Date: Jan 31, 2013 3:34 AM
Author: fom
Subject: Re: Matheology § 203

On 1/30/2013 3:55 PM, Virgil wrote:
> In article
> <0f579894-2a8a-47fb-9fbc-c7ad7ffa075e@l3g2000pbq.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>

>> On 30 Jan., 12:02, fom <fomJ...@nyms.net> wrote:
>>

>>> As for those "logical considerations," I mean that
>>> one can develop a hierarchy of definitions that
>>> depend on actual infinity. To say that mathematics
>>> is "logical" is to concede to such a framework. I
>>> do not believe that mathematics is logical at all.

>>
>> That is a very surprising statement. Why do you think so?

>
> He has probably been reading too many of WM's posts.
>


Perhaps you should investigate some things a little
more deeply.

You will find a standard account of identity in
the post

http://plato.stanford.edu/entries/identity-relative/

Oddly, it does not contain the axiom

AxAy(x=y <-> Ez(x=z /\ z=y))

found in Tarski and Monk.

If, using symmetry, one considers the ordered pairs of
this identity relation as a relation product,
one obtains

(z,y)*(y,x)*(x,z)=(z,z)

You will find the same relation product as an
essential premise for the metrization
lemma presented in the chapter on uniform spaces
in "General Topology" by Kelley.

It is related to the epsilon/3 arguments found
with uniform convergence or uniform continuity
from real analysis.

Moreover, you may look for Fischer's theory for
3-transposition groups. There, you will see that
a pair of transpositions such as

(x,z)(z,y)

generates some of the most interesting group
theory in mathematics. And, in particular,
the Mathieu groups are closely related to
the Golay codes of information theory.

All of this mathematics is ignored in
"the theory of identity" used in set theory.

My criticisms of mathematics and logic are
extremely subtle and revolve around the
fact that a logician's use of the sign of
equality differs from a mathematician's
use of the sign of equality. The corresponding
mathematics involves 12-dimensional codes
and a group with order 244,823,040.

Now, why don't you take a moment to
explain why you believe forcing models in
set theory should be taken seriously?

Should I, for example, simply trust a
set theorist like Woodin even though
the theory of identity on which his
forcing models are based is not the one
presented by Zermelo in 1908 (where
identity is interpreted relative to
denotations in the Fregean sense)?

Why?