FOM: FoM: Neo-Fregean reverse mathematics

Roger Jones RBJones at
Tue Mar 27 11:06:01 EST 2001

"A.P. Hazen" wrote:

> (This may not lead to anything interesting AT ALL; it's just something that
> caught my attention.)
>    As Charles Parsons points out, the two systems PA2 (Second-Order Peano
> Arithmetic) and "FA" ("Frege Arithmetic"=Second-order logic plus "Hume's
> Principle") are PROOF-THEORETICALLY equivalent.  Each can be interpreted in
> the other.
>    Looked at SEMANTICALLY (with the "standard" semantics, not the "Henkin"
> one) they are startlingly different: PA2 is categorical, and its only (up
> to isomorphism) model has a denumerable domain of individuals.  FA has
> models with individual domains of every infinite cardinality.  This has
> recursion-theoretic consequences.

My original gripe concerned the relationship between HP and the axiom of
infinity (implicitly in the context of second order logic).
2nd order logic +infinity also has lots of models.

I didn't follow your other point about the difference in the degree of the
validities in PA2 and FA.
Any chance of a fuller explanation of how you came to that conclusion?

In both cases surely, the validities are the formulae derivable from the pure
second order validities + one extra axiom, and hence have the same degree.

Roger Jones

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