Date: Feb 26, 2013 9:03 PM
Author: David Bernier
Subject: Re: Problems with Infinity?
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,

>>> Aleph-0, holds countable infinity. The other set,

>>> Aleph-1, holds continuum infinity, which includes

>>> Aleph-0, along with every possible arrangement of

>>> Aleph-0.

>

>> 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/1964-04.pdf?utm_medium=referral&utm_source=t.co>.

>

> (Followups set.)

>

> Brian

For non-constant 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 little-o 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 non-crucialness of strict/non-strict:

http://en.wikipedia.org/wiki/Partially_ordered_set#Strict_and_non-strict_partial_orders

].

David Bernier

--

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Please specify a *single* volume group to restore.