fom
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Registered:
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Re: Problems with Infinity?
Posted:
Apr 12, 2013 12:14 AM


On 4/11/2013 10:18 PM, Butch Malahide wrote: > 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 >
That is an interesting result.
It is almost like the logical hierarchy of the rationals to the reals in Cantor or Dedekind.
And, it makes sense that it would be related through the functions because what is involved with polynomials, extension fields, and the fundamental theorem of algebra.
Thank you for the link.

