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

[Ali Enayat]

This is a reply to a query of Richard Heck (May 11, 2016) who has
asked whether (*) below 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 answer is in the positive. Let me use Z_2 for the first order
theory whose models are precisely the Henkin models of second order
arithmetic. It has been known since Goedel's work on the class L of
constructible sets that every model M of Z_2 has a submodel  M' , with
the same numbers as M, in which "V=L" holds and in particular there is
parameter-free global well-ordering of the numbers and the reals of M'
(from the point of view of M').  By elementary model theory (Tarski's
test for elementarity), this implies that the collection M'' of
parameter-free definable elements of M' forms an *elementary submodel*
of M'.

The above shows that if T is an extension of PA (first order
arithmetic) such that *the numbers* of a model of Z_2 satisfy T, then
T also holds of *the numbers" of model of Z_2 in which everything
(numbers and reals) are parameter-free definable. This answer's Heck's
follow-up question (**) below, which asks :

(**) 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?

More specifically: the only restriction on theories of first order
arithmetic T at work here is the obvious one: the "numbers part" of a
model of Z_2 should satisfy T (or more explicitly: T should contain
the number theoretic consequences of Z_2).  In particular, T can be
all first order arithmetical truths [the meta-theory I am working
within is third order arithmetic or a stronger theory such as ZF].

All the best,

Ali Enayat