
Re: Problems with Infinity?
Posted:
Feb 26, 2013 9:03 PM


On 02/26/2013 04:23 PM, Brian M. Scott wrote: > On Tue, 26 Feb 2013 14:39:33 0500, Shmuel Metz > <spamtrap@library.lspace.org.invalid> wrote in > <news:512d0f75$10$fuzhry+tra$mr2ice@news.patriot.net> in > rec.arts.sf.written,sci.math: > >> In<20130225b@crcomp.net>, on 02/26/2013 >> at 12:51 AM, Don Kuenz<garbage@crcomp.net> said: > >>> Answering my own question, Cantor's conjectures concern >>> set theory and only tangentially touch on the infinities >>> of complex variables. Using beginner's language, Cantor >>> uses two sets to define two levels of infinity. One set, >>> Aleph0, holds countable infinity. The other set, >>> Aleph1, holds continuum infinity, which includes >>> Aleph0, along with every possible arrangement of >>> Aleph0. > >> No; Cantor's work on cardinality has nothing to do with >> Complex Analysis, > > Though there are results in complex analysis that depend on > the continuum hypothesis, e.g. > > <http://www.renyi.hu/~p_erdos/196404.pdf?utm_medium=referral&utm_source=t.co>. > > (Followups set.) > > Brian
For nonconstant entire functions f, g: C > C, say f '<' g when (Mf)(r) = o( (Mg)(r) ), r in [0, oo), where for r in [0, oo), (Mf)(r) = max_{z = r} f(z) and similarly for r in [0, oo), (Mg)(r) = max_{z = r} g(z), 'M' for maximal or maximal function.
If f(z) = z^2 and g(z) = z^3, (Mf) (r) = r^2, (Mg)(r) = r^3, and for r>0, (Mf) (r) = (1/r)*(Mg)(r) , so (Mf)(r) is o( (Mg)(r) ), o being littleo notation.
With the '<' strict partial order, which ordinals can be embedded in (U, '<'), with U being the set of all entire functions and '<' the strict partial order defined above?
[ cf. wikipedia for noncrucialness of strict/nonstrict: http://en.wikipedia.org/wiki/Partially_ordered_set#Strict_and_nonstrict_partial_orders ].
David Bernier
 dracut:/# lvm vgcfgrestore File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh Please specify a *single* volume group to restore.

