[FOM] Countable sets with imprediicative definitions

Daniel Mehkeri dmehkeri at gmail.com
Thu Nov 18 20:48:13 EST 2010

> Can anyone provide examples of countable sets with impredicative
> definitions and apparently no alternative predicative definitions? Just
> to clarify, my intended notion of countable in relation to a set refers
> to the existence of a surjection from the natural numbers (or from the
> natural numbers excluding zero) to that set.

 Kleene's O was mentioned, but it is constructed from below in a sense. It
common especially in constructive mathematics to call that predicative (a
Gamma_0 would be too).

But as far as I know nobody considers the power set of the natural numbers
predicative in any sense. Of course, that is uncountable. But Z_2,
arithmetic, is a formal theory. Assuming it has a model at all, then it has
countable model, by the downward Lowenheim-Skolem theorem. This model is
then a countable set with an impredicative definition, and no alternative
definition is known at this time. You might find this a simple and intuitive

As I recall, Z_2 also happens to be equiconsistent with the System F that

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