[FOM] 776: Logically Natural Examples 1

[Harvey Friedman]

On Thu, Dec 21, 2017 at 4:05 AM, 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.
>
Yes, technically interesting.

Of course, the significance of the results for ZFC depend on ZFC easily
formalizing all of normal mathematics. Part of normal mathematics is having
natural examples of infinite sets such as the set of all integers and the
set of all rationals and the set of all real numbers. So the system you are
talking about is not easily formalizing all of normal mathematics, as
indicated by, e.g., your Theorem (E).

FROM JOE SHIPMAN

Your examples where ZFC can’t prove that a definition specifies an object
> with the desired properties go away if V=L. What’s the weakest alternative
> to V=L that allows a well-ordering of the reals, or a nonmeasurable set, to
> be uniquely defined? From a definable well-ordering of the reals, one can
> define a non-measurable set, but can you get a definable well-ordering of
> the reals from a plausible set-theoretic axiom weaker than V=L? CH doesn’t
> count, it just says there exists a bijection with aleph_1 but does not
> provide one like V=L does.



​"​
can you get a definable well-ordering of the reals from a plausible
set-theoretic axiom weaker than V=L?
​"​


Yes, of course from "every set of integers is constructible". But not,
e.g., from diamond.

THEOREM. Let T be an extension of ZFC. The following are equivalent.
i. There exists phi(x) such that T proves that the unique solution to
phi(x) is a well ordering of the reals.
ii. T proves "every real number is ordinal definable".

Harvey Friedman
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