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  1. #41
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by clive
    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.
    That is how all actual process work, everything else is simply a multiple of that process.

    Quote Originally Posted by CLIVE
    (1a) Ask all guests to pack their belongings and enter the hallway. At this point, every room is empty.
    What significant difference is there in your 1A from my #1?

    Quote Originally Posted by CLIVE
    (1b) Ask all guests to move into the next room. From (1a), this room is empty.
    How is that significantly different from asking them to return to their rooms? In both cases the room was previously occupied by someone in list g1.

    By supposing that them leaving the room and entering a different room (all previously occupied by the list g1)
    You are saying that G1 is both sufficient and insufficient to occupy all the rooms in the hotel. That is a contradiction.


    Quote Originally Posted by CLIVE
    No. The list of rooms doesn't tell you which guest is in which room.
    That is not necessary information to know that the room is occupied.
    In other words you can refer to the guests by room number. IE Room # 3 wants room service.

    Quote Originally Posted by CLIVE
    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.
    Right.

    Quote Originally Posted by CLIVE
    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.
    agree.

    Quote Originally Posted by CLIVE
    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 would say that it isn't materially different. There is no reason to think that the list with John/Paul would be sufficient to occupy all rooms. but the list Paul/John is not sufficient. If the list is sufficient, then the order is irrelevant.

    Quote Originally Posted by CLIVE
    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?
    Yes, because they are really the same list only with a different order. Order is irrelevant to ability to fill a hotel.
    If a list of names is sufficient to occupy all rooms, then it doesn't matter in which order you put them. Because order is an irrelevant factor.

    Quote Originally Posted by CLIVE
    What makes you think ng1 is contained in g1 "when listed by room number"?
    Because g1 and r1 lists are equal and interchangeable.
    If Paul(g1) is in (r1) then we can do away with the G1 or R1 list and use them interchangeably.
    This is done in the practical way of referring to the room with the order of fish.

    As we know that g1 does not contain ng1, then to say that r1 contains ng1 is to say that ng1 is contained in the list of g1.


    Quote Originally Posted by CLIVE
    (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.
    Time issue is not my objection. My objection is that all possible solutions contain the same aspect, and by repeating to infinity a process in which no single instance solves the problem it is proclaimed to be solved.
    To serve man.

  2. #42
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    None of your distinctions make sense to me. What is an "imaginary set"? What is a "set in reality"? What makes a set "real"?
    Real, meaning the objects in the set are real, not representational. So the planets in the solar system are a set, and the set contains the 8 planets. The planets are actual planets, they are real things that exist. However I can make a set of the planets in the Alderan system. So far as we know those are imagined planets. they don't exist, they are just names of imaginary things.

    Between that is a set of real things that are only representative. For instance when dollars were backed by gold, you could say that any given dollar represented gold, but a stack of bills was a set of paper, not a set of gold. It was only representational a stand in.

    The set of us presidents is that, a set of names. The names were real people, but those people no longer actually exist as such thus the set cannot be realized, only represented.

    Why does all this matter?
    Because this is a discussion of the nature of reality. Using sets of imaginary things and then claiming them impossible to is not a way to show what reality is. Of you want to disclaim a description of reality by saying it is impossible, then you need to disclaim an actual real example, not an imagined or representative one.

    In reality, time does not add up in an infinite pile. One moment passes into the next in a single continuum of a changing universe. It is not an infinite set of separate moments that all co-exist and thus become impossible. It is a single unity of reality which changes and there is no practical limit to how many permutations it could have.
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  3. #43
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    That is how all actual process work, everything else is simply a multiple of that process.
    Please support this claim.

    Quote Originally Posted by MindTrap
    What significant difference is there in your 1A from my #1?
    In your example, there are only a finite number of rooms emptied at any one time. After my first step, every room is empty.

    How is that significantly different from asking them to return to their rooms? In both cases the room was previously occupied by someone in list g1.

    By supposing that them leaving the room and entering a different room (all previously occupied by the list g1)
    Yes, but your objection wasn't just that a new guest was entering a previously-occupied room. Your objection was that accommodating a new guest would result in an infinite chain of events where an old guest is being displaced. Both of my methods remove your objection.

    [quote]You are saying that G1 is both sufficient and insufficient to occupy all the rooms in the hotel. That is a contradiction.

    First, what does it mean to be "sufficient to occupy all the rooms in the hotel"?

    I propose the following definition:

    (D5) A set of guests G is sufficient to occupy all the rooms of a hotel Hif and only ifthere exists a room assignment R: G -> H such that every room in H is assigned a guest in G.

    What does it mean to be "insufficient to occupy all the rooms in the hotel"? I propose the following definition:

    (D6)
    A set of guests G is insufficient to occupy all the rooms of a hotel H if and only if G is not sufficient to occupy all the rooms of H according to definition (D5).

    That is, G is "insufficient to occupy all the rooms of H" if and only if every room assignment R: G -> H leaves at least one room unoccupied by a guest in G.

    According to these definitions, your statement that G1 is "insufficient to occupy all the rooms in the hotel" is false, since there exists at least one room assignment for G1 that fully occupies H.

    I would say that it isn't materially different. There is no reason to think that the list with John/Paul would be sufficient to occupy all rooms. but the list Paul/John is not sufficient. If the list is sufficient, then the order is irrelevant.
    I agree. The question is just about maps from the list of guests (without respect to order) to the set of rooms.

    Yes, because they are really the same list only with a different order. Order is irrelevant to ability to fill a hotel.
    If a list of names is sufficient to occupy all rooms, then it doesn't matter in which order you put them. Because order is an irrelevant factor.
    I agree.

    Because g1 and r1 lists are equal and interchangeable.
    If Paul(g1) is in (r1) then we can do away with the G1 or R1 list and use them interchangeably.
    This is done in the practical way of referring to the room with the order of fish.

    As we know that g1 does not contain ng1, then to say that r1 contains ng1 is to say that ng1 is contained in the list of g1.
    I have no idea what you're talking about. What precisely is your objection with regard to g1, ng1, and ng1+g1? To my understanding, all of the following statements are true:

    1) g1 and ng1 have no names in common;
    2) g1 is sufficient to fill the hotel;
    3) g1+ng1 is sufficient to fill the hotel;
    4) There are ways of accommodating every guest in g1 while leaving certain rooms in the hotel unoccupied;
    5) There are the same number of guests in each of the lists g1, ng1, and g1+ng1

    I don't see any contradiction in any of these statements. What precisely is your objection with regard to the lists g1, ng1, and g1+ng1?

    Time issue is not my objection. My objection is that all possible solutions contain the same aspect, and by repeating to infinity a process in which no single instance solves the problem it is proclaimed to be solved.
    This is a slightly separate discussion, but consider the problem of constructing an infinite set.

    Consider the finite set {0}. This set contains one element, the number 0. In general, the finite set {k} contains a single element, the integer k.

    Now, consider the union of all sets of the form {k} where k is an integer. This set is equal to the set of all integers, Z. Even though no finite union of finite sets results in an infinite set, an infinite union of finite sets can.



    You're talking about something slightly different, though. You're talking about a process, something like this:

    Consider the process of successively adding elements to form the set of non-negative integers.

    First, let S0 = {0}. Recursively define Sk+1 = Sk U {k+1} = {0,1,2, ..., k, k+1}

    Now, consider the infinite sequence {Sn} = {S0, S1, S2, ..., Sk, Sk+1, ...}

    Note that there is no point in this process where the "last" non-negative integer is added. None of the Sk's are infinite.




    Your argument is that any process like "successively adding elements" to a finite set will never result in an infinite set. I agree with that, assuming that by "never" you mean "there is never a step after which the result is infinite".
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  4. #44
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    Please support this claim.
    Well, my example had one guest leaving one room.
    Your example had infinite guests leaving infinite rooms.

    That is simply my example multiplied by infinity. Yes?

    That not withstanding, there is no substantive difference between one person leaving a room and a million or an infinite.
    The problem of one is not solved by multiplying or repeating.

    Quote Originally Posted by clive
    In your example, there are only a finite number of rooms emptied at any one time. After my first step, every room is empty.
    They are not significantly different, because every room would have previously had a person from g1 in it.
    The problem doesn't change because it is done "all at once".

    For example, Once they all exit their rooms(in step one) it is true that every room previously contained one from the g1 list.
    As it is true that any room they enter will be one previously occupied by a member of g1, then there is no difference if they re-entered their own previous room.
    They are equally true.
    What it does do is keep the problem in the immediate example instead of pushing it off into infinity.

    Quote Originally Posted by CLIVE
    Yes, but your objection wasn't just that a new guest was entering a previously-occupied room. Your objection was that accommodating a new guest would result in an infinite chain of events where an old guest is being displaced. Both of my methods remove your objection.
    By displacing them all at once.
    You are correct it wasn't "just" that, but it was included and very much essential to the point.
    You address a kind of timing issue, which is not the heart of my objection.

    Quote Originally Posted by CLIVE
    First, what does it mean to be "sufficient to occupy all the rooms in the hotel"?

    Quote Originally Posted by DEFINE SUFFICIENT
    1. adequate for the purpose; enough.
    2. Logic. (of a condition) such that its existence leads to the occurrence of a given event or the existence of a given thing. Compare necessary (def. 4c).
    3. Archaic. competent.
    In this case the second sense. g1 is sufficient for the hotel to have the status of "all rooms occupied" and it is also Insufficient for the hotel to have the status of "all rooms occupied".

    http://www.thefreedictionary.com/sufficient

    So the existence of g1 in the hotel and each with a room, leads to the occurrence and doesn't lead to the occurrence of having all the rooms in the hotel occupied.
    In the same sense. Here the sense is that they are each assigned a single room. The act of doing so will either fill the hotel or not.

    Quote Originally Posted by CLIVE
    I propose the following definition:

    (D5) A set of guests G is sufficient to occupy all the rooms of a hotel Hif and only ifthere exists a room assignment R: G -> H such that every room in H is assigned a guest in G.
    Can we define it as
    List G1 is sufficient to occupy all rooms if and only if the assignment of g1 to a room precludes any alternative assignments.

    Hence, a hotel can only be said to be "full" if it can not take on any new guests. When a hotel can assign a room to a new guest even if it requires the shuffling of other rooms, then it can not be said to be "full".

    Your definition does not account for the ability of a hotel to accommodate new guests. Which is inherently precluded by a definition of "full".


    Quote Originally Posted by DEFINE FULL
    1. Containing all that is normal or possible:
    http://www.thefreedictionary.com/full

    If there exists an order in which more can "fit" then it doesn't contain all that is possible and is thus not "full".

    Quote Originally Posted by CLIVE
    What does it mean to be "insufficient to occupy all the rooms in the hotel"? I propose the following definition: (D6) A set of guests G is insufficient to occupy all the rooms of a hotel H if and only if G is not sufficient to occupy all the rooms of H according to definition (D5).
    I would say see my last.

    Quote Originally Posted by CLIVE
    I agree. The question is just about maps from the list of guests (without respect to order) to the set of rooms.
    Right, so the contradiction is expressed (using my definitions) as this.
    That list G1 is sufficient to fill all the rooms in the hotel, but insufficient to fill the hotel, which is said to occur any time all the rooms are occupied.


    Quote Originally Posted by CLIVE
    I have no idea what you're talking about. What precisely is your objection with regard to g1, ng1, and ng1+g1? To my understanding, all of the following statements are true:

    1) g1 and ng1 have no names in common;
    2) g1 is sufficient to fill the hotel;
    3) g1+ng1 is sufficient to fill the hotel;
    4) There are ways of accommodating every guest in g1 while leaving certain rooms in the hotel unoccupied;
    5) There are the same number of guests in each of the lists g1, ng1, and g1+ng1

    I don't see any contradiction in any of these statements. What precisely is your objection with regard to the lists g1, ng1, and g1+ng1?
    I withdraw this point for now in the interest of stream lining.

    Quote Originally Posted by CLIVE
    This is a slightly separate discussion, but consider the problem of constructing an infinite set.

    Consider the finite set {0}. This set contains one element, the number 0. In general, the finite set {k} contains a single element, the integer k.

    Now, consider the union of all sets of the form {k} where k is an integer. This set is equal to the set of all integers, Z. Even though no finite union of finite sets results in an infinite set, an infinite union of finite sets can.
    Yea, being in construction I try to avoid building infinite things.. they take forever.


    Quote Originally Posted by CLIVE
    You're talking about something slightly different, though. You're talking about a process, something like this:

    Consider the process of successively adding elements to form the set of non-negative integers.

    First, let S0 = {0}. Recursively define Sk+1 = Sk U {k+1} = {0,1,2, ..., k, k+1}

    Now, consider the infinite sequence {Sn} = {S0, S1, S2, ..., Sk, Sk+1, ...}

    Note that there is no point in this process where the "last" non-negative integer is added. None of the Sk's are infinite.
    yea, it is a kind of what I'm saying.
    I'm not so much concerned about the process itself, only the results.

    If in the sequence, we know that there are no eggs contained in S0 and all of the sequence is the same, then the infinite set would not produce an egg.
    So I say that for every S- there exists a condition of 2 guests and one room. Then the speed at which it occurs is irrelevant. That it would occur simultaneously does not add an element which would change any of the individual facts.



    Quote Originally Posted by CLIVE
    Your argument is that any process like "successively adding elements" to a finite set will never result in an infinite set. I agree with that, assuming that by "never" you mean "there is never a step after which the result is infinite".
    Well, my argument is intended to convey that the whole set (infinite) does not solve the problem of it's individual parts.

    The only point of bringing up the successive parts, is to bring home that the whole set consists of the same problem.
    To serve man.

  5. #45
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    The short answer is...
    Yes or no. You don't get to re-phrase my question for me,and then answer it. The one I asked can be sufficiently answered by a simple yes or no. Your revision of it cannot.

    You know, I don't even need you to acknowledge the Grand Hotel is full. The simple truth here is that it is according to everybody I can find except you!

    The relevant part of this video is between 0:50 and 1:00


    I'm only allowed one video per post, so in this next I'm only going to give the url, and point out the relevant proposition is stated between 3:13 and 4:30.

    http://www.youtube.com/watch?v=bgvxVUyVdXk

    Here are some more textual sources online, all of them claiming all the rooms are occupied, entailing the hotel is full, where they do not expressly state it is full:

    https://www.maths.nottingham.ac.uk/p...d-Infinity.pdf (p. 17)

    http://wanda.uef.fi/matematiikka/kur...i/Infinite.pdf

    quoting from this source:

    1.1. Situation 1. The Hotel is full and a new guest arrives.

    1.2. Situation 2. The Hotel is full and each guest has one friend coming.

    1.3 Situation 3. The Hotel is full and each guest has nine friends coming.

    1.4 Situation 4. The Hotel is full and each guest has an infinite (sic) many friends coming.
    Every source I look at; source after source after source, says the Grand Hotel is full. I'm sure if David Hilbert were alive today, and I asked him, "Is the Grand Hotel full?", he not hesitate to say "yes". So why can't you?

  6. #46
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    Yes or no. You don't get to re-phrase my question for me,and then answer it. The one I asked can be sufficiently answered by a simple yes or no. Your revision of it cannot.

    You know, I don't even need you to acknowledge the Grand Hotel is full. The simple truth here is that it is according to everybody I can find except you!

    The relevant part of this video is between 0:50 and 1:00


    I'm only allowed one video per post, so in this next I'm only going to give the url, and point out the relevant proposition is stated between 3:13 and 4:30.

    http://www.youtube.com/watch?v=bgvxVUyVdXk

    Here are some more textual sources online, all of them claiming all the rooms are occupied, entailing the hotel is full, where they do not expressly state it is full:

    https://www.maths.nottingham.ac.uk/p...d-Infinity.pdf (p. 17)

    http://wanda.uef.fi/matematiikka/kur...i/Infinite.pdf

    quoting from this source:



    Every source I look at; source after source after source, says the Grand Hotel is full. I'm sure if David Hilbert were alive today, and I asked him, "Is the Grand Hotel full?", he not hesitate to say "yes". So why can't you?
    For the purposes of his thought experiment, it probably made sense to stipulate that the Hotel was full. My thought experiment does not make that stipulation, although it is a possible state of affairs within the domain of my thought experiment.

    The purpose of my thread is to analyze what WLC has claimed are "absurdities" that disqualify an infinite hotel from existing in actuality. These claimed absurdities follow from various properties of the Hotel that relate to accommodating various sets of guests. Stipulating that the hotel must always "begin" full when a "next" guest arrives is unnecessarily restrictive; if someone wants to consider such a case, they need only to frame their question properly, e.g. "If a hotel H is is "fully occupied" by a set of guests G, how could the hotel accommodate a new guest in addition to the guests in G?"

    With this in mind, your entire line of questioning is a pointless red herring. Nowhere have you shown that I have neglected to consider some important state of affairs with regard to the hotel; indeed, it would be impossible to show as such, since I have already specifically talked about how a "fully occupied" infinite hotel would accommodate an additional guest. Nowhere have you shown that any of my analysis is wrong.


    The only difference between my Hotel and Hilbert's so far is that I don't require that every analysis begin with the assumption that the Hotel is full...So what?

    ---------- Post added at 02:01 AM ---------- Previous post was at 01:34 AM ----------

    Quote Originally Posted by MindTrap028 View Post
    Well, my example had one guest leaving one room.
    Your example had infinite guests leaving infinite rooms.

    That is simply my example multiplied by infinity. Yes?
    Your example is done in sequence; as you state later, at no point in the sequence will every guest occupy their "new" room. My method addresses this issue.

    That not withstanding, there is no substantive difference between one person leaving a room and a million or an infinite.
    The problem of one is not solved by multiplying or repeating.
    What problem is that, again? You've mentioned something vague about there being more guests than rooms, but I'm not sure what the precise problem is.

    They are not significantly different, because every room would have previously had a person from g1 in it.
    The problem doesn't change because it is done "all at once".

    For example, Once they all exit their rooms(in step one) it is true that every room previously contained one from the g1 list.
    As it is true that any room they enter will be one previously occupied by a member of g1, then there is no difference if they re-entered their own previous room.
    They are equally true.
    What it does do is keep the problem in the immediate example instead of pushing it off into infinity.
    I have no idea what the problem is that you're talking about.

    By displacing them all at once.
    You are correct it wasn't "just" that, but it was included and very much essential to the point.
    You address a kind of timing issue, which is not the heart of my objection.
    Okay.

    In this case the second sense. g1 is sufficient for the hotel to have the status of "all rooms occupied" and it is also Insufficient for the hotel to have the status of "all rooms occupied".

    http://www.thefreedictionary.com/sufficient

    So the existence of g1 in the hotel and each with a room, leads to the occurrence and doesn't lead to the occurrence of having all the rooms in the hotel occupied.
    In the same sense. Here the sense is that they are each assigned a single room. The act of doing so will either fill the hotel or not.
    Wait, what? You refuse to be precise in your phrasing. Please be precise.

    What you're saying, if I've managed to guess correctly, is something like this:

    (1) The guests in G1 can fully occupy the hotel.
    (2) There are ways to assign rooms to guests in G1 in such a way that the hotel is not fully occupied.
    (3) (2) contradicts (1).

    But what is the contradiction? If (2) is the negation of (1), then we'd immediately have a contradiction. But (2) is not the negation of (1):

    (~1) The guests in G1 cannot full occupy the hotel. (I.e., every room assignment for guests in G1 fails to fully occupy the hotel.)



    Do you see the difference? It's about quantification.

    (2) merely asserts that at least one room assignment exists that has the property "at least one room is unoccupied".
    (~1) asserts that all room assignments have the property "at least one room is unoccupied".

    These are not logically equivalent. Since (2) is not logically equivalent to (~1), what is the contradiction that you're claiming exists?

    Can we define it as
    List G1 is sufficient to occupy all rooms if and only if the assignment of g1 to a room precludes any alternative assignments.

    Hence, a hotel can only be said to be "full" if it can not take on any new guests. When a hotel can assign a room to a new guest even if it requires the shuffling of other rooms, then it can not be said to be "full".
    What does it mean to "preclude any alternative assignments"? You couldn't assign rooms to guests in G1 in any other way?

    Your definition does not account for the ability of a hotel to accommodate new guests. Which is inherently precluded by a definition of "full".
    I assumed being "full" meant "having every room occupied", not "unable to accept more guests".

    Under your definition, infinite hotels are never "full" and thus cannot be both "full" and "not full" at the same time. I have already shown that an infinite hotel can always accommodate additional guests (so long as the total number of guests does not exceed the number of rooms in the hotel). So what's the contradiction?

    I would say see my last.


    Right, so the contradiction is expressed (using my definitions) as this.
    That list G1 is sufficient to fill all the rooms in the hotel, but insufficient to fill the hotel, which is said to occur any time all the rooms are occupied.
    This is the fallacy of equivocation. You are defining "full" to mean "incapable of accommodating more guests" and then later claiming that it means "every room is occupied". These definitions are not equivalent (although they are equivalent for hotels with a finite number of rooms). No infinite hotel is every full; thus "G1 is sufficient to fill all the rooms in the hotel" is false in your sense of the term fill.

    If in the sequence, we know that there are no eggs contained in S0 and all of the sequence is the same, then the infinite set would not produce an egg.
    So I say that for every S- there exists a condition of 2 guests and one room. Then the speed at which it occurs is irrelevant. That it would occur simultaneously does not add an element which would change any of the individual facts.
    I'm completely at a loss as to what the problem is. Some new guest N comes in, and takes G1's room. G1 takes G2's room. And so on. You're saying that during the time that N is moving into G1's room, there are "two guests" and "one room". No, there are an infinite number of guests and an infinite number of rooms.

    If everyone moves out of their room at the same time, how many empty rooms are there? Infinity. How many guests are to be accommodated now, with the additional new guest? Infinity. There are just as many rooms as guests, so each guest can have his or her own room. Can you please explain what the hell the problem is that you're talking about?
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    For the purposes of his thought experiment, it probably made sense to stipulate that the Hotel was full. My thought experiment does not make that stipulation, although it is a possible state of affairs within the domain of my thought experiment.
    Then my question is valid.

    Quote Originally Posted by CliveStaples
    The purpose of my thread is to analyze what WLC has claimed are "absurdities" that disqualify an infinite hotel from existing in actuality. These claimed absurdities follow from various properties of the Hotel that relate to accommodating various sets of guests. Stipulating that the hotel must always "begin" full when a "next" guest arrives is unnecessarily restrictive; if someone wants to consider such a case, they need only to frame their question properly, e.g. "If a hotel H is is "fully occupied" by a set of guests G, how could the hotel accommodate a new guest in addition to the guests in G?"
    Says who? Who died and made you the ruler over how to ask the question? Is there any conceptual difference between your hotel and the Grand? If not, all this is bluster to escape answering the question.

    Quote Originally Posted by CliveStaples
    With this in mind, your entire line of questioning is a pointless red herring. Nowhere have you shown that I have neglected to consider some important state of affairs with regard to the hotel; indeed, it would be impossible to show as such, since I have already specifically talked about how a "fully occupied" infinite hotel would accommodate an additional guest.
    The thing in your analysis you've consistently avoided (and gone to Herculean lengths, I would add, to avoid) is that being "full" is a property a hotel either possess or it doesn't possess. If it possess that property, then by the law of the excluded middle, so long as it does possess that property, it can't posses the complementary property "not being full", and therefore cannot accommodate any additional guests in reality.

    Now, as my sources have made clear, power set operations on infinite sets make it possible in the abstract world of mathematics for a completely full hotel to accommodate infinity upon infinity of additional guests. However, I take this fact as an argument against the actual, as opposed to the abstract reality of any infinite set. Furthermore, from what I've read concerning Hilbert's Grand Hotel, so did David Hilbert.

    Quote Originally Posted by CliveStaples
    Nowhere have you shown that any of my analysis is wrong.
    Don't need to until you answer my question to my satisfaction, which you've now had several opportunities to do, and instead took those opportunities to berate me for asking it. Well, you're entitled to your opinions, but they don't amount to an argument, and I'm not going to cater to you by pretending they do.

    Quote Originally Posted by CliveStaples
    The only difference between my Hotel and Hilbert's so far is that I don't require that every analysis begin with the assumption that the Hotel is full...So what?
    The law of non-contradiction is what. At some point in your analysis you have to have a full hotel. Tits on a bull matter more than whether or not your analysis "starts" with a full hotel, so your objection in that regard is our "red herring" if there is any in the vicinity, not my question about a property your hotel either has to have or lacks. All that matters is that at some point it becomes full in your analysis. Until then it's not a hotel with an infinite number of rooms all occupied with guests, but rather a hotel with an indefinite number of rooms, some of which are occupied by guests,some of which, for all anyone knows, are empty. And adding a guest to that kind of hotel is not worth discussing, much less debating.

    And get a grip, okay? Act like the adult I know you are. Printing bigger doesn't make what you say true, and it certainly doesn't make me more inclined to grant it any credence.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    Says who? Who died and made you the ruler over how to ask the question? Is there any conceptual difference between your hotel and the Grand? If not, all this is bluster to escape answering the question.
    I suppose you could say that there is a conceptual difference at work here. I've provided a framework for discussing how to accommodate an arbitrary set of guests in an arbitrary set of rooms. Within that framework, I presented WLC's particular challenges to the possibility of an actual hotel with an infinite number of rooms.

    I do not stipulate that every hotel discussed in this thread must be full by definition. I do not stipulate that every hotel discussed in this thread must have an infinite number of rooms.

    In this way, my framework is more general than Hilbert's Grand Hotel (which apparently is defined to be full--at all times? At some time? Can it ever be less than full? Or would it suddenly cease to be Hilbert's Grand Hotel and become some other kind of hotel?). Any statement about Hilbert's Grand Hotel can be constructed in my framework. And I've already done so in this very thread.


    The thing in your analysis you've consistently avoided (and gone to Herculean lengths, I would add, to avoid) is that being "full" is a property a hotel either possess or it doesn't possess. If it possess that property, then by the law of the excluded middle, so long as it does possess that property, it can't posses the complementary property "not being full", and therefore cannot accommodate any additional guests in reality.
    First, hotels do not possess or lack the property of being "full" per se. Sometimes they are full, and sometimes they are not. Whether or not a hotel has the property of being "full" depends only on the set of guests being accommodated and the existence of a room assignment for those guests which assigns a guest to every room in the hotel. It only makes sense to speak of a hotel being full under some room assignment of a set of guests. A hotel does not possess the property of being "full" except insofar as there are guests assigned to each of its rooms.

    Second, and this relates to the first point, I think you might be conflating the properties "Every room is occupied" with "No more guests can be accommodated". These properties are not equivalent.

    Third, I don't think you've given a very good argument here. Your argument goes something like this, so far as I can tell:

    1) A hotel is either full or not full.
    2) If a hotel is full, it can't be "not full".
    3) Therefore, if a hotel is full, it can't accommodate any additional guests in reality.

    How does (3) follow from (1) and (2)?

    Now, as my sources have made clear, power set operations on infinite sets make it possible in the abstract world of mathematics for a completely full hotel to accommodate infinity upon infinity of additional guests. However, I take this fact as an argument against the actual, as opposed to the abstract reality of any infinite set. Furthermore, from what I've read concerning Hilbert's Grand Hotel, so did David Hilbert.
    Cool story bro. Care to give me a reason to think that it's a good argument?

    Don't need to until you answer my question to my satisfaction, which you've now had several opportunities to do, and instead took those opportunities to berate me for asking it. Well, you're entitled to your opinions, but they don't amount to an argument, and I'm not going to cater to you by pretending they do.
    I've taken every opportunity to answer your question. Again, in case you missed it previously:

    While the topic of the thread is WLC's arguments regarding actual infinities, I am not restricting the universe of discourse to only infinite hotels that are filled with guests.

    The law of non-contradiction is what. At some point in your analysis you have to have a full hotel. Tits on a bull matter more than whether or not your analysis "starts" with a full hotel, so your objection in that regard is our "red herring" if there is any in the vicinity, not my question about a property your hotel either has to have or lacks. All that matters is that at some point it becomes full in your analysis. Until then it's not a hotel with an infinite number of rooms all occupied with guests, but rather a hotel with an indefinite number of rooms, some of which are occupied by guests,some of which, for all anyone knows, are empty. And adding a guest to that kind of hotel is not worth discussing, much less debating.
    How is the law of non-contradiction relevant? What contradiction follows from a fully-occupied infinite hotel being able to accommodate an additional guest?

    And get a grip, okay? Act like the adult I know you are. Printing bigger doesn't make what you say true, and it certainly doesn't make me more inclined to grant it any credence.
    Dude, you're asking the most irrelevant questions.

    Whether or not the infinite hotel is full by definition or its fullness occurs as a premise in an argument is completely irrelevant to anything going on in this thread. Harping on it, and pointing out how various other philosophers have gone about it, is a giant waste of time. A waste of your time and a waste of my time.

    You don't really have grounds to talk about "acting like an adult" while you're wasting the time of anyone who makes the mistake of reading, let alone responding to, your posts.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    I assumed being "full" meant "having every room occupied", not "unable to accept more guests".
    Why would you assume that?
    No matter, I have quoted the proper meaning for you so your assumption should change. Yes?

    Especially because it is a significant part of the objection.
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    Re: WLC's Argument Against an Actual Infinity

    Lets remember that we are all adults and that we are all doing this voluntarily gentlemen, I understand the frustrations that come with these kind of complex subjects, but lets keep the snark to professional level please?
    "Suffering lies not with inequality, but with dependence." -Voltaire
    "Fallacies do not cease to be fallacies because they become fashions.” -G.K. Chesterton
    Also, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.


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    Re: WLC's Argument Against an Actual Infinity

    Okay, it is as clear to me as you can make it you are not going to address arguments presented you that you do not wish to address. You've been given opportunity after opportunity to answer a simple question about a particular property of the hotel WLC argues cannot exist in reality, and you've passed on every single one of them.

    So I'm going to merely explain where what you have to say below again misses the mark, and from that point on ignore your blustering to the point of breaking ODN rules. I'm sure your blog is awesome, I'm sure you're the greatest mathematician on planet ODN, just as I'm sure you're avoiding my point with all your might. It's not enough for you to simply label it (more correctly malign it), you have to demonstrate why it's irrelevant, and do so without relying on irrelevancies yourself. You don't seem to want to shoulder that obligation, I can't force you to, so after this "last word" I'm out of here.

    Quote Originally Posted by CliveStaples View Post
    I suppose you could say that there is a conceptual difference at work here. I've provided a framework for discussing how to accommodate an arbitrary set of guests in an arbitrary set of rooms.
    Either that's a gross misrepresentation of what you've been doing in this thread, or you've grossly misrepresented your purpose in choosing it's title, "Re: WLC's Argument Against an Actual Infinity". Everyone who has ever read that argument, including me, knows one of the actual infinities WLC argues against is an actual hotel conceptually identical to Hilbert's Grand Hotel.

    Therefore, if your "framework" does not do the same, or even allows you not to do the same, you are not then using WLC's actual argument in analyzing it, and your entire thread has become a red herring! That's fatal flaw #1.

    Quote Originally Posted by CliveStaples
    I do not stipulate that every hotel discussed in this thread must be full by definition. I do not stipulate that every hotel discussed in this thread must have an infinite number of rooms.
    Which is a restatement of Flaw #1. WLC, to the extent that he treats hotels in arguing against an actual infinite relies exclusively on David Hilbert's Grand Hotel, which does have an infinite number of rooms at all times, and as a matter of fact is "full" during the entire argument examining whether or not an actual infinity can exist.

    Quote Originally Posted by CliveStaples
    In this way, my framework is more general than Hilbert's Grand Hotel (which apparently is defined to be full--at all times? At some time? Can it ever be less than full? Or would it suddenly cease to be Hilbert's Grand Hotel and become some other kind of hotel?).
    Now you're just being embarrassingly obtuse.

    Quote Originally Posted by CliveStaples
    Any statement about Hilbert's Grand Hotel can be constructed in my framework. And I've already done so in this very thread.
    Look, the only "hotel" of any interest in your thread is one with an infinite number of rooms, with a guest in each of them, and an additional guest wanting to check in. If your "framework" is "more general" than this, who gives a hoot? To the extent it is, to that extent it's not about WLC's argument against an actual infinity!

    Quote Originally Posted by CliveStaples
    First, hotels do not possess or lack the property of being "full" per se. Sometimes they are full, and sometimes they are not.
    This is incredible! If I were to say something like this to you in one of my threads that relied on the logic of properties, you'd be down my throat, and you'd be absolutely correct to do so.

    No one has ever even hinted, and certainly not me, that hotels have the property "being full" per se. The very idea is ludicrous. That said hotels described as having a guest in every room necessarily have that property. Because that is a necessary truth, your "model", "calculations", "framework", "argument", whatever you'd like it to be known as has only two options: accommodate this necessary truth, or be false. You apparently want it to be neither, and if you don't get your way, are going to fling yourself on the ground, stamp your feet, and yell.

    Quote Originally Posted by CliveStaples
    Whether or not a hotel has the property of being "full" depends only on the set of guests being accommodated and the existence of a room assignment for those guests which assigns a guest to every room in the hotel.
    True.

    Quote Originally Posted by CliveStaples
    It only makes sense to speak of a hotel being full under some room assignment of a set of guests. A hotel does not possess the property of being "full" except insofar as there are guests assigned to each of its rooms.
    False. Another way it makes perfect sense is to speak of a hotel that is full is to speak of one that has no empty rooms. We don't need to do a one-to-one correspondence between empty rooms and guest assignments to, in reality, decide whether or not a hotel is "full". And a hotel that is full, and full hotels cannot possess the property "having an empty room".

    Now, I understand why you don't want to take on this awkward truth in this thread. It's darned inconvenient for your "framework".

    Quote Originally Posted by CliveStaples
    Second, and this relates to the first point, I think you might be conflating the properties "Every room is occupied" with "No more guests can be accommodated". These properties are not equivalent.
    Well, you're wrong yet again. First of all, a hotel having the property "being such that no more guests can be accommodated" is not necessarily a hotel that has the property "being full". You're either being obtuse again, or hinting this isn't clear to me. But it is clear to me, which is why I chose to appeal to the latter property in critiquing your analysis, not the former you just suggested. Now, we can articulate the same state of affairs using different words, and this is the case with your "Every room is occupied" above. Any hotel that enjoys the property "being full" also has the property "being such that every room is occupied". These are not two distinct properties, but only one property expressed in two distinct ways.

    Quote Originally Posted by CliveStaples
    Third, I don't think you've given a very good argument here. Your argument goes something like this, so far as I can tell:

    1) A hotel is either full or not full.
    2) If a hotel is full, it can't be "not full".
    3) Therefore, if a hotel is full, it can't accommodate any additional guests in reality.

    How does (3) follow from (1) and (2)?
    Because a full hotel and one with no unoccupied rooms available are equivalent states of affairs by definition. I usually give people more respect than to condescend to quoting dictionary definitions to them in their mother tongue, but when faced with obtuseness presented as a challenge, you leave me little choice.

    Full:
    1. completely filled; containing all that can be held; filled to utmost capacity: a full cup (dictionary.com Unabridged)

    Quote Originally Posted by CliveStaples
    I've taken every opportunity to answer your question. Again, in case you missed it previously:

    While the topic of the thread is WLC's arguments regarding actual infinities, I am not restricting the universe of discourse to only infinite hotels that are filled with guests.
    Real mature. Bye-bye.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    Why would you assume that?
    No matter, I have quoted the proper meaning for you so your assumption should change. Yes?

    Especially because it is a significant part of the objection.
    If you define "full" to mean "unable to accommodate [in the sense of definition (D3)] any more guests", then I don't understand your objection at all.

    An infinite hotel is never "full" in your sense, since an infinite hotel can always accommodate a countably-infinite set of additional guests.

    If the contradiction you intend to demonstrate is something like, "An infinite hotel is both full and not full", then please show me how you can prove that an infinite hotel is ever "full" in your sense of the term.

    ---------- Post added at 01:25 AM ---------- Previous post was at 01:12 AM ----------

    Quote Originally Posted by cstamford View Post
    Okay, it is as clear to me as you can make it you are not going to address arguments presented you that you do not wish to address. You've been given opportunity after opportunity to answer a simple question about a particular property of the hotel WLC argues cannot exist in reality, and you've passed on every single one of them.

    So I'm going to merely explain where what you have to say below again misses the mark, and from that point on ignore your blustering to the point of breaking ODN rules. I'm sure your blog is awesome, I'm sure you're the greatest mathematician on planet ODN, just as I'm sure you're avoiding my point with all your might. It's not enough for you to simply label it (more correctly malign it), you have to demonstrate why it's irrelevant, and do so without relying on irrelevancies yourself. You don't seem to want to shoulder that obligation, I can't force you to, so after this "last word" I'm out of here.
    I've answered your question about 3 times so far.

    None of the hotels, as I've defined them, are ever defined to be full.

    Either that's a gross misrepresentation of what you've been doing in this thread, or you've grossly misrepresented your purpose in choosing it's title, "Re: WLC's Argument Against an Actual Infinity". Everyone who has ever read that argument, including me, knows one of the actual infinities WLC argues against is an actual hotel conceptually identical to Hilbert's Grand Hotel.

    Therefore, if your "framework" does not do the same, or even allows you not to do the same, you are not then using WLC's actual argument in analyzing it, and your entire thread has become a red herring! That's fatal flaw #1.
    My framework does allow the consideration of Hilbert's Grand Hotel, so your objection here is unfounded.

    Which is a restatement of Flaw #1. WLC, to the extent that he treats hotels in arguing against an actual infinite relies exclusively on David Hilbert's Grand Hotel, which does have an infinite number of rooms at all times, and as a matter of fact is "full" during the entire argument examining whether or not an actual infinity can exist.


    Now you're just being embarrassingly obtuse.



    Look, the only "hotel" of any interest in your thread is one with an infinite number of rooms, with a guest in each of them, and an additional guest wanting to check in. If your "framework" is "more general" than this, who gives a hoot? To the extent it is, to that extent it's not about WLC's argument against an actual infinity!



    This is incredible! If I were to say something like this to you in one of my threads that relied on the logic of properties, you'd be down my throat, and you'd be absolutely correct to do so.

    No one has ever even hinted, and certainly not me, that hotels have the property "being full" per se. The very idea is ludicrous. That said hotels described as having a guest in every room necessarily have that property. Because that is a necessary truth, your "model", "calculations", "framework", "argument", whatever you'd like it to be known as has only two options: accommodate this necessary truth, or be false. You apparently want it to be neither, and if you don't get your way, are going to fling yourself on the ground, stamp your feet, and yell.



    True.



    False. Another way it makes perfect sense is to speak of a hotel that is full is to speak of one that has no empty rooms. We don't need to do a one-to-one correspondence between empty rooms and guest assignments to, in reality, decide whether or not a hotel is "full". And a hotel that is full, and full hotels cannot possess the property "having an empty room".

    Now, I understand why you don't want to take on this awkward truth in this thread. It's darned inconvenient for your "framework".
    Rooms are only "empty" under some assignment of rooms to guests. I don't care if the room happens to be occupied by, say, a maid, or a bug, or a Room Inspector. The only kind of "empty" room that is relevant to WLC's argument is a room that has been left unoccupied under some particular room assignment being considered by the interlocutors.


    Well, you're wrong yet again. First of all, a hotel having the property "being such that no more guests can be accommodated" is not necessarily a hotel that has the property "being full". You're either being obtuse again, or hinting this isn't clear to me. But it is clear to me, which is why I chose to appeal to the latter property in critiquing your analysis, not the former you just suggested. Now, we can articulate the same state of affairs using different words, and this is the case with your "Every room is occupied" above. Any hotel that enjoys the property "being full" also has the property "being such that every room is occupied". These are not two distinct properties, but only one property expressed in two distinct ways.
    Well, MindTrap's definition of a hotel being "full" is precisely that the hotel is incapable of accommodating additional guests. So you can take that up with him.

    Because a full hotel and one with no unoccupied rooms available are equivalent states of affairs by definition. I usually give people more respect than to condescend to quoting dictionary definitions to them in their mother tongue, but when faced with obtuseness presented as a challenge, you leave me little choice.

    Full:
    1. completely filled; containing all that can be held; filled to utmost capacity: a full cup (dictionary.com Unabridged)
    Equivalent "by definition"? You might define them to be equivalent, but can you show it? Or are you asserting that there is an explicit contradiction?

    Consider the following set of propositions:

    (1) H has every room occupied
    (2) H is able to accommodate more guests

    Are you asserting that these two are explicitly contradictory? That is, that (1) is the negation of (2)?

    The negations of (1) and (2) are as follows:

    (1') H has an unoccupied room
    (2') H is unable to accommodate more guests

    I don't see how (1') and (2) are equivalent, unless you allow that a room "becomes" unoccupied for some amount of time after its old occupants have left and before its new occupants have arrived. (1) and (2') are not logically equivalent, since an infinite hotel can accommodate more guests (as I demonstrated earlier in the thread).

    Of course, you haven't exhibited any contradiction, nor presented any reason that I should believe so other than your statement that "fully occupied" and "unable to accommodate more guests" are contradictory by definition, which I suppose I am to merely take your word on.


    If you have any arguments or reasons for me to think that there is a contradiction, please present them. Don't just say there is one. Show it.

    Real mature. Bye-bye.
    Come back when you have an argument to offer.
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    Re: WLC's Argument Against an Actual Infinity

    Dear Everyone On This Thread:

    Everything that CliveStaples has said are very basic facts of mathematics that thousands of undergraduate students in mathematics and mathematical philosophy around the world are taught, including both CliveStaples and myself several years ago. There's absolutely nothing contentious or controversial about what he has said. Not understanding a technical detail does not make your argument valid. It means that you don't understand the technical detail, and this isn't undone or overcome by making an completely abject argument from incredulity.

    A fun historical fact that William Lane Craig got completely assbackwards is the part where he claimed that Hilbert used Hilbert's Hotel as an example of how actual infinities cannot exist. David Hilbert invented this thought experiment to demonstrate that properties of infinity are non-trivial and counter-intuitive, but are nevertheless true.

    --GP
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    Re: WLC's Argument Against an Actual Infinity

    Dear GP
    We appreciate your nonspecific generalizations of the thread and your disconnected from any specific argument critique as well as your expression of your personal opinions.

    Your opinion is duly noted. Thanks for stopping by.
    You may note that the posts have a "like" option for you to express your approval of posts you agree with or feel have reflected your thoughts well.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by GoldPhoenix View Post
    A fun historical fact that William Lane Craig got completely assbackwards is the part where he claimed that Hilbert used Hilbert's Hotel as an example of how actual infinities cannot exist. David Hilbert invented this thought experiment to demonstrate that properties of infinity are non-trivial and counter-intuitive, but are nevertheless true.
    Sometimes I'm amazed you say these things.

    You realize of course that Hilbert is talking about a countably infinite set. Craig is talking about a physical infinite set formed by successive addition. Those are two different things.


    Part of the problem with this thread, and I wouldn't expect GP to know this since he almost certainly has not read the work that generated it, is that Craig is not saying that infinities cannot exist. He uses Hilbert specifically to show that infinite sets are unintuitive, not that infinite sets are impossible. He uses Hilbert to show that an physical set of regressive events cannot be infinite because you cannot have an infinite history by successive addition.
    "Suffering lies not with inequality, but with dependence." -Voltaire
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347 View Post
    Part of the problem with this thread, and I wouldn't expect GP to know this since he almost certainly has not read the work that generated it, is that Craig is not saying that infinities cannot exist. He uses Hilbert specifically to show that infinite sets are unintuitive, not that infinite sets are impossible. He uses Hilbert to show that an physical set of regressive events cannot be infinite because you cannot have an infinite history by successive addition.
    Two reponses:

    1) WLC has claimed that an actual infinity is impossible due to the above "absurdities" an actual infinity would entail.
    2) You can get an infinite set from successive "addition" (which I take to mean the operation of union). Consider:

    N = {0,1,2,...}
    = {0} U {1} U {2} U ... U {k} U {k+1} U ...
    = U{n | n is a member of N}

    The last line is in set-builder notation. It can be expressed in English as "The union of all sets of the form {n}, where n is an element in N."

    The equalities above can be proved formally; to do so would require certain definitions (what a set union is, what it means to be a member of a union of sets, etc.). The proof is quite simple with these definitions in mind.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    1) WLC has claimed that an actual infinity is impossible due to the above "absurdities" an actual infinity would entail.
    In a physical environment yes. Specifically, he raises the objection I raised earlier in thread, that an actual infinitely constructed of successive addition could not have fusion producing stars, since all stars would have died out an infinitely long time ago.

    Quote Originally Posted by CS
    2) You can get an infinite set from successive "addition" (which I take to mean the operation of union).
    And to my knowledge Craig has no issue with that statement. He argues you can't get an actual infinite from successive addition, specifically referring to our temporal context.

    Regardless, I would be surprised if you could show where Craig argues that infinite sets are impossible in his works.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    Equivalent "by definition"? You might define them to be equivalent, but can you show it? Or are you asserting that there is an explicit contradiction?
    Against my better judgement, I'm going to give this one more try. And what follows does so baby step by baby step, so please don't jump ahead.

    What I said was, "Because a full hotel and one with no unoccupied rooms available are equivalent states of affairs by definition."

    Now, is there some logical flaw in that statement? Is it necessarily false? You're taking issue with it, but you're not explaining exactly why it's wrong. Instead, you reformulate it first, and then explain why the reformulation is wrong, and you've done this repeatedly. To wit, and yet again:

    Consider the following set of propositions:

    (1) H has every room occupied
    (2) H is able to accommodate more guests

    Are you asserting that these two are explicitly contradictory? That is, that (1) is the negation of (2)?
    I'm claiming that the H in (2) lacks a property the H in (1) has; i.e. the property "being full".

    Now, for me, when we have two Hs that do not share all their properties, they cannot be identical, and so it's irrelevant whether or not (1) and (2) are formally logically contradictory or not.

    What I'm saying is we have to re-write your above as:

    (1) H1 has every room occupied
    (2) H1 is full

    These are equivalent propositions, and so are

    (3) H2 does not have every room occupied
    (4) H2 is not full

    (3) is the negation of (1), and (4) is the negation of (2), thus the H1 and H2 designations.

    Now, the question is can an H1 and an H2 simultaneously be the same H in any physical reality? I don't think they can. (3) and (4) are the negations of (1) and (2) respectively. So we should be able to at least say that no H1 can ever be an H2.

    Now here is the problem as I see it. It doesn't matter what other properties we add to both H1 and H2, if we're doing that to try and make them equivalent, because they start off not being equivalent. So we can add to them both the property:

    (5) being an H with an infinite number of rooms

    which then gives us for H1

    (1) H1 has every room occupied
    (2) H1 is full
    (5) H1 being an H with an infinite number of rooms

    all of which are logically consistent with one another, and for H2:

    (3) H2 does not have every room occupied
    (4) H2 is not full
    (5) H2 being an H with an infinite number of rooms

    which is also a logically consistent set of properties. And the addition doesn't turn an H1 into an H2, because (5) does not entail that (3) and (4) are no longer negations of (1) and (2).

    Now considering all this, and taking H1 and H2 as described by (1,2,5) and (3,4,5) respectively, in the discussion there is something else to be considered, and that is a guest shows up at (5) looking for a room for the night. And you're telling me that for both H1 as given by (1,2,5), and H2 as given by (3,4,5) the hotel manager finds this new guest a room.

    Well, I'm sorry, CS, but to me that is logically impossible, as I believe I've just shown. For any H2 to become an H1 (3) and (4) must be changed to their negations, i.e., to (1) and (2), and (5) obviously doesn't have any capacity to do that, as we saw above. For when I added (5) to (1) and (2) and to (3) and (4) the distinction between H1 and H2 was unchanged.

    Therefore, or so I claim, whether a hotel has a finite, or a potentially infinite, or an infinite number of rooms, it is either an H1 or it is an H2. Furthermore, no H1 has an unoccupied room for an additional guest, and no H2 lacks an unoccupied room for a guest (and here I'm ignoring the quibble about an unoccupied room being necessary to accommodate a guest, but not sufficient, as I think it works against us both equally).

    Finally, in the Grand Hotel scheme, what I see happening is that when the new guest shows up at an H1, as given by (1,2,5), the manager of H1 first removes the guest in room 1, thus momentarily turning H1 into H2, as given by (3,4,5) with now two guests needing a room. He then places the new guest in room 1, which thus momentarily turns H2 back into H1, and the manager having a guest that needs a room in H1. So this process has one of two resultant states, it either results in an H1 and a guest without a room, or in an H2 with one unoccupied room, and two guests without a room. In either result there is always one more guest than room available. And since the process is absolutely unchanged throughout eternity, every result of every iteration of the process (may I say function here, and be correct?) results in one guest without a room. What I can't see, is how every guest gets a room in an H1 like the Grand via a process we've seen can only end in one of two states, neither of which accommodates all the persons needing a room in the scenario.
    Last edited by cstamford; August 22nd, 2013 at 02:27 PM.

  21. #59
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347 View Post
    In a physical environment yes. Specifically, he raises the objection I raised earlier in thread, that an actual infinitely constructed of successive addition could not have fusion producing stars, since all stars would have died out an infinitely long time ago.
    Not being an astrophysicist, and being unfamiliar with the formation of stars, I can't comment intelligently on whether there could have been an infinite number of stars. Without something preventing the formation of new stars, I don't see why an infinite regression of dying stars is impossible.

    And to my knowledge Craig has no issue with that statement. He argues you can't get an actual infinite from successive addition, specifically referring to our temporal context.

    Regardless, I would be surprised if you could show where Craig argues that infinite sets are impossible in his works.
    I think we have a misunderstanding, here. When Craig says something like, "Hilbert's Grand Hotel can't be actual because of the absurdities that follow from being able to accommodate additional guests", the absurdities he's talking about follow from how infinite sets work. He thinks that they are "impossible", in the sense that he thinks actual objects can't work like that.
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  22. #60
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    Against my better judgement, I'm going to give this one more try. And what follows does so baby step by baby step, so please don't jump ahead.

    What I said was, "Because a full hotel and one with no unoccupied rooms available are equivalent states of affairs by definition."

    Now, is there some logical flaw in that statement? Is it necessarily false? You're taking issue with it, but you're not explaining exactly why it's wrong. Instead, you reformulate it first, and then explain why the reformulation is wrong, and you've done this repeatedly. To wit, and yet again:
    My only problem is that I don't know what you mean when you say "full hotel". Mindtrap insists that a hotel has the property of being "full" if and only if it is incapable of accommodating additional guests. I'll refer to this property as "MT-full". That is, H has the property "MT-full" if and only if H is incapable of accommodating more guests.

    My initial interpretation of your statement is that you are defining "full" to mean "having no unoccupied rooms". I'll refer to this property as "C-full". That is, H has the property "C-full" if and only if H has no unoccupied rooms.

    I don't see any logical contradiction involved in the definition of "C-full".

    I'm claiming that the H in (2) lacks a property the H in (1) has; i.e. the property "being full".

    Now, for me, when we have two Hs that do not share all their properties, they cannot be identical, and so it's irrelevant whether or not (1) and (2) are formally logically contradictory or not.

    What I'm saying is we have to re-write your above as:

    (1) H1 has every room occupied
    (2) H1 is full

    These are equivalent propositions, and so are

    (3) H2 does not have every room occupied
    (4) H2 is not full

    (3) is the negation of (1), and (4) is the negation of (2), thus the H1 and H2 designations.
    This seems correct, if (2) and (4) are referring to the property "C-full".

    Now, the question is can an H1 and an H2 simultaneously be the same H in any physical reality? I don't think they can. (3) and (4) are the negations of (1) and (2) respectively. So we should be able to at least say that no H1 can ever be an H2.

    Now here is the problem as I see it. It doesn't matter what other properties we add to both H1 and H2, if we're doing that to try and make them equivalent, because they start off not being equivalent. So we can add to them both the property:

    (5) being an H with an infinite number of rooms

    which then gives us for H1

    (1) H1 has every room occupied
    (2) H1 is full
    (5) H1 being an H with an infinite number of rooms

    all of which are logically consistent with one another, and for H2:

    (3) H2 does not have every room occupied
    (4) H2 is not full
    (5) H2 being an H with an infinite number of rooms

    which is also a logically consistent set of properties. And the addition doesn't turn an H1 into an H2, because (5) does not entail that (3) and (4) are no longer negations of (1) and (2).
    I agree. If you're insisting that these properties be time-tensed, perhaps we should use the following notation:

    Let MT(H,t),C(H,t) be Boolean-valued functions (and thus take only the values TRUE and FALSE).

    MT(H,t) is true if and only if "H is MT-full at time t" is true.
    C(H,t) is true if and only if "H is C-full at time t" is true.

    It cannot be the case that both MT(H,t) and ~MT(H,t).

    Now considering all this, and taking H1 and H2 as described by (1,2,5) and (3,4,5) respectively, in the discussion there is something else to be considered, and that is a guest shows up at (5) looking for a room for the night. And you're telling me that for both H1 as given by (1,2,5), and H2 as given by (3,4,5) the hotel manager finds this new guest a room.

    Well, I'm sorry, CS, but to me that is logically impossible, as I believe I've just shown. For any H2 to become an H1 (3) and (4) must be changed to their negations, i.e., to (1) and (2), and (5) obviously doesn't have any capacity to do that, as we saw above. For when I added (5) to (1) and (2) and to (3) and (4) the distinction between H1 and H2 was unchanged.

    Therefore, or so I claim, whether a hotel has a finite, or a potentially infinite, or an infinite number of rooms, it is either an H1 or it is an H2. Furthermore, no H1 has an unoccupied room for an additional guest, and no H2 lacks an unoccupied room for a guest (and here I'm ignoring the quibble about an unoccupied room being necessary to accommodate a guest, but not sufficient, as I think it works against us both equally).
    Well, a hotel can go from full to not-full, yes? Suppose all the guests leave. You're not alleging that this state of affairs is contradictory, because (presumably) a hotel can be full at some times but not at others.

    So it isn't true that an H1 hotel can never have an empty room, or that an H1 hotel can never be an H2 hotel. When the Best Western is full on Monday and then has a vacancy Tuesday, it goes from being H1 to H2.

    The real issue is whether you're H1 and H2 at the same time.

    That is, MT(H,t) [similarly, C(H,t)] can differ from MT(H,t') [C(H,t')] so long as t =/= t', yes? (Not that t =/= t' implies that MT(H,t) =/= MT(H,t'), but rather that if MT(H,t) =/= MT(H,t'), it must be that t =/= t').


    What do you think it means to "accommodate an additional guest"? I gave my definition in this post, but my definitions are not tensed to time.

    Suppose that a manager of a hotel with 10 rooms, all currently occupied by different guests, is asked to accommodate an additional guest. If the guests in rooms 1 and 2 agreed to share the same room--say, room 1--then the new guest could be accommodated, do you agree?

    So let's think about room re-assignments. Room re-assignments take all the guests of a hotel and assign them to another room in the hotel (we could additionally require that the room being assigned is different from the room they currently have, but this is not necessary).

    First, we consider all possible ways of assigning rooms to the guests. That is, every function from G (the set of guests) to H (the set of rooms). Some of these room assignments involve guests sharing a room; others do not. Some of these room assignments result in every room being occupied; others do not.

    Let us say that a room assignment is licit if and only if no two guests share the same room, and illicit otherwise.

    Now, it is possible for a hotel to change room assignments without expelling any guests. Guests 1 and 2 exchange rooms, for instance. That is, it is possible to go from one room assignment to a different room assignment, and in particular it is possible to go from one licit room assignment to another, different licit room assignment.


    So when your manager looks to his list in an attempt to find a room for the new guest to occupy, it is possible (under certain circumstances) for him to "open up" a room by changing the current room assignment to a different one (for example, by double-booking a room). So it is possible for a manager to perform a room reassignment and produce an unoccupied room; this entails no contradiction. That is, a room reassignment can change a hotel from being MT-full [similarly, C-full] to not MT-full [not C-full], and this entails no contradiction.

    The relevant question here is whether it is possible to change a current, licit room assignment to another licit room assignment with the result that a room is unoccupied, while avoiding the case that both MT(H,t) [similarly, C(H,t)] and ~MT(H,t) [~C(H,t)] hold.


    Finally, in the Grand Hotel scheme, what I see happening is that when the new guest shows up at an H1, as given by (1,2,5), the manager of H1 first removes the guest in room 1, thus momentarily turning H1 into H2, as given by (3,4,5) with now two guests needing a room. He then places the new guest in room 1, which thus momentarily turns H2 back into H1, and the manager having a guest that needs a room in H1. So this process has one of two resultant states, it either results in an H1 and a guest without a room, or in an H2 with one unoccupied room, and two guests without a room. In either result there is always one more guest than room available. And since the process is absolutely unchanged throughout eternity, every result of every iteration of the process (may I say function here, and be correct?) results in one guest without a room. What I can't see, is how every guest gets a room in an H1 like the Grand via a process we've seen can only end in one of two states, neither of which accommodates all the persons needing a room in the scenario.
    Your process would (assuming that the time to change rooms doesn't converge to zero, or that if it does, it fails to converge quickly enough) never terminate after any finite amount of time, yes.

    But if we're considering an infinite hotel, then we're already okay with things like an infinite number of bedsheets, an infinite number of pillows, etc. Is it too much to presume that there are an infinite number of bells or telephones that a hotel employee could use to contact guests?

    For suppose that a manager was able to communicate simultaneously to all current guests that they were to exit into the hallway with all their belongings at precisely 10am the next morning. If all the guests comply, then every room is unoccupied. Every guest then moves into the next room "up" after some finite amount of time (some guests might be in wheelchairs, others might require assistance, etc.). After a long enough (but finite) amount of time, every room would be occupied, save room 1.



    Ultimately, I don't see the point of your last argument. Suppose that there is always a guest left without a room; where is the contradiction? It would be unfortunate for the hotel manager to keep track of who was precisely in which room precisely when, I suppose, but "An infinite hotel, in accommodating a new guest, will always have some prior guest in the process of finding a new room" is only contradicted by "An infinite hotel, in accommodating a new guest, will not always have some prior guest in the process of finding a new room." And you haven't shown how we can derive the latter.
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
    **** you, I won't do what you tell me

    HOLY CRAP MY BLOG IS AWESOME

 

 
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