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. KF is on the verge (finally!) of seeing his error:

    And I do not know how to be clearer and more specific than that, in stating that I have come to view endlessness as a pivotal part of the definition of the naturals.

    In that context the most I think we can reasonably say is that every counting set or natural number reached in successive +1 steps from 0 as actually taken will be finite, and similarly, every represented number such as place value or scientific notation that depends on such will be finite and will be succeeded by an onward endless succession that can be placed in 1:1 correspondence with the naturals from 0.

    Now he just needs to realize that there are no natural numbers other than the ones reachable by successive +1 steps starting from 0.

    The natural numbers are endless — there are infinitely many of them — yet each one is finite.

  2. keiths,

    Because for every element in the set, you remove that element and replace it with a new element.

    Nonsense. The 1 that replaced the 0 already existed. You didn’t add a new element.

    But thank you proving that you can’t even follow your own posts

  3. keiths: To spell it out: If instead of changing n to n+1, you insist on removing n and inserting n+1, JoeMath will still fail. Why? Because for every element in the set, you remove that element and replace it with a new element. It’s a one-for-one substitution, so the cardinality does not change.

    Again, getting technical, you have to be careful with this.

    If you replace 0 with 1, then you now have the set {1,2,3} which has a different cardinality. You would have to work in reverse order — first replace the 3 with 4 — in order for that argument to work.

  4. Neil,

    Again, getting technical, you have to be careful with this.

    If you replace 0 with 1, then you now have the set {1,2,3} which has a different cardinality. You would have to work in reverse order — first replace the 3 with 4 — in order for that argument to work.

    No, because the substitutions are done simultaneously.

  5. keiths:

    4. Take the set {0,1,2,3,…}. It has a cardinality. Add 1 to each of its elements without adding or removing any elements. You’ll obtain {1,2,3,4,…}.

    cupcake:

    No. If you step through from left to right, at some point in the process you have two ‘1’.

    LoL! Yeah like when you travel through infinity and end up back at the beginning. BWAAAAAAAHAHAHAHAHAHAHAHAHAHAHA

    Perhaps Richie is ignored by everyone and that is why no one has corrected him.

  6. Frankie: Perhaps Richie is ignored by everyone and that is why no one has corrected him.

    How’s that working out for you at UD?

  7. Neil,

    Then what you are really doing, is replacing the 0 with a 4.

    No, I’m replacing

    a) the 0 with a 1,
    b) the 1 with a 2,
    c) the 2 with a 3,

    …and so on.

  8. keiths:
    Neil,

    No, I’m replacing

    a) the 0 with a 1,
    b) the 1 with a 2,
    c) the 2 with a 3,

    …and so on.

    Actually you are just removing the 0. Everything else remained the same.

  9. Neil Rickert: Then what you are really doing, is replacing the 0 with a 4.

    keiths: No, I’m replacing

    a) the 0 with a 1,
    b) the 1 with a 2,
    c) the 2 with a 3,

    …and so on.

    Then you are not replacing them simultaneously.

    Initially, the set is {0,1,2,3}

    Replace “the 0 with a 1”.

    The set is now {1,2,3}

    Replace “the 1 with a 2”.

    The set is now {2,3}

    Replace “the 2 with a 3”.

    The set is now {3}.

    If you did it simultaneously, you started with {0,1,2,3}, and instanteously changed that to {1,2,3,4} which is just a change of the 0 to a 4 (the order is not relevant).

  10. Neil Rickert:
    Then you are not replacing them simultaneously.

    Initially, the set is {0,1,2,3}

    Replace “the 0 with a 1”.

    The set is now {1,2,3}

    Replace “the 1 with a 2”.

    The set is now {2,3}

    Replace “the 2 with a 3”.

    The set is now {3}.

    If you did it simultaneously, you started with {0,1,2,3}, and instanteously changed that to {1,2,3,4} which is just a change of the 0 to a 4 (the order is not relevant).

    Neil, this was the original claim:

    4. Take the set {0,1,2,3,…}. It has a cardinality. Add 1 to each of its elements without adding or removing any elements. You’ll obtain {1,2,3,4,…}. Has the cardinality changed?

  11. Frankie: Has the cardinality changed?

    The cardinality hasn’t changed. You are wrong about that. However, there are some technical problems with the argument that keiths was using.

  12. Neil Rickert: The cardinality hasn’t changed.You are wrong about that.However, there are some technical problems with the argument that keiths was using.

    Set subtraction says that the cardinality has changed. But yes there are technical problems with his argument

  13. Neil Rickert: Set subtraction shows that the set has changed.It does not follow that the cardinality has changed.

    Of course it shows that the cardinality has changed. With set subtraction the 0, which is an element, is left. That means one set has one more element than the other.

  14. Neil,

    If you did it simultaneously, you started with {0,1,2,3}, and instanteously changed that to {1,2,3,4} which is just a change of the 0 to a 4 (the order is not relevant).

    First of all, you were the one who insisted that elements cannot change, so your reference to “a change of the 0 to a 4” is nonsensical by your own (pedantic) standard.

    Second, the process of replacing each n in the set with n+1 is not the same as the process of replacing 0 with 4.

    This is trivial to demonstrate. Take the set {0,1,2}. Replace each n in the set with n+1, and the result is {1,2,3}. Now start with the original set again, and replace the 0 with 4. You get {1,2,4}. Two different processes, two different results: {1,2,3} vs {1,2,4}.

    The fact that the two processes give the same result in one particular case — when the set being operated on is {0,1,2,3} — does not mean that the processes are identical. It’s as silly as arguing that 2x and x^2 are the same function, because they give the same answer when x=2.

  15. Neil,

    You’re making the same mistake as Frankie, which ought to concern you. Like you, Frankie is assuming that if the results are the same, the process must be the same. Regarding my conversion of the set {0,1,2,3…} to {1,2,3,4…} by replacing every n with n+1, he writes:

    Actually you are just removing the 0. Everything else remained the same.

    He’s wrong for the same reason that you are wrong. The fact that the results are the same in one particular case does not mean that the processes are the same.

    Again, this is trivial to demonstrate. Take the set {0,2,4,6…} and replace each n in the set with n+1. The result is {1,3,5,7…}. Now take the original set and remove the 0, as Frankie suggests. The result is {2,4,6,8…}. Two different processes, two different results.

  16. keiths,

    The mistakes are all yours, keiths

    Like you, Frankie is assuming that if the results are the same, the process must be the same.

    That is your bullshit opinion and you can’t support it

    Actually you are just removing the 0. Everything else remained the same.

    That happens to be true.

    Now take the original set and remove the 0, as Frankie suggests.

    That is what you did. I didn’t have to suggest it

  17. keiths: First of all, you were the one who insisted that elements cannot change, so your reference to “a change of the 0 to a 4” is nonsensical by your own (pedantic) standard.

    Technical, not pedantic.

    We can change a 0 to a 4 in the set by removing the 0 and replacing it with the 4. It is, of course, a different set after that replacement.

    You can change the numeral “1” to the numeral “4” without removal or replacement, because numerals are just marks (material representations). But you cannot change the number 1 to the number 4 without removal and replacement. But if the number 4 is already in the set, you cannot add it to the set as a duplicate, since it is already there. There can be many numerals “4” but there is only one number 4.

    I am not being pedantic about change vs. replace. Rather, I am being technical about numbers being singular objects, rather than representations or marks.

    This may be technical. But the way that we talk about numbers requires it.

  18. keiths: You’re making the same mistake as Frankie, which ought to concern you. Like you, Frankie is assuming that if the results are the same, the process must be the same. Regarding my conversion of the set {0,1,2,3…} to {1,2,3,4…} by replacing every n with n+1, he writes

    No, I’m not making a mistake.

    Numbers are not numerals or marks. Sets are not buckets (or containers).

    The language of changing a 0 to a 1 might be good as an intuition pump, but it is not technically correct.

  19. Neil:

    No, I’m not making a mistake.

    Sure you are, and I’ve explained exactly what your error is:

    Second, the process of replacing each n in the set with n+1 is not the same as the process of replacing 0 with 4.

    This is trivial to demonstrate. Take the set {0,1,2}. Replace each n in the set with n+1, and the result is {1,2,3}. Now start with the original set again, and replace the 0 with 4. You get {1,2,4}. Two different processes, two different results: {1,2,3} vs {1,2,4}.

    The fact that the two processes give the same result in one particular case — when the set being operated on is {0,1,2,3} — does not mean that the processes are identical. It’s as silly as arguing that 2x and x^2 are the same function, because they give the same answer when x=2.

  20. keiths: Second, the process of replacing each n in the set with n+1 is not the same as the process of replacing 0 with 4.

    Which axioms of ZF (or ZFC) license the process that you claim to be using?

  21. Neil,

    Which axioms of ZF (or ZFC) license the process that you claim to be using?

    Heh. Nice try at changing the subject, but I’m not biting. Your question is irrelevant. You already asserted that my process — replacing n with n+1 for each n in a set — is the same as your process — replacing 0 with 4:

    If you did it simultaneously, you started with {0,1,2,3}, and instanteously changed that to {1,2,3,4} which is just a change of the 0 to a 4 (the order is not relevant).

    I’ve shown that your claim is incorrect:

    Second, the process of replacing each n in the set with n+1 is not the same as the process of replacing 0 with 4.

    This is trivial to demonstrate. Take the set {0,1,2}. Replace each n in the set with n+1, and the result is {1,2,3}. Now start with the original set again, and replace the 0 with 4. You get {1,2,4}. Two different processes, two different results: {1,2,3} vs {1,2,4}.

    The fact that the two processes give the same result in one particular case — when the set being operated on is {0,1,2,3} — does not mean that the processes are identical. It’s as silly as arguing that 2x and x^2 are the same function, because they give the same answer when x=2.

    Do you have a counterargument to offer?

  22. keiths: This is trivial to demonstrate. Take the set {0,1,2}. Replace each n in the set with n+1, and the result is {1,2,3}.

    Sigh.

    Start with {0,1,2}

    Replace 0 with 1. That gives {1,1,2}, which is just a badly described {1,2}.

    Replace 1 with 2. Now you have {2}

    Replace 2 with 3. Now you have {3}.

    Yes, I understand that you fail to get this. But that’s your misunderstanding.

    You can say “do it all simultaneously” and then ask me to look at the process. But if it all happens simultaneously, there isn’t a process. There’s only a final result.

    You are treating this as if you have billiard balls in a barrel, with numerals painted on them. But it doesn’t work that way with actual numbers.

    Yes, I understand that you were using an informal intuitive description. That’s fine. I don’t object to that. However, Frankie rejected your intuitive description and made a technicallly correct objection. You said that he was wrong. I was correcting the record on that.

  23. Neil,

    You can say “do it all simultaneously” and then ask me to look at the process.

    I’ve specified that the steps happen simultaneously, yet you bizarrely continue to take them as happening sequentially. Why?

    But if it all happens simultaneously, there isn’t a process. There’s only a final result.

    Good grief, Neil. The fact that the steps happen at the same time doesn’t mean that they don’t happen. Where do you think the result comes from?

    Replacing n with n+1 for all n in a set is a distinct process from replacing 0 with 4. You got confused because the two processes have the same result when the set in question happens to be {0,1,2,3}. You weren’t thinking like a mathematician, so you failed to notice that the two processes gave different results for other starting sets such as {0,1,2}.

    Yes, it’s an embarrassing mistake, but why prolong your embarrassment? Why not just acknowledge the mistake, resolve to think more carefully next time, and move on?

  24. Neil,

    You are treating this as if you have billiard balls in a barrel, with numerals painted on them. But it doesn’t work that way with actual numbers.

    The process of replacing each number n in a set with n+1 does not depend on “treating this as if you have billiard balls in a barrel, with numerals painted on them.”

  25. At UD, daveS sums up KF’s position:

    The successor function (on N) takes as input a finite set and outputs another finite set. It’s impossible for it to output an infinite set.

    You are saying something like “if we roll a 6-sided die infinitely many times, eventually a 7 will come up”.

  26. Meanwhile, Virgil is in trouble again:

    968

    News March 14, 2016 at 6:47 am

    News (as mod): Virgil Cain, we have received a complaint about some of your comments, particularly 837

    We appreciate your efforts to defend reason, evidence, and reality against the universal acid, but one must take care not to be contaminated oneself. We really must insist that you apologize for the things said there.

    UD is a family site in that teenagers and U students may be reading and participating. We try to model how mature adults debate a topic and welcome all efforts in that regard.

    I will watch this space.

    Virgil notpologizes:

    OK I apologize for calling people liars and cowards even though the posts they made contained lies, false accusations, nonsensical phrases and unsupported claims.

    However I still do not understand why the messenger is being picked on and not the people who posted the lies, false accusations, nonsensical phrases and unsupported claims.

    Every time KF is attacked he attacks back. Every time Barry is attacked he attacks back. Every time News is attacked News hits back.

    How many times does a person have to be attacked before he can defend himself with the truth? Why is it that people are allowed to lie but others are not allowed to point out their lies?

    I apologize for my actions but if I was debating these people face-to-face I would have dropped them like a bad habit. And they would have to eat from a straw for months. [Emphasis added]

    So I don’t understand why it is OK to provoke people into the type of posts I made but bad to respond to the diatribe and lies posted about me.

    Self-control is a far-off dream for Virgil.

  27. keiths: Good grief, Neil. The fact that the steps happen at the same time doesn’t mean that they don’t happen. Where do you think the result comes from?

    Good grief yourself.

    You seem to be thinking of this as if you had a physical barrel of billiard balls that you can change. But mathematics is abstract. Unless the axioms give an account of changing, then what you have is one set replaced by another set. There isn’t a process unless a process is defined. Loose talk of changing simultaneously is not a definition.

    Replacing n with n+1 for all n in a set is a distinct process from replacing 0 with 4.

    Unless you can define “replace” in accordance with the axioms, then the talk of replacement is just extra-axiomatic talk and of zero relevance. What you have is a different set than you started with. The process of replacement might be going on in your mind, but is not available to anybody who doesn’t read your mind. It’s made up bullshit.

    Yes, it’s an embarrassing mistake, but why prolong your embarrassment?

    It is your embarrassment, not mine.

    Why not just acknowledge the mistake, resolve to think more carefully next time, and move on?

    I fully acknowledge that keiths has made a fundamental mistake here. Moreover, I acknowledge that keiths doesn’t understand his mistake and will probably continue to make a big fuss about it.

    I’m just walking away, as I usually do when keiths gets onto the wrong side of an argument.

  28. Neil,

    Okay, I see you’ve chosen to prolong your embarrassment. As you wish.

    You seem to be thinking of this as if you had a physical barrel of billiard balls that you can change.

    No, as I’ve already explained:

    The process of replacing each number n in a set with n+1 does not depend on “treating this as if you have billiard balls in a barrel, with numerals painted on them.”

    Neil:

    But mathematics is abstract. Unless the axioms give an account of changing, then what you have is one set replaced by another set.

    Whether you take 0 as being changed to 1 within a set, replaced by 1 within the set, or mapping to 1 in a new set makes no difference for the purposes of this discussion. Success in abstract thinking requires an understanding of which details matter and which can be abstracted away. What matters is that you start with the set {0,1,2,3}, you apply the specified process, and you end up with {1,2,3,4}.

    There isn’t a process unless a process is defined.

    I defined the process. For each n in the set, you change it to (or equivalently, replace it with or map it to) n+1. That’s different from changing 0 to 4 (or equivalently, replacing 0 with 4 or mapping 0 to 4), as I demonstrated.

    You made a silly mistake. As I said earlier, it’s like concluding that 2x and x^2 are the same function because they have the same value when x=2.

    Unless you can define “replace” in accordance with the axioms, then the talk of replacement is just extra-axiomatic talk and of zero relevance. What you have is a different set than you started with. The process of replacement might be going on in your mind, but is not available to anybody who doesn’t read your mind. It’s made up bullshit.

    You were the one who introduced replacement into the conversation:

    1 is a different number from 0. So changing 0 to 1 entails removing 0 and inserting 1.

    And you have been discussing replacement throughout the thread. For example:

    If you replace 0 with 1, then you now have the set {1,2,3} which has a different cardinality. You would have to work in reverse order — first replace the 3 with 4 — in order for that argument to work.

    By your own standard, then, you have been spouting “made-up bullshit”.

    How does that help your case?

    You’re tripping over your own shoelaces, Neil.

  29. LOL.

    Virgil:

    You are unable to grasp anything that I have said so far. And I am sure that what you ask can be done as nothing prevents it. Even YOU could do what you ask of me given everything I have told you. Well, that is if you were half the mathematician you want everyone to believe you are.

    But if you want to continue to make this personal I suggest we get off the internet, face each other and get it over with.

    [Emphasis added]

    ellazimm patiently explains Virgil’s error:

    Beating people up doesn’t establish mathematical ideas.

  30. keiths,

    Except Virgil wasn’t talking about mathematical ideas at that time. He was talking about ellazimm’s continued lies about the utility of the concept being debated. All ellazimm has done is bluff and lie. But I understand why that would appeal to you.

  31. keiths,

    LoL! The people who spew lies and bluffs have lost self-control. But I can understand why you would agree with that tactic and disagree with anyone exposing it.

  32. KF:

    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.

    daveS:

    I say there is no chance of proving this statement. You have a function which accepts finite inputs and can produce only finite outputs. How would it ever generate an infinite output?

    Will it finally sink in?

  33. daveS, to KF:

    Anyway, I agree this is not very interesting. I really would like you to prove this:

    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.

    In other words, prove that the smallest set S such that:

    1) {} ∈ S

    2) If n ∈ S, then n ∪ {n} ∈ S

    contains an infinite element m. That is, m is in 1-1 correspondence with a proper subset of itself.

    Once again, I’m asking for a proof.

    It’s remarkable that Mr. “Principles of Right Reason” doesn’t see the need to justify his intuitions.

  34. A commenter at AtBC brings up an interesting problem for Virgil’s cardinality “theory”, although Virgil is banned now at UD so I won’t post this there. But this is nice.

    1. Virgil argues as follows: Let A = the naturals, B = the evens, and C = the odds. Then, since A – B = C, this proves that B has fewer elements than A – presumably 1/2 as many.

    2. But, on the other hand, start with A = {1,2,3,…} and multiply each element by 2 to get set D = {2,4,6,…}. Set D has the same number of elements as A because all we did was double each element. However D is just all the evens, so this proves the evens have the same number of elements as N

    Therefore, arguments 1 and 2 contradict each other, one saying that the cardinality of E is less than N and one saying the cardinality of E is the same as N. Both can’t be right, so there must be a flaw in one of the two arguments.

    Wonder what Virgil would say to this?

  35. aleta,

    Wonder what Virgil would say to this?

    That’s the homework I gave to Joe/Virgil/Frankie back in 2013:

    Some exercises for math genius Joe G:

    1. Take any set A of integers. A has a cardinality. Leave A alone for five minutes without adding or removing any elements. Has A’s cardinality changed?

    2. Take any set B of integers. B has a cardinality. Add 1 to each of the elements of B without adding or removing any elements. Has B’s cardinality changed?

    3. Take any set C of integers. C has a cardinality. Multiply each element of C by 17 without adding or removing any elements. Has C’s cardinality changed?

    4. Take the set {0,1,2,3,…}. It has a cardinality. Add 1 to each of its elements without adding or removing any elements. You’ll obtain {1,2,3,4,…}. Has the cardinality changed?

    5. Take the set {0,1,2,3,…}. It has a cardinality. Multiply each element by 17 without adding or removing any elements. You’ll obtain {0,17,34,51,…}. Has the cardinality changed?

    6. Flounder about uselessly, trying to explain why your method says that the cardinality changes in scenarios 4 and 5.

    Also:

    Joe,

    After you’ve told us where the missing element went, here’s another question for you.

    You claim that {1,2,3,…} is twice as big as {2,4,6,…}. If that were true, then it would be impossible to map {1,2,3,…} to {2,4,6,…} using F(n) = 2n, because we would exhaust the elements of {2,4,6,…} before all of the elements in {1,2,3,…} had been mapped.

    Which elements of {1,2,3,…} are left over when we attempt this “impossible” mapping, Joe?

    Recommence flailing.

  36. aleta,

    2. But, on the other hand, start with A = {1,2,3,…} and multiply each element by 2 to get set D = {2,4,6,…}. Set D has the same number of elements as A because all we did was double each element.

    That doesn’t follow at all. By doubling each element you have also removed elements.

  37. keiths,

    LoL! We have already addressed your points, keiths. Merely repeating what we have refuted is a sign of desperation. In #4 you did remove the 0

    You claim that {1,2,3,…} is twice as big as {2,4,6,…}. If that were true, then it would be impossible to map {1,2,3,…} to {2,4,6,…} using F(n) = 2n,

    That doesn’t follow at all. You just make shit up and post it as fact. Mapping converts one set into the other. It just shows the sets are countable.

    Pathetic

  38. No you haven’t, Frankie. 1 became 2, 2 became 4, 3 became 6. You haven’t removed elements, you’ve changed the value of the elements.

    Let’s make it simple: let A = {1,2,3,… 1 billion} and double each element to get B = {2,4,6,… 2 billion}. Set A has 1 billion elements and set B has 1 billion elements. The number of elements didn’t change.

  39. aleta:
    No you haven’t, Frankie. 1 became 2, 2 became 4, 3 became 6. You haven’t removed elements, you’ve changed the value of the elements.

    Let’s make it simple: let A = {1,2,3,… 1 billion} and double each element to get B = {2,4,6,… 2 billion}. Set A has 1 billion elements and set B has 1 billion elements. The number of elements didn’t change.

    Of course elements were removed as they are no longer there. All of the odds have been removed. And all the evens already existed. And yes finite sets are not the same as infinite sets, are they?

  40. Frankie, to Aleta:

    By doubling each element you have also removed elements.

    I explained this to you three years ago, “Frankie”:

    In desperation, you are now making this argument:

    Take set {0,1,2,3,…} and add 1 to each element.

    That would mean that I have to remove the 0, yet keiths sez without adding or removing any elements.

    No, Joe, you don’t have to remove the 0 in order to add 1 to it. And even if you did, it wouldn’t help your case at all. You could still convert {0,1,2,3,…} to {1,2,3,4,…} by removing one element at a time, operating on it, and adding it to a result set. It would work like this:

    1. Start with A = {0,1,2,3,…} and B = {}.
    2. Remove an element from A.
    3. Add 1 to the element.
    4. Add the element to B.
    5. Repeat steps 2-4 for each remaining element of A.

    Here’s how the sets would evolve, iteration by iteration:

    A = {0,1,2,3,…} B = {}
    A = {1,2,3,4,…} B = {1}
    A = {2,3,4,5,…} B = {1,2}
    A = {3,4,5,6,…} B = {1,2,3}
    …and so on.

    After you’ve done this for all the elements of set A, you’ll have:
    A = {} B = {1,2,3,4,…}

    At each iteration, you removed exactly one element from A and added exactly one element to B. Yet according to “Joe math”, the new set B is one element smaller than set A was.

    Where did the missing element go, Joe? Did the Designer poof it out of existence? Can you identify the step where it went missing?

  41. keiths,

    No, Joe, you don’t have to remove the 0 in order to add 1 to it.

    LoL! Once you add 1 to the 0 the 0 is gone.

    You could still convert {0,1,2,3,…} to {1,2,3,4,…}

    By removing the 0 from the first set

    After you’ve done this for all the elements of set A

    Lol! It is impossible to remove all of the elements from an infinite set

  42. The result is even more amusing in this case:

    5. Take the set {0,1,2,3,…}. It has a cardinality. Multiply each element by 17 without adding or removing any elements. You’ll obtain {0,17,34,51,…}. Has the cardinality changed?

    JoeMath says that the cardinality of the result is only 1/17 of the cardinality of the original set.

    Where did all of those elements go, Joe?

  43. keiths,

    5. Take the set {0,1,2,3,…}. It has a cardinality. Multiply each element by 17 without adding or removing any elements. You’ll obtain {0,17,34,51,…}. Has the cardinality changed?

    You are removing elements, again

    Where did all of those elements go, Joe?

    Who cares? They are definitely no longer in the set

  44. keiths thinks that you can remove all of the elements in an infinite set. How can anyone take him seriously when it comes to math and infinity?

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