Date: Feb 26, 2013 11:52 AM
Author: David DeLaney
Subject: Re: Problems with Infinity?
Frederick Williams <freddywilliams@btinternet.com> wrote:

>Don Kuenz wrote:

>> beginner's language, Cantor uses two sets to define two levels of

>> infinity. One set, Aleph-0, holds countable infinity. The other set,

>> Aleph-1, holds continuum infinity,

>

>That the cardinality of the continuum (c = 2^{aleph_0}) is equal to

>aleph_1 is Cantor's continuum hypothesis which modern set theory settles

>neither one way nor the other.

Stronger than that - it's provable that it CAN'T settle it one way or another,

unless you extend the theory one way or another (Axiom of Choice, etc., as

Brian already noted).

>> which includes Aleph-0, along with

>> every possible arrangement of Aleph-0. The infinities of complex

>> variables belong to both sets, as does every other common infinity

>> found in mathematical literature.

>

>Surely not.

Right - "large cardinals" are extremely bigger than either one, and set

theory these days contains a good bit of work on them.

Dave

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