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  1. #1
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    WLC's Argument Against an Actual Infinity

    I. Introduction

    The modern philosopher and prolific debater William Lane Craig is well-known for renewing and updating an old argument for the existence of God, called the Kalaam Cosmological Argument. One premise of the KCA, as amended by WLC, is the following:

    (INF) An actual infinity is impossible.

    To support (INF), WLC usually offers a few supporting reasons. Chief among these are the following:

    (1) Referring to Hilbert's famous thought experiment of the Grand Hotel, which involves a hypothetical hotel with a countably infinite number of rooms. Craig describes various resulting properties of the Hotel as "absurd" or "against reason" or "contradictory". The chief complaint is against the following counter-intuitive result: the Hotel can simultaneously have every room occupied--but can accommodate additional guests.


    (2) Referring to transfinite arithmetic, Craig states that when we consider "infinity minus infinity", we arrive at contradiction. For example, the integers without the even numbers leaves the odd numbers, an infinite set, hence "infinity minus infinity" in this case is infinity; the integers without any number greater than 5 or less than 1 leaves only the set {1,2,3,4,5}, a finite set, hence "infinity minus infinity" in this case is 5. Craig argues that these results are contradictory, and that if there was an actual infinity, such operations must be able to be performed, and thus we would arrive at a contradiction.


    These objections are essentially the same: infinite sets result in absurdity or contradiction. The objection in (1) is to the possibility of "adding" a set to an infinite set without increasing the infinite set's cardinality; the objection in (2) is to the inability to unambiguously determine the result of "subtracting" a set from an infinite set.

    Now, WLC does not claim that these examples involve logical contradictions. Rather, he claims that these examples show that "infinity" is necessarily constrained to the realm of imagination or ideas.

    NOTE ON NOTATION: Although technically there are infinitely many "sizes" of infinity, within the context of this post I will use the word "infinity" to refer specifically to aleph-null, the cardinality of the set of natural numbers.

    II. Problems

    I don't see how any of his examples forestall the existence of an actual infinity.


    IIa. Hilbert's Grand Hotel

    That Hilbert's Grand Hotel can always accommodate (countably) more guests may seem counter-intuitive, but this result in and of itself does not entail that an actual Grand Hotel could not exist. There might be other limitations that would prevent the construction of a Grand Hotel; it would require an infinite amount of energy/mass/etc.

    But with regard to the capacity of a Grand Hotel, I don't see how the ability to accommodate additional guests somehow renders an otherwise unobjectionable Grand Hotel contradictory. What rule of reason requires a careful thinker to reject the possibility of an actual Grand Hotel? Other than WLC's (and others') say-so, I don't see one.


    IIb. Arithmetic with Infinity


    Craig argues that because there is no sensible answer to the question, "What is infinity minus infinity?", there cannot be actual infinities. Here is where more rigorous formalization is needed.

    When we talk about "infinity minus infinity", what we're actually talking about are set complements. Suppose that S is a subset of U. This means that every element in S is also in U. Then the complement of S in U, written U\S, is defined to be the set of elements in U but not in S. If we consider the set N = {0, 1, 2, ..., k, k+1, ...}, then N\{0} = {1,2, ..., k, k+1, ...}. Call this set A; thus, A = N\{0} = {1, 2, ..., k, k+1, ...}. Then N\A = {0}.

    So what is meant by "X minus Y"? Roughly speaking, it means "how many things are left over when you take away X many things from Y?" The labels "X" and "Y" refer to what are called cardinal numbers. Intuitively, cardinal numbers answer the question "how many?" The cardinality of a set S, written |S|, refers to the "number" of elements contained in S. In the example above, |N| = |A| = infinity, while |{0}| = |{1}| = ... = |{k}| = |{k+1}| = 1, etc.

    In order to answer the question, "What is X minus Y?", first you find sets S, T such that: (1) |S| = X; (2) |T| = Y. Then X - Y is defined to be |S/T|. In order for this operation to be well-defined, your choice of S and T should not change the result of |S\T|. That is, if the operation is well-defined, then no matter which sets you use as S and T, the result |S/T| must be the same so long as |S| = X and |T| = Y.

    This definition of "subtraction"--the cardinality of a set complement--is how Craig derives what he calls 'contradictions' or 'absurdities'. When Craig gives examples of what infinity "minus" infinity equals, he is using precisely this definition. For example, using the notation above, N - 2N = N\{0, 2, 4, ..., 2k, 2(k+1), ...} = {1, 3, 5, ...}, and thus "infinity minus infinity" equals infinity. But since |A| is infinity, "infinity minus infinity" must also equal |N\A| = |{0}| = 1. Thus infinity = 1, a contradiction. This shows that subtraction cannot be well-defined when both sets are infinite.

    What does this mean? It means there is no unique solution to the following equation:

    - = ___

    But what contradiction does this entail? It makes it more difficult to figure out what |S\T| is, if we know that both S and T are infinite; we'd need to know the actual elements of S and T in order to determine |S\T|. But I don't see how the inability to determine |S/T| knowing only |S| and |T| results in absurdity, should an actual infinity exist.

    III. Conclusion

    I see no reason to conclude from Hilbert's Grand Hotel and from the ambiguity of subtraction on infinite numbers that an actual infinity cannot exist. Anyone who thinks such reasons exist, please present them here.
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  3. #2
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    Re: WLC's Argument Against an Actual Infinity

    My primary objection to Craig on this account is his use of moments in time as arithmetic entities like rooms in a hotel.

    Rooms in a hotel are co-existent, one room must necessarily exist with the next room such that one different guest can occupy each.
    But a moment in time, an event, does not co-exist with another. They don't accumulate into a set. All members of a set must co-exist in order to satisfy any of Craig's objections. To add or subtract from a set all members of the set must exist. Events in time cease to exist the moment after they occur, they are no longer part of any set and cannot add to a set or remove from it.

    They are in this sense entirely imaginary. Once a moment in time "passes" it exists only by imagination or facsimile. And of imaginary things you certainly can have an infinite.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Sigfried View Post
    My primary objection to Craig on this account is his use of moments in time as arithmetic entities like rooms in a hotel.

    Rooms in a hotel are co-existent, one room must necessarily exist with the next room such that one different guest can occupy each.
    But a moment in time, an event, does not co-exist with another. They don't accumulate into a set. All members of a set must co-exist in order to satisfy any of Craig's objections. To add or subtract from a set all members of the set must exist. Events in time cease to exist the moment after they occur, they are no longer part of any set and cannot add to a set or remove from it.
    Your requirement that every member of a set must exist contemporaneously with one another is bizarre. In what way to Craig's objections rely on each of the sets he considers having the property that they only contain contemporaneously-existing objects?

    They are in this sense entirely imaginary. Once a moment in time "passes" it exists only by imagination or facsimile. And of imaginary things you certainly can have an infinite.
    This is one theory of time. It is not the only theory of time.
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  6. #4
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    Your requirement that every member of a set must exist contemporaneously with one another is bizarre. In what way to Craig's objections rely on each of the sets he considers having the property that they only contain contemporaneously-existing objects?
    If we didn't have that restriction the grand hotel would only need one room to accommodate infinite guests. There is an implicit assumption in the example that all the rooms and guests are co-existent in time. Whatever the comparison any set has to have a relationship between the set boundaries and the members of the set and with each-other. They also have some quantitative property. Rooms take up space as do people and so forth. Moments don't, only what they describe changing does.

    This is one theory of time. It is not the only theory of time.
    So? Do you have a superior one? Can you provide evidence that yesterday still exists as it did before it became now? We can imagine all manner of things but what counts is what we can demonstrate. We have a wealth of evidence to say that all moments of reality are not demonstrated as co-existing and that instead there is one that changes definition but persists through change.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Sigfried View Post
    If we didn't have that restriction the grand hotel would only need one room to accommodate infinite guests. There is an implicit assumption in the example that all the rooms and guests are co-existent in time. Whatever the comparison any set has to have a relationship between the set boundaries and the members of the set and with each-other. They also have some quantitative property. Rooms take up space as do people and so forth. Moments don't, only what they describe changing does.
    This still doesn't make any sense to me.

    First, the Grand Hotel thought experiment is about various ways of combining infinite sets; whether or not the rooms and people actually exist, and in particular actually exist at the same time is irrelevant.

    Second, a set is defined by its members. Whether or not these members possess "some quantitative property" is irrelevant. The fact that moments don't "take up space" does not somehow disqualify them from being contained in a set.

    So? Do you have a superior one? Can you provide evidence that yesterday still exists as it did before it became now? We can imagine all manner of things but what counts is what we can demonstrate. We have a wealth of evidence to say that all moments of reality are not demonstrated as co-existing and that instead there is one that changes definition but persists through change.
    I am not interested at all in comparing various theories of time. Your criticism depends on your assumption that sets must contain only members that are contemporaneous with one another, which is completely false. Your last sentence doesn't make any sense to me.
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  8. #6
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CS
    But with regard to the capacity of a Grand Hotel, I don't see how the ability to accommodate additional guests somehow renders an otherwise unobjectionable Grand Hotel contradictory. What rule of reason requires a careful thinker to reject the possibility of an actual Grand Hotel? Other than WLC's (and others') say-so, I don't see one.
    As I understand it the Hotel shows how infinite would make an "fully occupied hotel" = "not fully occupied".
    Because what one could do with the math, can not be done with the actual hotel. If all the rooms in the hotel are occupied, then you can not (by definition) add more. Yet the arithmetic says you could add an INFINITE number more.

    It is not simply "counter intuitive" it is illogical, and thus counted as impossible.
    I apologize to anyone waiting on a response from me. I am experiencing a time warp, suddenly their are not enough hours in a day. As soon as I find a replacement part to my flux capacitor regulator, time should resume it's normal flow.

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  10. #7
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    As I understand it the Hotel shows how infinite would make an "fully occupied hotel" = "not fully occupied".
    Because what one could do with the math, can not be done with the actual hotel. If all the rooms in the hotel are occupied, then you can not (by definition) add more. Yet the arithmetic says you could add an INFINITE number more.

    It is not simply "counter intuitive" it is illogical, and thus counted as impossible.
    Well, here's where we need to be very precise. In order to show that there is a contradiction, you must prove that there is a proposition p such both p and ~p can be derived.

    So let's go with some definitions. For any given hotel, let H designate the set of distinct rooms in the hotel. So if the hotel has 8 rooms, H = {h1, h2, ..., h7, h8}.

    What does it mean to accommodate a set of guests G = {g1, g2, ...}? A set of guests is said to be accommodated if (and only if) each guest has his/her own room in the hotel.

    Do you agree with these definitions?
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  11. #8
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    Well, here's where we need to be very precise. In order to show that there is a contradiction, you must prove that there is a proposition p such both p and ~p can be derived.

    So let's go with some definitions. For any given hotel, let H designate the set of distinct rooms in the hotel. So if the hotel has 8 rooms, H = {h1, h2, ..., h7, h8}.

    What does it mean to accommodate a set of guests G = {g1, g2, ...}? A set of guests is said to be accommodated if (and only if) each guest has his/her own room in the hotel.

    Do you agree with these definitions?
    Yes, I think I got it, and I agree.

    If I may, you are probably about to do some math work. IE move G1 into H2 .. G2 to H3 .etc in order to "accommodate" a new guest.

    But I think we can really shorten this process.

    --Explanation #1---

    The basic question is.
    1) Is the Hotel "fully occupied". In that for every room is their a guest in it? That answer should be "yes" because that is part of the set up of the Hotel.
    2) Is it possible to accommodate a guest into any hotel that is also fully occupied? That answer is necessarily "no".
    3) both 1 and 2 apply to this hotel, thus no new guest can be accommodated.

    So before you go into your figures, please tell me which of the above you disagree with.

    So mathematically, you may be able to establish that it is possible to create an empty room by manipulating the math(not inappropriately mind you), however the meaning of a fully occupied hotel, dictates that such a proposition must be false.
    Thus we have the contradiction of a hotel that is both fully occupied but has vacancies. (Here "vacancies would be defined as "having an unoccupied room".

    -----Explanation #2 ---
    Let me re-state the above in a different way.

    Any hotel that actually exists has actual rooms. In the Hotel example, there is no actual room, that does not also have an actual person in it. So just as the idea of moving people does not create any new rooms now, so to in an actual infinite, the moving of people from room to room does not change the fact that every room already has a person in it.

    I have refereed to this as the "hall-way fix" in past discussions. Because the effect of adding 1 new occupant, is to simply put some other occupant into the hall-way.
    I apologize to anyone waiting on a response from me. I am experiencing a time warp, suddenly their are not enough hours in a day. As soon as I find a replacement part to my flux capacitor regulator, time should resume it's normal flow.

  12. #9
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    Yes, I think I got it, and I agree.

    If I may, you are probably about to do some math work. IE move G1 into H2 .. G2 to H3 .etc in order to "accommodate" a new guest.

    But I think we can really shorten this process.
    You just need to assign rooms to guests. Actually physically "moving guests in" isn't an important part of the process.

    --Explanation #1---

    The basic question is.
    1) Is the Hotel "fully occupied". In that for every room is their a guest in it? That answer should be "yes" because that is part of the set up of the Hotel.
    2) Is it possible to accommodate a guest into any hotel that is also fully occupied? That answer is necessarily "no".
    3) both 1 and 2 apply to this hotel, thus no new guest can be accommodated.

    So before you go into your figures, please tell me which of the above you disagree with.

    So mathematically, you may be able to establish that it is possible to create an empty room by manipulating the math(not inappropriately mind you), however the meaning of a fully occupied hotel, dictates that such a proposition must be false.
    Thus we have the contradiction of a hotel that is both fully occupied but has vacancies. (Here "vacancies would be defined as "having an unoccupied room".
    (2) is demonstrably wrong; it fails for hotels with an infinite number of rooms. (2) does hold for all hotels with a finite number of rooms, but that doesn't make (2) necessarily true.

    No hotel can be fully occupied and have vacancies. It can't be that every room has an occupant but there is a room without an occupant. A hotel with an infinite number of rooms can be fully occupied and accommodate more guests, but it has no vacancies according to the definition you give of "vacancy".

    -----Explanation #2 ---
    Let me re-state the above in a different way.

    Any hotel that actually exists has actual rooms. In the Hotel example, there is no actual room, that does not also have an actual person in it. So just as the idea of moving people does not create any new rooms now, so to in an actual infinite, the moving of people from room to room does not change the fact that every room already has a person in it.

    I have refereed to this as the "hall-way fix" in past discussions. Because the effect of adding 1 new occupant, is to simply put some other occupant into the hall-way.
    The question is about the room assignment, not physically moving the guests. You could just have all the guests move at the same time. And being physically "out in the hallway" while you're walking to your room has no bearing on whether there is a room assigned to you.
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  13. #10
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    You just need to assign rooms to guests. Actually physically "moving guests in" isn't an important part of the process.
    I'm not objecting on the level of time to move or something like that.


    Quote Originally Posted by CLIVE
    (2) is demonstrably wrong; it fails for hotels with an infinite number of rooms. (2) does hold for all hotels with a finite number of rooms, but that doesn't make (2) necessarily true.
    I will let you demonstrate it
    (2) Holds in every instance which the conditions are met. The objections are not based on a number of rooms, they are based on the concepts of an occupied room and a hotel that has all of it's rooms occupied. "All" is not a specific numerical value.

    Quote Originally Posted by CLIVE
    A hotel with an infinite number of rooms can be fully occupied and accommodate more guests, but it has no vacancies according to the definition you give of "vacancy".
    Well, I disagree
    Does it, or does it not require an empty room in order to accommodate a new guest? (yes it does).
    Does the hotel have any empty rooms? (No it does not).
    Does any mathematical process create more actual rooms? (No, because moving someone doesn't magically create a new room in the real world).

    Thus, all rooms are full, and there is no place to accommodate any number of new guests.
    That is why the solution of "moving people" is insufficient in a real world of a full hotel... no matter how many rooms there are.


    Quote Originally Posted by CLIVE
    The question is about the room assignment, not physically moving the guests. You could just have all the guests move at the same time. And being physically "out in the hallway" while you're walking to your room has no bearing on whether there is a room assigned to you.
    I believe I have made the objection known, and look forward to your answers to the questions above.
    If I have not answered something, Let me know. otherwise, by all means make your case, as I sort of jumped to the end. (those reading will appreciate it I'm sure).
    I apologize to anyone waiting on a response from me. I am experiencing a time warp, suddenly their are not enough hours in a day. As soon as I find a replacement part to my flux capacitor regulator, time should resume it's normal flow.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    I'm not objecting on the level of time to move or something like that.



    I will let you demonstrate it
    (2) Holds in every instance which the conditions are met. The objections are not based on a number of rooms, they are based on the concepts of an occupied room and a hotel that has all of it's rooms occupied. "All" is not a specific numerical value.

    Well, I disagree
    Does it, or does it not require an empty room in order to accommodate a new guest? (yes it does).
    Does the hotel have any empty rooms? (No it does not).
    Does any mathematical process create more actual rooms? (No, because moving someone doesn't magically create a new room in the real world).

    Thus, all rooms are full, and there is no place to accommodate any number of new guests.
    That is why the solution of "moving people" is insufficient in a real world of a full hotel... no matter how many rooms there are.
    So let's be precise, here.

    I. Definitions: What does it mean for a hotel that already accommodates a set of guests G1 to accommodate an additional set of guests G2?


    Again, let H = {h1, h2, ...} denote the set of rooms in H. Let G1 = {ga1, ga2, ga3, ...} denote the set of guests already accommodated by the hotel. Let G2 = {gb1, gb2,gb3, ...} denote the set of guests to be accomodated in addition to G1.

    Remember, the definition we agreed on for a hotel to accommodate a set of guests is that each guest has his/her own room. This means that there's a pairing between a set of accommodated guests G1 and the set of rooms H such that each guest in G1 is paired up to their own room in H. Since each guest has his/her own room, no two guests have the same room. (We're ignoring the difference between an individual guest and, say, a family that wishes to rent out a single room.)

    One way of representing this pairing is with a map F: G1 -> H from the set of accommodated guests to the set of rooms, where F(gak) is defined to be the room that guest gak has. Since every guest in G1 is accommodated, we know that F(gak) is defined for all gak in G1. And since every guest has his/her own room, we know that two different guests must always have different rooms; that is, gak =/= gaj => F(gak) =/= F(gaj).

    But we could also look at the pairing from the opposite side, and define a map F': H -> G1 from the set of rooms in H to the set of accommodated guests G1. We define F'(hk) to be the guest in G1 that has room hk. Unlike the map F, which was defined for all its inputs, F' might be undefined for certain inputs; a room might fail to be occupied by a guest. However, since no two guests have the same room, F' will never map the same room to two different guests.


    I propose the following definitions:
    (D3) A hotel that currently accommodates a set of guests G1 can accommodate an additional set of guests G2 if and only if there is a room assignment such that every guest in G1 has his/her own room and every guest in G2 has his/her own room.

    (D4) A hotel is said to be fully occupied under a room assignment R: G -> H if and only if for every hk in H there exists a gj in G such that R(gj) = hk

    In plain English, (D4) says that for a hotel to be "fully occupied" according to some room assignment, every room must be occupied by some guest.



    II. Proof: A fully occupied hotel can accommodate an additional set of guests

    With (D3) and (D4) in mind, we can now answer this question.

    Let H = {h1, h2, ...} be an infinite set. Suppose that H is fully occupied (I will use the notions of "hotel" and "set of rooms" interchangeably).

    Then there is some set of accommodated guests G = {g1, g2, ...} and a map R: G -> H such that for any hk in H, there exists a gj in G such that R(gj) = hk.

    Let A = {a1, a2, ...} be a set of guests distinct from G.

    In order for H to accommodate A in addition to G, under (D3) we must show that there are room assignments R1: G -> H and R2: A -> H with the following properties:

    (1) For every g in G there is an h in H such that R1(g) = h
    (2) No two different guests in G have the same room under R1
    (3) For every a in A there is an h in H such that R2(a) = h
    (4) No two different guests in A have the same room under R2
    (5) No guest in A has the same room under R2 as a guest in G under R1.

    If such room assignments exist, then H can accommodate A in addition to G (by definition (D3)).

    Let us define the following room assignments:

    R1: G -> H where R1(gk) = h2*k
    R2: A -> H where R2(aj) = h2*j+1

    These room assignments meet the 5 conditions above:

    (1): Let g be a guest in G. Then g = gk for some integer k. Note that h2*k is an element in H. By definition, R1(gk) = h2*k.
    (2): Suppose that gk and gj are two different guests in G. Then k =/= j (otherwise they'd be the same guest). Thus 2k =/= 2j, and thus h2k =/= h2j. Hence gk and gj are sent to different rooms under R1.
    (3): Let a be an element in A. Then a=aj for some integer j. Note that h2*k+1 is an element in H. By definition, R1(aj) = h2*k+1
    (4): Suppose that ak and aj are two different guests in A. Then k =/= j (otherwise they'd be the same guest). Thus 2k+1 =/= 2j+1 and thus h2*k+1 =/= h2*j+1. Hence ak and aj are sent to different rooms under R2.
    (5): Every guest in A is sent to an odd-numbered room. Every guest in G is sent to an even-numbered room. Since there is no number that is both even and odd, no guest from A has the same room as a guest from G.


    Therefore, H can accommodate A in addition to G.




    III. Conclusions


    IIIa. Irrelevance of room assignment chronology

    H's ability to accommodate A in addition to G did not depend in any way on how H happened to accommodate G "without" A. Accommodating A along with G is just the same as accommodating all the guests of A and G put together.

    In order to understand this point properly, I'm going to use some standard set theory notation. The union of A and B, written A U B, is defined to be the set of all elements that are in A together with all the elements that are in B.

    So suppose that H is fully occupied by the guests in G. The question you're asking is whether H can also accommodate the guests in A. This is precisely the same question as whether H can accommodate the guests in G U A.



    It does not matter at all how the guests in G were assigned to their rooms "before" A. This is why your requirement that there be "empty rooms" is not necessary; you could simply move your previous guests into their "new" assigned rooms, leaving some previous room "temporarily" unoccupied, then move the "new" guest into the "unoccupied" room. The only thing that matters is the room assignment itself, not who gets moved when or when the rooms are empty. All we need to talk about are room assignments.



    IIIb. The real problem with your (2)

    The proper statement that necessarily holds in place of your (2) is the following:

    (2')
    If |
    G| > |H|, then H cannot accommodate G.

    In plain English, you cannot accommodate a number of guests that is larger than your number of rooms.


    Finite sets

    With finite sets and only with finite sets, any time you add elements you end up with a strictly larger cardinality. Adding a new element is sufficient to increase the cardinality of any finite set.

    When H and G are finite, and H is fully occupied by G, then you know |H| = |G| = k, for some (finite) integer k. In order to accommodate even one more guest, call him x, you'd need to accommodate the set G U {x}. But |G U {x}| > |G|, since |G| = k and |G U {x}| = k + 1, which is strictly greater than k. Since |G U {x}| > |H|, H cannot accommodate |G U {x}|.

    This means that for finite sets, my (2') is equivalent to your (2).


    Infinite sets

    But when you get to infinite sets, adding new elements is no longer sufficient to increase cardinality. The set {1, 2, 3, ...} is infinite. Adding the new element "0" results in the set {0,1, 2, ...}, but these sets have identical cardinality.

    However, even for infinite sets my (2') holds. I could prove this, but I don't know if you'd care to read the proof:

    Theorem: A hotel with countably infinite rooms cannot accommodate an uncountably infinite number of guests.



    TL;DR Accommodating a new guest does not require finding an empty room, it only requires finding a room assignment that assigns rooms to all your previous guests and all your new guests.
    Last edited by CliveStaples; July 30th, 2013 at 01:02 AM.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    You just need to assign rooms to guests. Actually physically "moving guests in" isn't an important part of the process.
    It would be important to a real hotel with real guests. Hoteliers don't get paid for just assigning rooms.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    It would be important to a real hotel with real guests. Hoteliers don't get paid for just assigning rooms.
    I don't see how the cost of managing Hilbert's Grand Hotel is relevant to WLC's arguments about the impossibility of an actual infinity. Can you elaborate?
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    This still doesn't make any sense to me.

    First, the Grand Hotel thought experiment is about various ways of combining infinite sets; whether or not the rooms and people actually exist, and in particular actually exist at the same time is irrelevant.

    Second, a set is defined by its members. Whether or not these members possess "some quantitative property" is irrelevant. The fact that moments don't "take up space" does not somehow disqualify them from being contained in a set.
    It is not irrelevant. Mathematics is a means of describing reality. Two apples is a quantity of apples that if realized means two apples exist at the same time. Otherwise there is truly only one apple and to call it two apples is just a false description. A set of three distinct apples is not three distinct apples if only one apple ever exists in the set at a given viewing of it. Then its just a set of one apple where the apple in the set changes upon observation.

    Sets are meaningless without some assurance that the members of the set are in the set together. 1,2,3 is not a 3 member set if 1 and 2 can't both exist in the set together. Then its just 1,3 or 2,3 which would be a set of 2 rather than a 3 count set.

    To create a set of moments when no two moments can co-exist is simply creating an imaginary set. It may as well be a squared circle or a zombie squirrel god. Either illogical given one set of principles or utterly fanciful given an unfounded set of principles.

    Consider an infinite series of 0. Add them all up and its still 0. An infinite number of infinitesimal moments is still nothing. We know moments are real in a sense, but its pretty clear there is only ever one moment, it is a singular, not a set. The moment of "now" is the only existent moment that ever was, is, or will be.

    ]I am not interested at all in comparing various theories of time. Your criticism depends on your assumption that sets must contain only members that are contemporaneous with one another, which is completely false. Your last sentence doesn't make any sense to me.
    It doesn't make sense when you don't have a rational view of time. Understand time as an observed change in the nature of the singularity of existence and you can see that the idea of a set of moments is irrational.
    Feed me some debate pellets!

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Sigfried View Post
    It is not irrelevant. Mathematics is a means of describing reality. Two apples is a quantity of apples that if realized means two apples exist at the same time. Otherwise there is truly only one apple and to call it two apples is just a false description. A set of three distinct apples is not three distinct apples if only one apple ever exists in the set at a given viewing of it. Then its just a set of one apple where the apple in the set changes upon observation.

    Sets are meaningless without some assurance that the members of the set are in the set together. 1,2,3 is not a 3 member set if 1 and 2 can't both exist in the set together. Then its just 1,3 or 2,3 which would be a set of 2 rather than a 3 count set.

    To create a set of moments when no two moments can co-exist is simply creating an imaginary set. It may as well be a squared circle or a zombie squirrel god. Either illogical given one set of principles or utterly fanciful given an unfounded set of principles.

    Consider an infinite series of 0. Add them all up and its still 0. An infinite number of infinitesimal moments is still nothing. We know moments are real in a sense, but its pretty clear there is only ever one moment, it is a singular, not a set. The moment of "now" is the only existent moment that ever was, is, or will be.



    It doesn't make sense when you don't have a rational view of time. Understand time as an observed change in the nature of the singularity of existence and you can see that the idea of a set of moments is irrational.
    So the set {Abraham Lincoln, George Washington, Superman} doesn't exist?
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    Re: WLC's Argument Against an Actual Infinity

    @CLIVE

    Last post I said this.

    Quote Originally Posted by mt
    Does it, or does it not require an empty room in order to accommodate a new guest? (yes it does).
    Does the hotel have any empty rooms? (No it does not).
    Does any mathematical process create more actual rooms? (No, because moving someone doesn't magically create a new room in the real world).

    From your post, you are denying the truth of the first point.
    You also deny the truth of the second point.

    Is that correct?



    ---To #1---
    To the first you said that we simply had to "assign a room to each guest". I think this runs an immediate risk.
    It changes the Hotel into an abstract again, instead of the actual rooms being actually occupied. (which is sort of the point).



    ----From SC's & Clive
    Quote Originally Posted by CLIVE
    I don't see how the cost of managing Hilbert's Grand Hotel is relevant to WLC's arguments about the impossibility of an actual infinity. Can you elaborate?
    I don't think it is cost management issue he is bringing up.
    It is more along the lines of.

    Your the manager of the hotel that has infinite rooms all of which are occupied. A new customer comes in.
    So you take out a sheet of paper and write down what you did in your last response to me.
    When you turn around.. there will still not be a key hanging on the wall.
    Because what you wrote was an abstract, and the fact that there are no keys is a reality.

    The reality remains that in order to accommodate a new guest, you must first remove one of the existing guests.

    --------

    Overview objection.
    The problem is not that you can't combine the lists on paper, but HOW do you combine even a single new guest into a hotel where every room already has a guest.
    you must ask one to leave his room ... correct?
    I apologize to anyone waiting on a response from me. I am experiencing a time warp, suddenly their are not enough hours in a day. As soon as I find a replacement part to my flux capacitor regulator, time should resume it's normal flow.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    @CLIVE

    Last post I said this.




    From your post, you are denying the truth of the first point.
    You also deny the truth of the second point.

    Is that correct?
    I deny the truth of the first point, and I've proved that the first point is false.
    I don't deny the second point, which is that the hotel doesn't have any "empty" rooms.

    The "(2)" that I was objecting to in my post referred to your earlier list:

    1) Is the Hotel "fully occupied". In that for every room is their a guest in it? That answer should be "yes" because that is part of the set up of the Hotel.
    2) Is it possible to accommodate a guest into any hotel that is also fully occupied? That answer is necessarily "no".
    3) both 1 and 2 apply to this hotel, thus no new guest can be accommodated.

    ---To #1---
    To the first you said that we simply had to "assign a room to each guest". I think this runs an immediate risk.
    It changes the Hotel into an abstract again, instead of the actual rooms being actually occupied. (which is sort of the point).
    No, it doesn't. The room assignment can be a list on a computer or piece of paper or in your head. The ability of the hotel to admit more guests depends only on whether or not certain room assignments are possible. Whether or not the actual guests happen to make their way to the actual hotel and get their actual key isn't really the issue.

    ----From SC's & Clive

    I don't think it is cost management issue he is bringing up.
    It is more along the lines of.

    Your the manager of the hotel that has infinite rooms all of which are occupied. A new customer comes in.
    So you take out a sheet of paper and write down what you did in your last response to me.
    When you turn around.. there will still not be a key hanging on the wall.
    Because what you wrote was an abstract, and the fact that there are no keys is a reality.

    The reality remains that in order to accommodate a new guest, you must first remove one of the existing guests.

    --------
    Not remove from the hotel, but change their room assignment, yes.

    It doesn't matter, though. You could simply kick everyone out for an hour, take back all their keys, then create a new room assignment for the complete set of all guests, old and new. It doesn't change whether you're able to accommodate the complete set of guests.

    Overview objection.
    The problem is not that you can't combine the lists on paper, but HOW do you combine even a single new guest into a hotel where every room already has a guest.
    you must ask one to leave his room ... correct?
    Sure, but I don't see how this is relevant.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    So the set {Abraham Lincoln, George Washington, Superman} doesn't exist?
    Only by representation. They are not a set in reality, they are an imaginary set. The actual people are not in a set, but we can represent them by reference in one.

    So we can imagine a set with all the moments in history, but the set is not real, its only imaginary, and in imagination it can easily be potentially infinite. In reality, there is no such set.
    Feed me some debate pellets!

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Sigfried View Post
    Only by representation. They are not a set in reality, they are an imaginary set. The actual people are not in a set, but we can represent them by reference in one.

    So we can imagine a set with all the moments in history, but the set is not real, its only imaginary, and in imagination it can easily be potentially infinite. In reality, there is no such set.
    None of your distinctions make sense to me. What is an "imaginary set"? What is a "set in reality"? What makes a set "real"?
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    I don't see how the cost of managing Hilbert's Grand Hotel is relevant to WLC's arguments about the impossibility of an actual infinity. Can you elaborate?
    I didn't say a word about cost. I said in a real hotel, with real guests, moving them into their rooms would be as important as assigning them rooms. Wouldn't you agree?

 

 
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