Date: Feb 5, 2013 10:53 AM Author: fom Subject: Re: WMytheology § 203 On 2/5/2013 4:11 AM, WM wrote:

> On 5 Feb., 01:26, Virgil <vir...@ligriv.com> wrote:

>

>> If, as WM claims, there are at most countably many ways of accessing

>> reals and, as Cantor claimed, there are more than countably many reals,

>> then ...

>

> Then the axiom that every real can be put in trichotomy with every

> rational number is contricted. Then ZFC contradicts analysis.

In the logical type hierarchy that *defines* a real number,

the order relation of the rationals is inherited. The

rationals that are reals are distinct from the rationals

from which the real numbers are defined. In this construction,

the identity of a real number is tied to the trichotomy of

the underlying rationals and to the fact that in the complete

space any given pair of irrationals taken to be distinct are

linearly separated by a rational.

The problem of identity of a real number as part of the

real number system relative to identity within

Zermelo-Fraenkel set theory is a pseudo-metrization

problem. Classical set theory treats identity

philosophically as in the work of Carnap. Hence,

one must think of identity relative to model-theoretic

considerations. Such a construction involves relations

on a Cartesian product. The topological spaces that

support such a construct are called uniform spaces and

are derived from systems of relations called uniformities.

In general, uniform spaces have topological properties

independent from the constraints that permit

pseudo-metrization. But, it is instructive to read

the proof of the metrization lemma in Kelley. In effect,

the Cantorian fundamental sequence is grounded by the

Dedekind cut.

The relationship between Kelley's assumptions in the metrization

lemma and identity in logic do not appear in the literature

until Tarski's axioms for cylindrical algebras in 1971.

Kelley's metrization lemma uses relation products to

support a non-numeric sense of the triangle inequality.

Thus, one has something like

R_n*R_n*R_n c R_(n+1)

which translates to

(w,x)*(x,y)*(y,z)=(w,z) e R_(n+1)

Tarski's axiom is given by

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

which, using symmetry, translates to

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

and is part of the assumption in Kelley's metrization

lemma since the system of relations must all contain

the model-theoretic diagonal.

Ultimately, the problem with this account is

the philosophical position of logicism which,

significantly, is based upon a misstatement of

Leibniz' principle of the identity of indiscernibles.