<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Larry Burford</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[Skarp]
I was wondering how it is that a person could show up to rent a room in Hilberts hotel with an infinity of rooms that were occupied by an infinity of people. Surely that person already has a room rented.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">
[123]
There would be no person left to show up at the hotel if they are all in a room already.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
We can't be sure of that. And, there can be lots of people left.
Both of you seem to be assuming that an infinite number of (hotel rooms, people, etc) is equivalent to all (hotel rooms, people, etc). This is not true.
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Imagine three meter sticks. Each can be divided (conceptually) into an infinite number of fraction-of-a-meter distance intervals.
Let the distance intervals on the first meter stick correspond to the infinite number of rooms in the hotel.
Let the distance intervals on the second correspond to the infinite number of people occupying those rooms.
=
Let the distance intervals on the third correspond to some other people that have not yet rented a room.
Let the distance intervals on the third also correspond to some people that are left over.
Note that there can be a fourth meter stick. And so on.
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You guys have a long way to go. But it is a fun trip. Better than drugs in many ways. Please read some books and ask some more questions.
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Here is an interesting question that might serve as an inducement to do some studying.
"Is this Hilbert Hotel full, or empty?"
LB
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The Hibert hotel is neither full nor empty. It keeps growing in size whenever a guest shows up, but there are no more person guests
since they are all in the rooms, that's why they started adding
horses and cows.
Cantor's diagonal proof of different cardinalities of infinities and his one to one correspondence discriminating between countable and uncountable infinities is pretty controversial actually. A brief search in google and sci.math will demonstrate this. For example,
read the page in this link, and in particular note the arguments presented in the section "Cantor's Error":
en.wikipedia.org/wiki/Talk:Cantor's_diagonal_argument
In my opinion, sets of finite numbers (like naturals
and rationals) are by definition countable since how can you not
put a number tag to different finite objects- that's one of the purposes for the natural numbers. Similiarly, by definition, any set of numbers which have members that have infinite non-repeating digits (like the irrational numbers) are uncountable- it's impossible to put a place tag to an irrational. In this respect, I think Cantor's proof is redundant. But more than redundant, he commits a couple of logical fallacies
when he tries to show how the set of reals has a higher cardinality than the naturals. He commits the error of logical contradiction
(assumes an infinite list is completed) which then led to a circular argument (since he assumes, instead of having proven, that he has generated a new number using his diagonal method to prove that the number is not in the original list). But really, anyone can easily see it's nonsense by the fact that he's trying to attach a number tag to a bunch of nonrepeating infinite numbers- like how do you locate one nonrepeating infinite decimal number in the first place?
A brief search on google will find scores of mathematicians who disagree with Cantor and have gone as far as using his diagonal method (by arranging the digits in the form of Cantor's proof
for the larger cardinality of the reals), to prove that the set of even or odd integers for example is uncountable with respect to the set of naturals, demonstrating how silly an argument it is.
