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Re: Problems with Infinity?
Posted:
Feb 26, 2013 9:03 PM
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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
--
dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.
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