[FOM] first/second order logic

[Richard Heck]

On 11/6/19 10:27 AM, Harvey Friedman wrote:
> QUESTION. Is there an interesting completeness theorem for nontrivial
> fragments of second order logic? Obviously, first order logic is a
> nontrivial fragment that does have a completeness theorem. But what if
> we look at SIMPLE fragments of second order logic. Maybe there are
> really interesting such with a completeness theorem. Or if there has
> been a good start on this, then how far can it be pushed?

Well, predicative second-order logic is natural and is complete, isn't
it, with respect to some reasonably natural notion of what a model is?
The natural extension would be to \Delta_1^1 second-order logic, but
you'd probably know more about that situation than I would. Still, some
of the work on fragments of Frege's "Basic Laws", as summarized in John
Burgess's /Fixing Frege/, might be of interest here.


> There has been a lot of confusion about the use of first order logic,
> and second order logic, in f.o.m. (foundations of mathematics), or
> even what second order logic is.

There will shortly be published (in a volume for Crispin Wright) a
previously unpublished paper by George Boolos, from the early 1970s,
that addresses many such issues.

Riki


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Richard Kimberly (Riki) Heck
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Brown University

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