
Re: Problems with Infinity?
Posted:
Apr 11, 2013 11:18 PM


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:protoE08F93.20403111042013@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/196404.pdf

