Re: Questions for Uncountability Deniers



MoeBlee wrote:
> If you deny uncountability, then for me to understand your position, I
> should know exactly what it is you deny.

It is not so much that I deny uncountability (e.g. of reals). Rather,
it is not valid to ask how many reals there are, i.e., there is no
"cardinality" for the reals. See below for a brief explanation.

[...]
>
> Or, if you do not deny any of the above,
>
> (7) Do you believe that, in classical first order logic, from the axiom of
> Zermelo set theory, you have proven a theorem that is the negation of
> another theorem of Zermelo set theory? If so, what is your theorem and what
> is its proof?

No.

>
> (8) Do you reject Zermelo set theory or Zermelo-Fraenkel set theory? If so,
> do you propose axioms for mathematics? If so, what are they?

Yes. I question the axiom of infinity. I propose the theory of finite
sets, with classes. See math.LO/0506475 for a brief explanation.

>
> (9) Do you reject classical first order logic?

Yes, and also intuitionistic/constructive logics.

>If so, what logistic system
> do you prefer and what logistic system and meta-theory do you prefer for
> that logistic system?

I have proposed a logic NAFL. See math.LO/0506475 (and also other
references cited therein) for more details. Briefly, infinite sets do
not exist in consistent NAFL theories. But NAFL theories that prove the
existence of infinitely many objects identifiable by a property in the
language (e.g. natural numbers in Peano arithmetic) must also prove the
existence of the corresponding infinite (proper) class. Thirdly,
quantification over proper classes is banned. E.g. the reals, being
proper classes themselves, do not constitute a class and so cannot be
quantified over. That means that Cantor's argument cannot be formulated
in NAFL; you cannot "stack" real numbers one after the other as
required for diagonalization. There is no "cardinality" for the reals.

Regards, R. Srinivasan

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