```Date: Jan 11, 2013 4:16 AM
Author: Zaljohar@gmail.com
Subject: Finitely definable reals.

Lets say that a real r is finitely definable iff there is a predicateP that is describable by a Finitary formula that is uniquely satisfiedby r.Formally speaking:r is finitely definable <-> Exist P for all y. (P(y) <-> y=r)where of course P(y) is a Finitary formula.Of course NOT all reals are finitely definable in the above manner.This is an obvious corollary of Cantor's arguments of uncountabilityof reals.Also it is obvious that we have only COUNTABLY many finitely definablereals.Other kinds of reals can be "infinitely" definable, this can beachieved in a language that encounters infinitely long strings ofsymbols, and many known first order languages are of that sort andthey are proven to be consistent and even supportive of a proofsystem.However one must understand that when we say that we have countablymany finitely definable reals then we are accepting the existence of abijection between the naturals and the finitely definable reals andthat this bijection is itself not finitely definable!This is also acorollary of Cantor's arguments. Also the diagonal on the list of allfinitely definable reals IS also non finitely definable real! since itis defined after the bijection between the naturals and the set of allfinitely definable reals, and that bijection as said above is notfinitely definable.Finitely definable reals are definitely very interesting kinds ofreals, they are superior to those that are non finitely definable ofcourse, but however that doesn't mean that the later ones do notexist, nor does it mean that the later ones cannot be spoken about, wecan still speak of those kinds of reals by using formulas that do notuniquely hold of one of them, and still those sentences can illustrateinteresting pieces of mathematics that might possibly find someapplication one day. However it is expected of course that finitelydefinable reals would be of more importance no doubt and thereforethey would have the leading stance among reals.Zuhair
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