[FOM] second-order logic once again

[Kevin Watkins]

On Wed, Sep 5, 2012 at 6:20 AM, Robert Black <mongre at gmx.de> wrote:
> The tempting move now is to say that different models of first-order ZF have
> different sets of 'second-order validities', but that in every such model
> the 'set of second-order validities' is not r.e., and that's a statement of
> incompleteness that doesn't assume the determinacy of second-order logic,
> since everything is relativized to models. But I think it does assume the
> determinacy of second-order logic in its naïve quantification over *all*
> models of first-order ZF (of which, of course, the fattest ones will be
> initial segments of the models of genuinely second-order ZF).

I agree with the first part: different models of ZFC might have
different sets of second-order validities, and in every case the set
of second-order validities is not r.e.  These observations belong to a
discussion regarding second-order validity relative to different
models of ZFC.

But I think this is beside the point when it comes to the statement
and proof of incompleteness for second-order logic.  When someone
states and proves this theorem, they are just talking about sets.
They are using the language of ZFC to talk about them.  They are
assuming the facts about sets that are embodied in the axioms of ZFC.
The sets they are talking about encode various gadgets in the syntax
and semantics of second-order logic--formulas, proofs, (standard)
models, the satisfaction relation, etc.--according to the usual
paradigm by which all mathematical objects are reduced to sets in ZFC.

This latter discussion (talking about sets that encode gadgets in the
semantics of second-order logic, using the language and axioms of ZFC)
does not in any way involve a quantification over models of ZFC.
There is no step at which models of ZFC even enter the discussion, let
alone are quantified over.

Think of PA--if someone proves the fundamental theorem of arithmetic
using the language of PA, and assuming the facts about numbers
embodied in the axioms of PA, they are just talking about numbers.
They are not talking about models of PA, or relativizing to models of
PA.  They are not quantifying over models of PA.

Kevin