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Author: Frydaze1 Big gold star, 5000 posts Old School Fool Add to my Favorite Fools Ignore this person (you won't see their posts anymore) Number: of 195874  
Subject: Re: Heaven is real Date: 10/15/2012 4:16 PM
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Actually, there are. Some infinite numbers are bigger than other infinite numbers.


http://en.wikipedia.org/wiki/Cardinality


Cool, thanks for the link! Though I admit that a bunch of it is over my head.

On the other hand, it doesn't appear to say what you're claiming (see above disclaimer that I don't really understand a lot of it). For example there's this section:

Infinite setsOur intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel.

The reason for this is that the various characterizations of what it means for set A to be larger than set B, or to be the same size as set B, which are all equivalent for finite sets, are no longer equivalent for infinite sets. Different characterizations can yield different results. For example, in the popular characterization of size chosen by Cantor, sometimes an infinite set A is larger (in that sense) than an infinite set B; while other characterizations[which?] may yield that an infinite set A is always the same size as an infinite set B.

For finite sets, counting is just forming a bijection (i.e., a one-to-one correspondence) between the set being counted and an initial segment of the positive integers. Thus there is no notion equivalent to counting for infinite sets. While counting gives a unique result when applied to a finite set, an infinite set may be placed into a one-to-one correspondence with many different ordinal numbers depending on how one chooses to "count" (order) it.


Which seems to me to say that some scholars agree with you and some with me. But there isn't a definitive answer.

In fact, it links to the Paradox of the Grand Hotel, which quite nicely explains AngelMay's bff's statement about even numbers:
Some find this state of affairs profoundly counterintuitive. The properties of infinite "collections of things" are quite different from those of finite "collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of Transfinite Numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is no smaller than total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets, this cardinality is called (aleph-null).

Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.



And, yes, I'm weird enough to think this is a lot of fun. ;-)

Frydaze1
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