Hilberts hotel

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Skarp</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Imagine three meter sticks. Each can be divided (conceptually) into an infinite number of fraction-of-a-meter distance intervals<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">There is no such animal as an infinite number. There are no fractions of a meter in an infinitely devided meter.
I say again - How could anyone show up to rent a room if everyone already has a room? I'll go as far to say - How could there be any people if there is an infinity of rooms? I.E We already have an infinity of entities (rooms) - No more entities are possible.
You might say that you can add to infinity, but I'd have to say that it's already been done - Infinitely.
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Now that I rethink it, I still think it's correct to say that if an infinite number of persons are in the rooms already, there will be no person left to check in the hotel (otherwise it would be like saying the set of all naturals - which is represented by all the
persons in the room- was not infinitely large). But I think it's more correct to say that there cannot ever be an infinite number of persons in the room already since an infinite set cannot ever be completed.

jrich,

My tongue is bleeding. You up for some company?

LB

123,

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">But I think it's more correct to say that there cannot ever be an infinite number of persons in the room already since an infinite set cannot ever be completed.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

In other words, for every customer arriving at the hotel there is a room available. I have yet to see such hotel.

I have to disagree with you Larry on this one.

Everyone who said the premise excludes the possibility that anyone could show up to rent a room is correct. I have yet to make up my own mind about Cantor's theories beyond aleph-0, but up to that point I think they are neccessarily true. The key part of the question is the fact that the infinite set of rooms are given to be already occupied by an infinite set of guests. The cardinality of the set of rooms in the hotel and the set of guests is the same, aleph-0. More importantly, the cardinality of the set of guests who already have rooms and that of the set of all possible guests is also aleph-0. Even though both are infinite, they are exactly the same. Therefore, every possible guest already has a room, and so there is noone who has not already rented a room.


JR

Larry or jrich

okay my answer was not what you were after,so what is wrong with it?

i happen to have the book 123...infinity ( had it for years) i'm not the most mathematical person in the world,perhaps this is why i never got very far in it. i'm stuck from pg.19 to pg.24. i just cannot understand why he's doing what he does(combining two numbers? decimal)what on earth is he doing!! i have read this over and over to no avail,i'm i frustrated YES.i think it is the rules that confuse me,what,why and how is he able to do these things? simply I DON"T GET IT!!

perhaps another forum would be more appropriate, for this in some ways is off topic.

appreciate what ever help you can give.

north

Woops! I made a mistake. With infinite sets, a set and a proper subset of that set can have the same cardinality. Therefore, the equivalence of the cardinalities of the sets of guests and possible guests cannot be used to show the sets are equivalent. So Larry you are correct after all. I don't know why I ever doubted you.


JR