[FOM] first/second order logic

Richard Heck richard_heck at brown.edu
Sat Nov 9 04:12:20 EST 2019

On 11/8/19 8:39 AM, sambin at math.unipd.it wrote:
> Quoting Richard Heck <richard_heck at brown.edu
> <mailto:richard_heck at brown.edu>>:
>> 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? 
> Dear Riki,
> what do you have in mind for "predicative second-order logic"?
There is no doubt a good deal of precision here I've not considered. But
what I mean here is what Albert Visser has in mind when he talks about
the predicative extension of a given theory, and proves various results
about it. Roughly, the predicative second-order extension of some
first-order theory T is T plus all predicative comprehension axioms for
T (where the language of T has been extended in the obvious way).

> And hence also the question: what is the natural notion of model for it?
Let M be a model for a first-order theory T. Let PV(T) be the
predicative extension of T (a la Visser). Let M be a model of T. Then
let PV(M) be the extension of M to a second-order model of PV(T) where
the domain of the second-order variables consists of the sets S
definable over M, i.e., for which there is some formula A(x) of the
language of T such that S is the extension of A(x) in M (possibly with
parameters, if we wish to allow for that, though that is inessential).
What I just said describes a natural notion of 'predicative second-order
extension of a model of T', and it's clear (yes? not thought through
this part in detail...) that predicative second-order logic is complete
with respect to that class of models.

I.e. and roughly: If T is a first-order theory, then let PV(T) be T plus
predicative second-order logic. Then if M is any model of T, then there
is an extension of M---where the second-order domain contains exactly
the (first-order) definable sets over M---that verifies PV(T). That's a
kind of completeness theorem, at least.



Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University

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