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