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
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.