[FOM] 776: Logically Natural Examples 1

[Epstein, Adam]

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.

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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.