Re: Question about Frege's Theorem

Peter_Smith wrote:
lugita15@xxxxxxxxx wrote:
By Frege's Theorem, I mean the result that second-logic plus Hume's
Principle is sufficient to prove second-order arithmetic. ...
My question is, what semantics is required? Would Henkin
Semantics suffice? Or is the Standard Semantics needed?

Any help would be greatly appreciated.
Thank You in Advance.

Frege's Theorem is a claim about what can be deduced in a certain
*second-order deductive system*. It's a claim about a syntactic proof
relation. So Frege's Theorem doesn't "require" a semantics.

True, it wouldn't be so interesting if the proof-rules weren't
inituitively sound :-)) But they are.

If I recall Henkin semantics restricted to faithful models is (more
than?) enough to warrant the relevant rules.
I'm not sure what a faithful model is. My question is, is there a
"second-order deductive system" sound with respect to Henkin Semantics
under which Hume's Principle implies second-order arithmetic? If not,
is there a second-order deductive system sound (I know it can't
additionally be complete) with respect to Standard Semantics under
which Hume's Principle implies Second-Order Arithmetic?

Any further help would be greatly appreciated.
Thank You in Advance.

.

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