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?