FOM: FoM: Neo-Fregean reverse mathematics
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
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
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