[FOM] What is second order ZFC?

[Richard Heck]

On 09/11/2013 11:17 AM, Colin McLarty wrote:
>
> And towards the end of his post Richard offers the following as if it 
> disagreed with Harvey:
>
>     Nonetheless, there are lots of ways in which logic can be used in
>     the study of mathematics, mind, and world, and one should not
>     confuse unsuitability for one's own purposes with unsuitability
>     tout court. Solipsism, after all, is false.
>
>
> But Harvey had already said what I quote from him above: "I'm not 
> talking about arbitrary interesting foundational purposes."  And he is 
> obviously not talking about other non-foundational purposes.  This is 
> not a point of disagreement.  Taking it to be a disagreement may be 
> one source of confusion in this discussion.

Well, we can ask Harvey what he had in mind, but it seems pretty obvious 
to me that he has made some pretty strong claims in other places, that 
seem to border on claiming that there is something fundamentally unclear 
about "truly" second-order theories---a viewpoint that is not exactly 
unknown in the history of logic. Indeed, just preceding the quoted 
remark, Harvey said that he was talking about "any of the usual 
classical foundational purposes". I'm not sure what that includes 
exactly, but it's a pretty broad claim. I was simply trying to point out 
that plenty of people interested in what one surely would have thought 
counted as such have been interested in (truly) second-order theories, 
and without simply ignoring the kinds of questions Harvey is asking.

I'm all for finite representations of proofs. And I think axiomatization 
is a wonderful thing. I think it's a very nice question, for example, 
what kinds of axioms we should accept for the ancestral. The fact that 
we know no complete axiomatization is possible doesn't mean we shouldn't 
produce partial axiomatizations, and we can study them in the usual way. 
(To say that I am then treating the theory as "first-order" would be 
misleading at best and terminological quibbling at worst.) What it means 
is that we can never regard any such axiomatization as final and so that 
it will remain difficult to say, in some cases, whether some formula A 
does or does not follow, in ancestral logic, from some other formula B.

Apparently, it is supposed to follow that ancestral logic is useless 
"for any of the usual classical foundational purposes". Why? Yes, it's 
useless for one particular salient purpose, and I'm sure there are 
others, too. But any stronger claim wants argument.

Richard Heck

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