[FOM] Question About Henkin Models for Second-order Arithemetic

[Richard Heck]

Does anyone know whether the following is true or false?

(*) There is a Henkin model for second-order PA in which every set in
the domain of the second-order quantifiers is defined (in the sense of
the model) by some formula A(x) in which x is the only free variable.

The fact that the model satisfies comprehension gives the converse of
(*), and in fact something even stronger, since comprehension permits
parameters. That suggests that maybe the question I'm asking is closely
related to the question whether, if we restrict comprehension to
formulae in which only x is free (no parameters), we get the same theory.

If (*) is true, are there restrictions on the theories for which it is
true? E.g., does it continue to hold if we expand the theory by adding
as 'axioms' all first-order arithmetical truths?

Cheers,
Richard Heck

-- 
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Richard G Heck Jr
Professor of Philosophy
Brown University

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