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 <> 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


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.