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 predicate

P that is describable by a Finitary formula that is uniquely satisfied

by 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 uncountability

of reals.

Also it is obvious that we have only COUNTABLY many finitely definable

reals.

Other kinds of reals can be "infinitely" definable, this can be

achieved in a language that encounters infinitely long strings of

symbols, and many known first order languages are of that sort and

they are proven to be consistent and even supportive of a proof

system.

However one must understand that when we say that we have countably

many finitely definable reals then we are accepting the existence of a

bijection between the naturals and the finitely definable reals and

that this bijection is itself not finitely definable!This is also a

corollary of Cantor's arguments. Also the diagonal on the list of all

finitely definable reals IS also non finitely definable real! since it

is defined after the bijection between the naturals and the set of all

finitely definable reals, and that bijection as said above is not

finitely definable.

Finitely definable reals are definitely very interesting kinds of

reals, they are superior to those that are non finitely definable of

course, but however that doesn't mean that the later ones do not

exist, nor does it mean that the later ones cannot be spoken about, we

can still speak of those kinds of reals by using formulas that do not

uniquely hold of one of them, and still those sentences can illustrate

interesting pieces of mathematics that might possibly find some

application one day. However it is expected of course that finitely

definable reals would be of more importance no doubt and therefore

they would have the leading stance among reals.

Zuhair