09-18-2005, 02:29 PM #1
books asked about multiple universe info and I guess more might be interested and thats a good article
09-18-2005, 04:16 PM #2
My response is Hilberts Hote
lPerhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.
But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.
1. Whatever begins to exist has a cause of its
2. The universe began to exist.
2.1 Argument based on the impossibility of an
2.11 An actual infinite cannot exist.
2.12 An infinite temporal regress of
events is an actual infinite.
2.13 Therefore, an infinite temporal
regress of events cannot exist.
2.2 Argument based on the impossibility of
the formation of an actual infinite by
2.21 A collection formed by successive
addition cannot be actually infinite.
2.22 The temporal series of past events
is a collection formed by successive
2.23 Therefore, the temporal series of
past events cannot be actually
3. Therefore, the universe has a cause of its
10-08-2005, 01:16 PM #3
well there is certanly nothing mathematicly impossible with hilberts hotell.
this article however makes a etter point that I ever could.
A supertask is a task involving an infinite number of steps, completed in a finite amount of time. I would argue supertasks are impossible, for there are only finite beings, and a being which performed a supertask would become thereby an infinite being.
Hilbert's Grand Hotel: A Psuedo-Paradox
Hilbert's Supposed Paradox of the Grand Hotel concerns the existence of a hotel with a countably infinite number of rooms. With an ordinary hotel, which has a finite number of rooms, it is impossible to add more guests to the hotel after it is full (unless you make it overfull, by forcing guests to share rooms, or depriving them of rooms.) However, in an infinite hotel, according to Hilbert's paradox, this is possible. Although this claim is widely believed, I will now argue that it is false -- once an infinite hotel is full, it is only possible to add more guests if the guests can move abritrarily fast, and an infinite task (shifting an infinite number of guests to the next room) can be completed in a finite amount of time (i.e. a supertask.) However, Hilbert's scenario does not assume either that the guests can move arbitrarily fast, nor does it assume that they can perform an infinite number of tasks at once.
Let us suppose every room is full, and a new guest arrives. There is apparently no more room; but we can instruct the new guest to go to room #1, and tell that guest to move to room #2, and then that guest in room #2 will move to room #3 and tell that guest to move to room #4. Note that, while this process is going on, the hotel is actually overfull, since some guests (however temporarily) are deprived of rooms, or forced to share the same room. For example, when the guest moves in to a new room, and forces the guest already in there to move to the next, they temporarily share a room; and while a guest is in motion to the next room, they are temporarily deprived of the room. But, assume this entire process could be completed in an finite amount of time (which would require the guests to be able to pack up and move arbitrarily fast); then we would have a supertask, and the possibility of supertasks is not assumed in the original description of Hilbert's problem.
Another example, commonly given, is that if a countably infinite number of guests arrive, an infinity of new guests can be fit into an already full infinite hotel. Send the guest in room numbered n to the room numbered 2n. Now, only even-numbered rooms are filled; move the new guests into the odd-numbered rooms. However, again, without supertasks, both in moving the existing guests and moving in the new guests, and infinite number of steps must be performed, and thus without supertasks the hotel would always be overfull.
The Actual Infinite
The Paradox of Hilbert's Grand Hotel has been used as an argument for why an actual infinite is impossible, by opponents of that notion. These opponents are mainly opponents of the possibility of an infinite temporal regress (most commonly, theists attempting to use a cosmological argument for the existence of their God.) For example, William Lane Craig [http://www.leaderu.com/truth/3truth11.html] discusses Hilbert's hotel as an example of an actual infinite, a number of "absurdities" which he claims would arise if the Hotel actually existed, and therefore concludes "Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things."
The alleged absurdities he discusses are basically two: the addition of new guests to an empty hotel (as discussed above), and that some subset of an infinite set can be the same size as the original set.
The first alleged absurdity he deals with, as I have discussed above, involves not merely an actual infinity, but a particular type of actual infinity, a supertask; as such, it may be an argument against supertasks, but is not an argument about actual infinities that do not amount to supertasks. In particular, the actual infinity he is really arguing against, an infinite temporal regress.
The second alleged absurdity he deals with, is simply a well-known mathematical fact, although some of the examples he discusses also involve supertasks (he has an infinite number of guests leave the hotel "[a]t a single stroke").
His argument concerning the second absurdity is as follows:
An infinite subset of an infinite set can be the same size as its superset.
But that's absurd!
Therefore, no such infinite set could actually exist.
The only evidence he produces for (2) is his intuitions, intuitions he assumes to be shared by his readers. As such, this is a very flimsy argument. The mere fact that humans (or more accurately, some humans) feel something to be absurd, does not prove that it actually is. This is especially true, given that as finite beings, we are well-used to dealing with finite quantities throughout our lives; but our experience with infinite quantities is limited to mathematics or philosophy. Since most people have limited experience at all with the latter, most people have no experience at all dealing with the infinite. Intuitions are often wrong when they are based on experiences of situations fundamentally different from that at hand.
Through our experiences of sets of finite size, we learn certain assumptions which hold true for finite sets. Such, for example, that adding more elements to a set makes it bigger. But, having no experience with infinite sets, it is easy for us to assume that this property is an inherent part of the notion of "size", rather than being simply part of the notion of infinite sizes. Thus, upon hearing that it does not apply to infinite sizes, many think that there is something absurd about infinite sets. On the contrary, it is merely that their understanding of "size" is incorrect, but that it this incorrectness only manifests itself in situations that they do not have to deal with normally.
If Craig is going to argue against the actual infinite on the basis that it does not agree with his intuitions, then I can argue against his God on the grounds that it does not agree with my intutitions. But both arguments are not very forceful.
Finally, even if we intuitively feel Hilbert's hotel to be impossible, it does not follow that an infinite temporal regress is impossible. For there are a number of differences between the two, which may be significant. One, already discussed, is the issue of supertasks. Another is that Hilbert's hotel is a distinguishable infinite object in an infinite universe, while an infinite temporal regress involves an infinity of the universe itself. Both of which, at least one defender of an infinite temporal regress (myself) holds to be impossible, while still believing in an infinite temporal regress.
And finally, an infinite Hotel is a somewhat amusing example invented by a mathematician -- for this very reason, to some it feels intuitively impossible -- but is this intuition due to the underlying principles behind the problem, or merely its surface amusement? This is another problem with arguments from intuition -- where the example used as the subject for the intuitions (an infinite Hotel) differs from what is really being argued about (an infinite temporal regress), it is difficult to tell whether the intuitions are due to general features shared by both examples, or by features specific to one.
© Simon Kissane 2003. Licensed to all under the GNU FDL.
10-08-2005, 01:51 PM #4
Multiuniverse is supposed to be a closed circut.
You don't need a creator you only need +, - and motion and it will exist forever.
It's quit logical.
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