FOM: FoM: Neo-Fregean reverse mathematics

A.P. Hazen a.hazen at
Mon Mar 26 23:22:06 EST 2001

(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.  The sets of (semantically) valid
sentences of the two systems are both non-recursive (and non-r.e.), but the
degree of the set of validities of FA is much higher than that of the set
of validities of PA2.   ... Relating this to the proof-theoretic point: I
guess the point is that the interpretation of PA2 in FA is FAITHFUL, but
that of FA in PA2 is not,  since it translates non-theorems of FA (for
instance: any two concepts that don't have natural numbers as cardinalities
have the same cardinality) into theorems of PA2.
   I don't know what follows from this, but it seems as if it's the sort of
thing that OUGHT to be interesting!  (O.k., I'll shut up.)
   Allen Hazen
   Philosophy Department
   University of Melbourne

More information about the FOM mailing list