FOM: Neo-Fregean reverse mathematics

[Harvey Friedman]

Reply to Urquhart 10:26AM 3/27/01.

>Following on from Charles Parsons's and Allen Hazen's
>postings, let me try to make my earlier query more precise.
>Let PA2 (2nd order Peano arithmetic) and FA be as in
>Allen Hazen's posting.  Then it appears that they
>are mutually interpretable.  Now is the stronger claim
>true that they are synonymous?  Here I am taking
>"synonymous" to be defined as in the paper of Karel
>de Bouvere ("Theory of Models", ed. by Addison, Henkin
>and Tarski, 1965, p. 402).  If they are not, then
>they provide an interesting non-artificial example of
>two mutually interpretable theories that are not
>synonymous (De Bouvere mentions an unpublished example
>of this phenomenon due to David Kaplan).

Your posting reminded me of some unfinished business of mine that I should
report on in detail in a numbered posting on FOM.

Years ago, I proved a number of related results to the effect that for a
wide variety of systems (interpreting weak arithmetic), interpretability is
the same as relative consistency. A technique for creating interpretations
between theories interpreting weak arithmetic, using nonstandard models,
was introduced.

But I also proved some results about showing that different notions of
interpretability in this context are the same. I believe I can show that
under very general conditions, mutually interpretable is equivalent to
synonymity. More later, if you are interested.