number finite, despite the fact that there are infinitely many of them?

    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.

  10. keiths:

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

    Sal:

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

    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.

  11. keiths:

    What do you think, Frankie? Is every natural number finite, despite the fact that there are infinitely many of them?

    Virgil:

    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.

    It’s okay, Virgil. Just put the pencil down and tell me what you think the answer is — not what it “should be”.

  12. 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.

  13. Frankie: OK, so you are also too stupid to understand my answer. Figures.

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

  14. 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

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

    And you have to take into account the nonsense I am responding to

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

  16. Frankie:
    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

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

  17. Frankie,

    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

    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,

    AND on the journey down the number line a train picking up all non-negative integers will always have twice as many numbers as a train picking up all positive even numbers (going down the same number line). And that is always and forever-> meaning for the entire journey.

    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.

  18. Aleta:

    Well I’ll be darned – I agree with Sal

    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:

    So are my numbers correct given P, T, etc.?

    I’m not asking about CSI, IC, FSCO/I, IDOW, SFOD-D, MmIG, WMDs, MIGs, BUFFs, AWACS, VLSI, DicNavAb, etc.

    I’m asking about the standard state entropy of the Aluminum content of:

    A. 747
    B. Broken 747
    C. Soda can

    A simple yes or no, would be helpful to everyone. You’ve been very verbose, and I’m not asking you to print more than 3 characters for a response of “yes”, 2 characters for a response of “no”, and 12 charcters to say “I don’t know”.

    You don’t have to print a dissertation that doesn’t answer the question I pose.

    If you don’t want to answer the question, say so. “I don’t want to answer the question. I want to talk about something else.” (that would be 73 characters for a response). I’ll accept that but please offer me the courtesy of saying that you prefer to talk about something else rather than answering a question I’ve posed more than a few times in this discussion.

    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

    A warm living human has substantially more thermodynamic entropy than a lifeless ice cube. This can be demonstrated by taking the standard molar entropies of water and ice and estimating the entropy of water in a warm living human vs entropy of water in a lifeless ice cube.

    http://en.wikipedia.org/wiki/Water_(data_page)

    Std Molar Entropy liquid water: 69.95 J/mol/K
    Std Molar Entropy ice: 41 J/mol/K

    A human has more liquid water, say 30 liters, than an ice cube (12 milliliters).

    Let S_humum be the entropy of a human, and S_ice_cube the entropy of an ice cube.

    Order of magnitude entropy numbers:

    S_human > 30 liters * 55.6 mol/liter * 69.95 J/K = 116,677 J/K

    S_ice_cube ~= 0.012 liters * 55.6 mol/liter * 41 J/K = 27 J/K approximately (ice is a little less dense than liquid water, but this is inconsequential for the question at hand).

    Thus warm living human has more entropy than a lifeless cube of ice.

    So why do creationists worry about entropy increasing in the universe as precluding evolution? Given that a warm living human has more entropy than an ice cube, then it would seem there are lots of cases where MORE entropy is beneficial.

    Ergo, the 2nd law does not preclude evolution. Other lines of reasoning should be used by ID proponents to criticize evolution, not the 2nd law.

    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.

  19. 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”?

  20. 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.

  21. Aleta,

    It really is astounding how some people can not know, and be so oblivious to their not-knowing.

    Astounding — and entertaining.

  22. 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.

  23. stcordova: I tried to gently set JoeG straight here becaue I knew TSZ would be merciless.

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

  24. Mung: Rather like you coming in to rescue keiths.

    Can you give *one* example of that happening?

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

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

  25. Mung:

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

    Rich:

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

    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?

  26. keiths,

    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.

    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 me

    Just 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.

  27. keiths,

    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}.

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

  28. keiths,

    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?

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

  29. 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?

  30. aleta:
    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.

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

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

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

    Frankie:

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

    I explained this to you in 2013:

    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.

  33. keiths,

    You were wrong in 2013:

    You seem to think that finite sets, if they are growing and will never stop growing, are already infinite

    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.

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

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

    daveS:

    An infinite Turing Machine tape with a single end is a good model for the HH, with each cell representing a single room, and the “last” cell representing the front desk.

    Given such a tape, how many cells are infinitely many steps from the last cell? None, right?

    daveS:

    Now, can you answer my simple yes/no question: Are any cells on the infinite Turing Machine Tape infinitely far from the end?

    You can explain all you want, but I’m requesting that you first respond with either “Yes” or “No”.

    daveS:

    Ok let’s be very clear. Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    This is a yes/no question.

    daveS:

    So again, are there any cells in this tape infinitely many steps from cell 0?

    My question (for the third time) is:

    Ok let’s be very clear. Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    This is a yes/no question.

    daveS:

    Gotta run now, but is that a “yes”?

    Aleta:

    We know all this. What is your answer to dave’s question?

    Aleta:

    No, you haven’t “given and emphatically underscored” an answer. You’ve repeated your concerns, but I can’t tell whether your answer is yes or no…

    Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    Yes or No

    Aleta:

    Then your answer is No: there are no cells which are infinitely far from the end.

    Is this correct?

    hrun0815:

    Must. Not. Ever. Answer. Simple. Question!

    Aleta:

    So your answer is “Yes”???

    Or is it “No”

    Please, answering the question with a paragraph with lots of rhetorical questions isn’t useful.

    Just answer, with one word, Yes or No.

    Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    daveS:

    I have yet to see a yes-no answer to my question. My best guess is that now you are saying that the thing I’ve been calling an “infinite Turing Machine tape” is not actually infinite??

    daveS:

    When Aleta and I ask for yes/no answers, that’s exactly what we mean. We would like you to literally type “Y-E-S” or “N-O” in your reply, because we have such a hard time figuring out what you are saying.

    At this point, I’m still not clear what your position is. If you would actually answer yes or no, then it would save us a lot of guesswork.

  36. Mung: In what sense is it a set if it keeps having members added to it?

    Collections can be added to, just not for infinity.

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