On Thursday, December 5, 2013 4:24:40 AM UTC-7, WM wrote: > Am Dienstag, 3. Dezember 2013 18:53:38 UTC+1 schrieb Zeit Geist: > > > Where in the Axiom Set for the Real Numbers are "Defined Numbers" mentioned? > > They are not mentioned because undefined numbers are not numbers. For undefined numbers we cannot find out the sum or, in Cantor's list, the digit a_nn. > > > Where in you posts have you given a a Formal Conception of "Definable Numbers"? > > There is no formal defintion because formal language is incapable of many necessary things. But every definable number is definable by a finite word that does not defina another number. >
But there is a Formal Definition. One that does result in a Contradiction. You just don't know, don't understand or refuse to acknowledge it.
You are right, kind of, since FOL cannot "define" Definability.
> > > No. ZFC has uncountability. Definable numbers have not. > > > Could you give an example of a Model of ZFC in which all Real Numbers are definable and that contains a Bijection between N and R. > > Could you give a model of ZFC at all? >
I give two.
ZFC + V=L
ZFC + "There Exists an Inaccessible Cardinal."
> > > > Some Models contain them, and some don't > > > > The latter must be countable. > > > Really? Show me! > > You seem so convinced of it. > > You should be able to Prove it. > > I did. You seem to have a very bad memeory.
You NEVER supplied a Valid Proof.
> Look: Every definition in mathematics is a finite word in English or any formal language. The set of all possible finite words in English or any formal language is countable. Proof left as an exercise. >
First of all, I AM looking, if I weren't I couldn't be reading your crap.
> A subset of a countable set is countable. >
And the Set of All Definitions, when properly And Logically implemented, cannot be put into Bijection with N.