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  1. #21
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    Re: WLC's Argument Against an Actual Infinity

    To help clarify WLC's position...

    Hilbert's Hotel

    That's a link to Blackwell Companion of Natural Theology in which WLC provides a more academic treatment.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    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?
    For the sake of providing excellent customer service, sure. But the question is about the Hotel's ability to accommodate guests; the impossibility of an infinite hotel being real is not based on the difficulty of moving guests into their assigned rooms, but rather on the "absurdity" that an infinite hotel can always accommodate additional guests (so long as the total set of guests to be accommodated is not a "larger" infinity than the number of rooms in the hotel).
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  4. #23
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    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.
    I must object to this.
    What is on paper could be abstract and incomparable with reality.
    That is why dealing with the actual guests is important. The entire point of the example is to show that their is a disconnect between the math on the paper and the People occupying the rooms.

    Quote Originally Posted by CLIVE
    I don't deny the second point, which is that the hotel doesn't have any "empty" rooms.
    Good, because it is exactly that point that stands in stark opposition to the proposed solutions.
    A hotel with no empt rooms by definition can not accommodate any new guests, unless new rooms are created.

    Quote Originally Posted by CLIVE
    Not remove from the hotel, but change their room assignment, yes.
    Think a step deeper past the simple assignment. That unless an occupant leaves a room then there is no place for the new guest.
    When one person leaves a room, there is still a 1 to 1 ratio of people and rooms. If they walk out and walk back in, it is no different than moving, because no new rooms are created.

    Quote Originally Posted by CLIVE
    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.
    yes, it does in the actual world.

    Because, Again, re above with walking out of the room.

    That is where the contradiction in reality occurs from the math.

    Quote Originally Posted by CLIVE
    Sure, but I don't see how this is relevant.
    Because greater numbers of people do not create new rooms.


    ----How about this. (Please.. I apologize for the poor use of math language here and ask that you try your best to see my objection and mentally correct, if you can, for user error as best you can)

    Do you agree, that there are different size infinite sets?

    Hotel A has all of it's rooms occupied.
    and it's guests are represented by the infinite set of g1.. g2 etc.

    Then a new list of guests want to be accommodate they comprise of list (new guests) ng1 ..ng2. etc

    This new list (new list) contains all the guests of G1 and all the guests of NG1. (correct?)
    So list g1 and list ng1 when combined would be a larger infinite set than just G1. .. correct?


    So then the hotel now houses G1 and NG1.


    The absurdities...(contains repeating concepts)
    1) The hotel has a set of rooms that is fixed because it exists in reality. So the set size of the hotel rooms is the same size infinite as the list of people that ACTUALLY occupy the rooms. So that they are equal by definition. On the original list no one from ng1 list can be found in that set. Yet, it is large enough to also contain ng2. So the hotel is both properly occupied and properly has vacancies. (IE The hotel is both the = to g1 and = to g1+ng1 While g1 both equal to the size of the hotel,as all the rooms are occupied but is not the size of g1+ng1. A contradiction.


    2) Now(the lists are combined).. all the guests on list NG1 leave the hotel... does the hotel have any empty rooms? Yes. (half in fact)
    This creates the contradiction that the hotel is fully occupied, and also is half empty while not changing sizes(IE creating new rooms).
    So the hotel can not contain the list of G1 without vacancies if it is actually the larger set of g1 and Ng1.

    So Hotel with only g1 = no unoccupied rooms and Hotel with only g1 has unoccupied rooms. even though no change to the hotel has occurred. (a contradiction).

    3)Through the process of merging lists, the hotel is made to change sizes evidenced by #2, which (see 4) is not logical.
    If g1 moved back to their old rooms after ng1 leaves, the hotel would have had half its occupants leave, but still have no vacancies. (a contradiction)

    4) The process of assigning rooms is independent of the size of the hotel, because the process of assigning rooms does not effect the number of actual rooms. (IE no new rooms are created or destroyed).
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    Good, because it is exactly that point that stands in stark opposition to the proposed solutions.
    A hotel with no empt rooms by definition can not accommodate any new guests, unless new rooms are created.
    I cannot have a constructive dialogue with someone who maintains a claim that has been proven false. Please defend your claim. I have shown that under my definition of "accommodate", a fully occupied hotel can accommodate new guests.
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  6. #25
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    I cannot have a constructive dialogue with someone who maintains a claim that has been proven false. Please defend your claim. I have shown that under my definition of "accommodate", a fully occupied hotel can accommodate new guests.
    As I have offered valid challenges to your "dis-proof", my statement is in the process of being defended.
    My last post is exactly what you are now demanding of me. .. a defense of my claim, by rebutting and questioning the validity of your supposed "dis-proof".

    Is it that you simply stopped reading my post at the first instance of my re-asserting the claim? Or is it that you do not understand how my last post supports my claim by bringing valid objections to your "dis-proof"?
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    As I have offered valid challenges to your "dis-proof", my statement is in the process of being defended.
    My last post is exactly what you are now demanding of me. .. a defense of my claim, by rebutting and questioning the validity of your supposed "dis-proof".

    Is it that you simply stopped reading my post at the first instance of my re-asserting the claim? Or is it that you do not understand how my last post supports my claim by bringing valid objections to your "dis-proof"?
    I stopped reading your post when I saw that you were just repeating exactly what you had said before. I don't really see this discussion going any further between you and I.
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  8. #27
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    Re: WLC's Argument Against an Actual Infinity

    Then you are not qualified to say my position is not supported(I mean, if you didn't read my post).
    Thanks.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    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.
    Well, if you don’t mind, I’m going to check out of the H. Hotel for a bit, because it’s getting a little too crowed for me. But toward your argument of Craige's idea that infinity cannot exist, the way I understand his argument, and I think this point has been presented on ODN in the past, is that time can indeed move on for infinity, however at no point in time could there have ever existed infinite time.
    Last edited by eye4magic; July 30th, 2013 at 10:01 PM.
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  10. #29
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    But the question is about the Hotel's ability to accommodate guests
    Okay, but the "Hotel" is defined as "full". A "full" hotel and a hotel with empty rooms cannot logically be the same hotel, right?

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    Re: WLC's Argument Against an Actual Infinity

    I think there is a more practical objection to Clive's concern here that gets more to what WLC was actually intending. It is that an infinite, whatever that may actually entail, is incompatible with our observed universe. The paradoxical outcomes he laid out were to illustrate that point. The hotel example shows, in part the problem with assigning a point to an infinite past.

    In essence, WLC's response to the classical objection to the CA (an infinite universe) is that it isn't possible for us to be in one.

    To rephrase it, "When was an infinitely old universe 10 billion years old?" Since there clearly is no point where that is true, since all points in a countable infinite have infinite previous points, then there is no point where it could be said that stellar activity operated. At all points on the timeline in an infinite universe the entirety of stellar matter would have burned out infinitely long ago.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by SQUATCH
    I think there is a more practical objection to Clive's concern here that gets more to what WLC was actually intending. It is that an infinite, whatever that may actually entail, is incompatible with our observed universe. The paradoxical outcomes he laid out were to illustrate that point. The hotel example shows, in part the problem with assigning a point to an infinite past.

    In essence, WLC's response to the classical objection to the CA (an infinite universe) is that it isn't possible for us to be in one.

    To rephrase it, "When was an infinitely old universe 10 billion years old?" Since there clearly is no point where that is true, since all points in a countable infinite have infinite previous points, then there is no point where it could be said that stellar activity operated. At all points on the timeline in an infinite universe the entirety of stellar matter would have burned out infinitely long ago.
    That is true. Of course that is a different example than the H-Hotel.
    The hotel is an example of getting inherently contradictory answers to real questions that hotels would face.

    For example the list of Original guests is both sufficient and insufficient to fill the hotel in the same sense based only on movement.
    Further that movement requires them to move out of one previously occupied room and into anther previously occupied room, and yet results by producing empty rooms.
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  13. #32
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    Okay, but the "Hotel" is defined as "full". A "full" hotel and a hotel with empty rooms cannot logically be the same hotel, right?
    The Hotel is not defined as "full". The Hotel is defined according to the number of rooms it has; for the Grand Hotel, the number of rooms is infinite.

    A hotel is only said to be "full" under some room assignment. A hotel could be full under one room assignment R1 but not full under some room assignment R2.

    ---------- Post added at 04:01 PM ---------- Previous post was at 03:46 PM ----------

    Quote Originally Posted by MindTrap028 View Post
    Good, because it is exactly that point that stands in stark opposition to the proposed solutions.
    A hotel with no empt rooms by definition can not accommodate any new guests, unless new rooms are created.
    Support this, please. If you're referring to definition (D3) of "accommodate", I have already proved your statement false.

    Think a step deeper past the simple assignment. That unless an occupant leaves a room then there is no place for the new guest.
    When one person leaves a room, there is still a 1 to 1 ratio of people and rooms. If they walk out and walk back in, it is no different than moving, because no new rooms are created.
    All of this is true. I don't see how it is relevant to whether the hotel can accommodate more guests.

    yes, it does in the actual world.

    Because, Again, re above with walking out of the room.

    That is where the contradiction in reality occurs from the math.
    Support, please. Tell me explicitly what your claimed contradiction is; that is, the proposition p is such that both p and ~p are true.

    Because greater numbers of people do not create new rooms.
    How is "creating new rooms" relevant?

    ----How about this. (Please.. I apologize for the poor use of math language here and ask that you try your best to see my objection and mentally correct, if you can, for user error as best you can)

    Do you agree, that there are different size infinite sets?
    Yes.

    Hotel A has all of it's rooms occupied.
    and it's guests are represented by the infinite set of g1.. g2 etc.

    Then a new list of guests want to be accommodate they comprise of list (new guests) ng1 ..ng2. etc
    Okay.

    This new list (new list) contains all the guests of G1 and all the guests of NG1. (correct?)
    Okay.

    So list g1 and list ng1 when combined would be a larger infinite set than just G1. .. correct?
    No. The union of a countably infinite set with another countably infinite set is countably infinite. The "size" of infinity does not increase.

    So then the hotel now houses G1 and NG1.


    The absurdities...(contains repeating concepts)
    1) The hotel has a set of rooms that is fixed because it exists in reality. So the set size of the hotel rooms is the same size infinite as the list of people that ACTUALLY occupy the rooms. So that they are equal by definition. On the original list no one from ng1 list can be found in that set. Yet, it is large enough to also contain ng2. So the hotel is both properly occupied and properly has vacancies. (IE The hotel is both the = to g1 and = to g1+ng1 While g1 both equal to the size of the hotel,as all the rooms are occupied but is not the size of g1+ng1. A contradiction.
    But g1 = g1+ng1. Your statement is false.

    2) Now(the lists are combined).. all the guests on list NG1 leave the hotel... does the hotel have any empty rooms? Yes. (half in fact)
    This creates the contradiction that the hotel is fully occupied, and also is half empty while not changing sizes(IE creating new rooms).
    So the hotel can not contain the list of G1 without vacancies if it is actually the larger set of g1 and Ng1.
    "Half empty while not changing sizes". How is that a contradiction?

    So Hotel with only g1 = no unoccupied rooms and Hotel with only g1 has unoccupied rooms. even though no change to the hotel has occurred. (a contradiction).
    What is the contradiction? It seems like you're assuming something like this: "For a given set of guests, a hotel must always have the same number of unoccupied rooms under any room assignment of those guests." This statement holds for hotels with a finite number of rooms, but obviously does not hold for an infinite number of rooms.

    3)Through the process of merging lists, the hotel is made to change sizes evidenced by #2, which (see 4) is not logical.
    If g1 moved back to their old rooms after ng1 leaves, the hotel would have had half its occupants leave, but still have no vacancies. (a contradiction)
    The hotel never changes size. No rooms are created or destroyed; the hotel is precisely the same under any room assignment.

    You have still failed to identify a contradiction. You are just making a true statement, and then saying the words "a contradiction". This does not amount to proof.

    4) The process of assigning rooms is independent of the size of the hotel, because the process of assigning rooms does not effect the number of actual rooms. (IE no new rooms are created or destroyed).
    I have no idea what this sentence means. Independent in what way? We know that a hotel with a finite number of rooms, say k, cannot accommodate (in the sense of definition (D3)) a set containing more than k guests.

    This holds true for hotels with an infinite number of rooms as well. A hotel with a countably infinite set of rooms cannot accommodate an uncountable number of guests.


    The process of assigning rooms is not independent of the size of the hotel, because the existence of a 'legal' room assignmentis logically equivalent to whether the cardinality of guests exceeds the cardinality of rooms.

    ---------- Post added at 04:01 PM ---------- Previous post was at 04:01 PM ----------

    Quote Originally Posted by eye4magic View Post
    Well, if you don’t mind, I’m going to check out of the H. Hotel for a bit, because it’s getting a little too crowed for me. But toward your argument of Craige's idea that infinity cannot exist, the way I understand his argument, and I think this point has been presented on ODN in the past, is that time can indeed move on for infinity, however at no point in time could there have ever existed infinite time.
    How do you know that?
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  14. #33
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    The Hotel is not defined as "full". The Hotel is defined according to the number of rooms it has; for the Grand Hotel, the number of rooms is infinite.
    Then you are apparently at odds with:

    princeton.edu:

    Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest.
    And The Math Book: From Pythagoras tothe 57th Dimension, 250 Milestones in the History of Mathematics:

    Imagine an ordinary hotel with 500 rooms, all of which are occupied by guests...Next, imagine a hotel in which there are an infinite number of rooms, each of which is occupied. Although the hotel is full, the clerk can give you a room.
    And psu.edu:

    Imagine a hotel with infinitely many rooms, all of which are full.
    And the book summary of Hilbert's Paradox of the Grand Hotel, authored by Frederic P. Miller, Agnes F. Vandome, and McBrewster John:

    Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied that is to say every room contains a guest.
    And Floyd Merrell, in his book Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics:

    Imagine a hotel with an infinity of rooms, all of which are full of guests.
    So which is it? Are you appealing to a hotel with an infinitely countable number of rooms that is in some other fashion conceptually distinct from Hilbert's famous Grand Hotel, or are all these sources misinformed, and you correct to claim the hotel is not defined as "full", or are you wrong to claim the hotel is defined by "room assignment", and not as "full"?

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    Then you are apparently at odds with:

    princeton.edu:



    And The Math Book: From Pythagoras tothe 57th Dimension, 250 Milestones in the History of Mathematics:



    And psu.edu:



    And the book summary of Hilbert's Paradox of the Grand Hotel, authored by Frederic P. Miller, Agnes F. Vandome, and McBrewster John:



    And Floyd Merrell, in his book Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics:



    So which is it? Are you appealing to a hotel with an infinitely countable number of rooms that is in some other fashion conceptually distinct from Hilbert's famous Grand Hotel, or are all these sources misinformed, and you correct to claim the hotel is not defined as "full", or are you wrong to claim the hotel is defined by "room assignment", and not as "full"?
    I'm saying that the Hotel I'm considering is defined merely as a set of rooms. In these terms, the statements you've cited are understood in terms of room assignments of sets of guests. I never claimed that your sources were misinformed.

    Your line of questioning here is a red herring. There's no meaningful difference between considering a hotel that's defined as full and considering room assignments under which a hotel is full.

    It's like saying, "Consider a tall man" versus "Consider a man who is tall". It's a distinction without a difference.
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  16. #35
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    Support this, please. If you're referring to definition (D3) of "accommodate", I have already proved your statement false.
    Because your D3 requires that people leave their rooms and thus be "re-assigned". That process inherently involves an "empty room".

    If you agree then I have sustained my contention that accommodating new guests in ANY full hotel(limited or infinite) requires an occupied room to have it's occupant leave.

    Quote Originally Posted by CLIVE
    All of this is true. I don't see how it is relevant to whether the hotel can accommodate more guests.
    It is based on my sustaining the previous point above.

    Quote Originally Posted by CLIVE
    Support, please. Tell me explicitly what your claimed contradiction is; that is, the proposition p is such that both p and ~p are true.
    That the first list of guests is sufficient and insufficient to fill the hotel in the same sense and at the same time.

    Quote Originally Posted by CLIVE
    How is "creating new rooms" relevant?
    Because if all rooms are occupied the only way to accommodate new guests is to create an empty room.
    If the hotel is full, then a New room is required.

    Quote Originally Posted by CLIVE
    No. The union of a countably infinite set with another countably infinite set is countably infinite. The "size" of infinity does not increase.
    O.K. I withdraw that point. But list g1 does not contain any guest form list ng1. Agree?

    Quote Originally Posted by CLIVE
    But g1 = g1+ng1. Your statement is false.
    If you are simply refering to size, then you are correct(noting your correction above)
    However as ng1 is not within the list of guests of g1, it is demonstrably false to equate the list in that way.
    Because if ng1 was contained in the list g1, then they would not be "new guests".

    Further the contradiction expresses itself like this.
    If g1 = g1+ng1.
    Then g1 = full hotel & g1 = half full hotel.
    Because if all the people in list ng1(after merging) were to leave, then all the odd rooms would be vacant, and the hotel would be half empty.

    That is an inherent contradiction to say that list g1 is sufficient and insufficient in the same sense.

    Quote Originally Posted by CLIVE
    "Half empty while not changing sizes". How is that a contradiction?
    Because g1 is both sufficient and insufficient to fill the hotel in the same sense.

    Quote Originally Posted by CLIVE
    What is the contradiction? It seems like you're assuming something like this: "For a given set of guests, a hotel must always have the same number of unoccupied rooms under any room assignment of those guests." This statement holds for hotels with a finite number of rooms, but obviously does not hold for an infinite number of rooms.
    No, it does not "obviously hold". It isn't obvious at all.

    It may be asserted to not hold, but it seems to be more of an assumption than a proof.

    Quote Originally Posted by CLIVE
    The hotel never changes size. No rooms are created or destroyed; the hotel is precisely the same under any room assignment.

    You have still failed to identify a contradiction. You are just making a true statement, and then saying the words "a contradiction". This does not amount to proof.
    That is because we are dealing with actual rooms. And a list that your position dictates the list exists in a 1 to 1 ration and not a 1 to 1 ratio, at the same time and in the same sense.
    That is a contradiction. Nor have you forwarded why moving people out of rooms for which they exist in a 1 to 1 ratio, would change the ratio and allow for new members.

    You are literally forwarding the concept that by leaving your room and walking back in(or the logical equivalent) that you have changed the ratio of room to guests. There is no reason to think that would be the case in any real world example.

    Quote Originally Posted by CLIVE
    I have no idea what this sentence means. Independent in what way? We know that a hotel with a finite number of rooms, say k, cannot accommodate (in the sense of definition (D3)) a set containing more than k guests
    That was simply to say that room assignment doesn't change the rooms or the hotel itself.
    It is to say that if I had a hotel with 10 rooms, and I give out 11 keys, a new room is not created.

    Quote Originally Posted by CLIVE
    This holds true for hotels with an infinite number of rooms as well. A hotel with a countably infinite set of rooms cannot accommodate an uncountable number of guests.
    The problem is that list g1 does not contain anyone on the list of ng1.
    It is one thing to say that list g1 is the same size as g1 and ng1, but that does not mean it contains the people in the list of ng1.
    Otherwise you would literally be saying that the guests trying to check in, were already a guest of the hotel.

    Quote Originally Posted by CLIVE
    The process of assigning rooms is not independent of the size of the hotel, because the existence of a 'legal' room assignmentis logically equivalent to whether the cardinality of guests exceeds the cardinality of rooms.
    So your saying that in the real hotels that we are used to(which are limited in ultimate size), that the guy handing out keys has no control over how large the hotel is.
    But.
    In an infinite hotel, the guy handing out keys directly dictates the size of the hotel?



    --sorry for the repeats --
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    I'm saying that the Hotel I'm considering is defined merely as a set of rooms.
    Okay, fine. Then when the next person wanting a room in your Hotel shows up, are the rooms already all full or not? A simple yes or no answer is all that is needed here, as I'm sure you're more than enough of a logician to know.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    Okay, fine. Then when the next person wanting a room in your Hotel shows up, are the rooms already all full or not? A simple yes or no answer is all that is needed here, as I'm sure you're more than enough of a logician to know.
    They are if you say they are. If you want to talk about how to accommodate an extra guest when the hotel is already accommodating a certain set of guests under some room assignment, go right ahead. That's what we've been talking about. But it isn't the only possible state of affairs to consider with regard to a hotel with an infinite set of rooms.

    ---------- Post added at 01:02 AM ---------- Previous post was at 12:38 AM ----------

    Quote Originally Posted by MindTrap028 View Post
    Because your D3 requires that people leave their rooms and thus be "re-assigned". That process inherently involves an "empty room".

    If you agree then I have sustained my contention that accommodating new guests in ANY full hotel(limited or infinite) requires an occupied room to have it's occupant leave.
    Yes. What you consider an "empty" room is unclear. It's empty for a certain amount of time, and then "full" again--although when is it full? When you assign it to a new guest? When the new guest arrives at the hotel? When the new guest's bags are put into the room? When the guest physically enters the room? When the guest tips the service staff? When the guest feels at home?


    It is based on my sustaining the previous point above.


    That the first list of guests is sufficient and insufficient to fill the hotel in the same sense and at the same time.
    You're going to have to be more precise. I don't know what your statement means.

    It is true that there are different possible room assignments for the guests; some assignments result in every room being occupied. Some assignments leave a finite number of rooms unoccupied. Some assignments leave an infinite number of rooms unoccupied. But I don't see a contradiction; no particular assignment results in the hotel being full and not full at the same time.

    Are you tacitly assuming that for any set of guests G, every room assignment from G to the set of rooms in the hotel must leave the same number of rooms unoccupied.


    Because if all rooms are occupied the only way to accommodate new guests is to create an empty room.
    If the hotel is full, then a New room is required.
    You have to be more precise. What is a "new" room? If a guest relocates to another room, is his previous room a "new" room?

    O.K. I withdraw that point. But list g1 does not contain any guest form list ng1. Agree?
    Yes, they are disjoint.

    If you are simply refering to size, then you are correct(noting your correction above)
    However as ng1 is not within the list of guests of g1, it is demonstrably false to equate the list in that way.
    Because if ng1 was contained in the list g1, then they would not be "new guests".
    I never said ng1 was contained in g1...????? What are you talking about?

    Further the contradiction expresses itself like this.
    If g1 = g1+ng1.
    Then g1 = full hotel & g1 = half full hotel.
    Because if all the people in list ng1(after merging) were to leave, then all the odd rooms would be vacant, and the hotel would be half empty.

    That is an inherent contradiction to say that list g1 is sufficient and insufficient in the same sense.
    How is it a contradiction? You're giving some vague criteria that every list of guests cannot be both sufficient and insufficient in the same sense. I have no idea what it means for a list of guests to be "sufficient". You mean "sufficient to fill the hotel"? Is that what you're talking about?


    Because g1 is both sufficient and insufficient to fill the hotel in the same sense.
    "In the same sense". What does that mean? So far as I can tell, different room assignments will leave different amounts of rooms empty. Where's the contradiction? No room assignment leaves the hotel "full and not full at the same time".

    No, it does not "obviously hold". It isn't obvious at all.

    It may be asserted to not hold, but it seems to be more of an assumption than a proof.
    It is obvious to anyone with a college freshman education in mathematics.


    Here's a proof:

    In general, there are infinitely many injections from a countably infinite set into itself. Take any countably infinite set, and number its elements S = {s1, s2, ...}. Consider fk: S -> S where f(s1) = sk and f(sj) = sj+1. That is, fk sends the first element in S to the kth element in S, the 2nd element to the (k+1)th element, and so on. For every natural number k, the map fk is an injection. Consider gk: S-> S where g(sk) = s2k. This sends the first element in S to the second element, the second element to the fourth, etc. This map is also an injection.

    Why does this matter? Because the room assignments from a set of guests G to a set of rooms H are precisely the injections from G to H. Every room assignment is a function, since every guest is assigned a room; and since no two guests are assigned the same room, the room assignment function is injective. Conversely, every injection from G to H defines a room assignment; if f: G -> H is an injection, then you can create a room assignment by assigning guest g to room f(g).

    So, each of the injections I gave above--the family of injections fk and the injection gk--each "leave out" or "skip over" a different number of elements of S. Or, in the language of hotels, each of the room assignments induced by these injections leaves a different number of rooms empty, while the number of guests remains constant.

    That is because we are dealing with actual rooms. And a list that your position dictates the list exists in a 1 to 1 ration and not a 1 to 1 ratio, at the same time and in the same sense.
    That is a contradiction. Nor have you forwarded why moving people out of rooms for which they exist in a 1 to 1 ratio, would change the ratio and allow for new members.
    I have no idea what this "ratio" is you're talking about, or why it would have to change to allow for new members.

    You are literally forwarding the concept that by leaving your room and walking back in(or the logical equivalent) that you have changed the ratio of room to guests. There is no reason to think that would be the case in any real world example.
    The ratio of rooms to guests? That doesn't make any sense. What is the ratio of rooms to guests? The number of guests divided by the number of rooms? How do you do division with infinite numbers?

    That was simply to say that room assignment doesn't change the rooms or the hotel itself.
    It is to say that if I had a hotel with 10 rooms, and I give out 11 keys, a new room is not created.
    Okay. Why are we talking about creating new rooms?

    The problem is that list g1 does not contain anyone on the list of ng1.
    It is one thing to say that list g1 is the same size as g1 and ng1, but that does not mean it contains the people in the list of ng1.
    Otherwise you would literally be saying that the guests trying to check in, were already a guest of the hotel.
    The size of g1, ng1 and the combination of g1+ng1 are all identical. The lists all have the same number of people on them.

    So your saying that in the real hotels that we are used to(which are limited in ultimate size), that the guy handing out keys has no control over how large the hotel is.
    But.
    In an infinite hotel, the guy handing out keys directly dictates the size of the hotel?
    No. In an infinite hotel, "handing out keys" doesn't change the number of rooms in the hotel. Where are you getting this from???
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    They are if you say they are.
    You're simply stalling. Yes or no. Is your hotel "full" or not when the next guest wanting a room for the night shows up?

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    Yes. What you consider an "empty" room is unclear. It's empty for a certain amount of time, and then "full" again--although when is it full?
    Then I feel you are not grasping the objection.

    You have a hotel with infinite rooms, all of which are occupied. This means there is one person in each room.
    A new guest arrives and there are no empty rooms. So the solution is to do the following.

    1) Ask guest G1 in R1 to exit his room
    2) Ask G2 to exit his room(R2) so that G1 may enter.
    You will notice that in the first step, you have 2 rooms and 3 people.
    This practical and very real problem persists to infinity. Where we wave a magic wand and say "we did it instantly".
    There is no point where the above conditions change, so there is no reason to think repeating it any number of times would produce a different result.
    Thus at any given time there will be two guest for the single empty room(R1).


    When one writes what I quote below. It is to say that after some amount the above becomes false. But it isn't apparent of when it would be. Unless what I have written above is not the actual process that occurs in the execution/applcation of what your write below.
    Quote Originally Posted by CLIVE
    Here's a proof:

    In general, there are infinitely many injections from a countably infinite set into itself. Take any countably infinite set, and number its elements S = {s1, s2, ...}. Consider fk: S -> S where f(s1) = sk and f(sj) = sj+1. That is, fk sends the first element in S to the kth element in S, the 2nd element to the (k+1)th element, and so on. For every natural number k, the map fk is an injection. Consider gk: S-> S where g(sk) = s2k. This sends the first element in S to the second element, the second element to the fourth, etc. This map is also an injection.

    Why does this matter? Because the room assignments from a set of guests G to a set of rooms H are precisely the injections from G to H. Every room assignment is a function, since every guest is assigned a room; and since no two guests are assigned the same room, the room assignment function is injective. Conversely, every injection from G to H defines a room assignment; if f: G -> H is an injection, then you can create a room assignment by assigning guest g to room f(g).

    So, each of the injections I gave above--the family of injections fk and the injection gk--each "leave out" or "skip over" a different number of elements of S. Or, in the language of hotels, each of the room assignments induced by these injections leaves a different number of rooms empty, while the number of guests remains constan

    ----

    Quote Originally Posted by CLIVE
    I never said ng1 was contained in g1...????? What are you talking about?
    Right, that is why I had the qualifier of "if you are talking about size then you are correct". Hinting that what followed would not apply.

    What is the difference between a list of Guests in a room, and a list of rooms?
    Isn't it the same list going by different names?
    If ng1 is not contained in g1, then how can it be contained in g1 when listed by room number?
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    You're simply stalling. Yes or no. Is your hotel "full" or not when the next guest wanting a room for the night shows up?
    The short answer is, "The hotel isn't 'full' until you give me a set of guests and a way to assign them rooms in the hotel." If you want to consider the scenario where a "full" hotel must accommodate a new guest, you can do so using my model. And I've done so already.



    The long answer is:

    Your question is like asking, "In the Bohr model of the atom, is the atom a hydrogen atom?" The model of the atom is of any atom, not just one particular atom. You can consider Bohr models of hydrogen atoms if you wish to; nothing prevents this. The Bohr model allows you to model a hydrogen atom.

    Similarly, my model is of a hotel in general and guest assignments in general. The question is about the capacity of a hotel to accommodate guests, and how guests can be accommodated in a hotel. So starting with a hotel that's already "fully booked" is unnecessary. You can just consider hotels in general, without requiring that the hotels are always full or always start full.

    ---------- Post added at 10:56 AM ---------- Previous post was at 10:42 AM ----------

    Quote Originally Posted by MindTrap028 View Post
    Then I feel you are not grasping the objection.

    You have a hotel with infinite rooms, all of which are occupied. This means there is one person in each room.
    A new guest arrives and there are no empty rooms. So the solution is to do the following.

    1) Ask guest G1 in R1 to exit his room
    2) Ask G2 to exit his room(R2) so that G1 may enter.
    You will notice that in the first step, you have 2 rooms and 3 people.
    This practical and very real problem persists to infinity. Where we wave a magic wand and say "we did it instantly".
    There is no point where the above conditions change, so there is no reason to think repeating it any number of times would produce a different result.
    Thus at any given time there will be two guest for the single empty room(R1).
    All you've shown is that there's one way of moving guests around that will result in someone always being out in the hallway. That is, there will never be a point after which every guest is in their newly-assigned room for your particular method of moving people.

    You haven't shown that your way is the only way to move people. You haven't shown that any way to relocate guests results in absurdity.


    Consider the following methods:


    (1a) Ask all guests to pack their belongings and enter the hallway. At this point, every room is empty.
    (1b) Ask all guests to move into the next room. From (1a), this room is empty.

    (2a) Suppose you want to move in your new guest at time 1. Start by moving guest 1 out of room 1 at time 0.
    (2b) Move guest 2 out of room 2 at time 1/2.
    (2c) Move guest 3 out of room 3 at time 1/4.
    (2d) Move guest n out of room 3 at time 1/(2^n).
    (2e) By time 1, every room will be empty.


    What is the difference between a list of Guests in a room, and a list of rooms?
    Isn't it the same list going by different names?
    If ng1 is not contained in g1, then how can it be contained in g1 when listed by room number?
    No. The list of rooms doesn't tell you which guest is in which room.

    I don't even understand your last question. ng1 is not contained in g1, and g1 is not contained in ng1. In fact, they have no elements in common. These are lists of guests.



    If you're just talking about lists of names, then g1 never lists any guests in ng1 regardless of which order you decide to list the guests in.


    But apparently you want to talk about lists of guests in a room--that is, the first guest on the list is in room 1, etc. In this case, order matters. The list {John, Paul} is different than the list {Paul, John} because the first list assigns Paul to room 1 and the second assigns Paul to room 2.

    I don't understand what "containing" means for ordered lists. Does {John, Paul} contain {Paul, John}, even though they're different lists? Does {Mark, John, Paul} contain {John, Paul} even though they're different lists?

    What makes you think ng1 is contained in g1 "when listed by room number"?
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