Date: Apr 11, 2013 11:18 PM
Author: Butch Malahide
Subject: Re: Problems with Infinity?
On Apr 11, 7:52 pm, "Brian M. Scott" <b.sc...@csuohio.edu> wrote:

> On Thu, 11 Apr 2013 20:40:31 -0400, Walter Bushell

> <pr...@panix.com> wrote in

> <news:proto-E08F93.20403111042013@news.panix.com> in

> rec.arts.sf.written,sci.math:

>

> > In article <512CA332.AD4F7...@btinternet.com>,

> > Frederick Williams <freddywilli...@btinternet.com> wrote:

> >> 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.

> > Does anyone care.

>

> Yes.

>

> > That is do any important results hang on either one?

>

> There are results in a variety of fields, from commutative

> algebra through functional analysis and topology to complex

> analysis, that depend on CH. Some of them involve questions

> of a fairly fundamental character.

In complex analysis, Paul Erdos showed that the continuum hypothesis

is *equivalent* to the existence of an uncountable family F of entire

functions such that {f(z): f in F} is countable for each complex

number z.

http://www.renyi.hu/~p_erdos/1964-04.pdf