[FOM] 776: Logically Natural Examples 1

[Robert Solovay]

Can you give a reference for this result that in Zermelo set theory there
is no definable infinite set?

-- Bob Solovay

On Dec 21, 2017 6:17 PM, "Epstein, Adam" <A.L.Epstein at warwick.ac.uk> wrote:

> Consider Zermelo Set Theory (ZF without Replacement) where we take as
> Axiom of Infinity the existence of a (Tarski) infinite set.
>
> In this setting there is no natural example of an infinite set:
>
> THEOREM (E) There is no formula phi(x)
> in the language of set theory, with only the free variable x, such
> that the above theory proves that the unique solution to phi(x) is
> infinite.
>
> -------------------------------
>
>
>
>
>
>
>
> Turning to set theory, we know, and it is very important to know, in
> full blown set theory, that there is a well ordering of the set R of
> all real numbers. Is there a natural example of a well ordering of R?
> It is completely compelling that provable ZFC definability is
> necessary for logical naturalness.
>
> THEOREM (Cohen)? Assume ZFC is consistent. There is no formula phi(x)
> in the language of set theory, with only the free variable x, such
> that ZFC proves that the unique solution to phi(x) is a well ordering
> of R.
>
> And the above Theorem can be greatly strengthened to replace ZFC by
> any of the usual extensions of ZFC by large cardinals.
>
> Another famous one.
>
> THEOREM (Solovay) Assume ZFC is consistent. There is no formula phi(x)
> in the language of set theory, with only the free variable x, such
> that ZFC proves that the unique solution to phi(x) is a non Lebesgue
> measurable subset of R.
>
> which also can be greatly strengthened by using any of the usual
> extensions of ZFC by large cardinals.
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