FOM: Franzen query of March 5

[hartry field]

The sort of theory I had in mind as a candidate for "our maximal theory" was
ZFC with ones favorite large cardinal axioms etc. (call this ZFC+),
supplemented with the sort of axiomatized truth theory discussed in Friedman
and Sheard's "An axiomatic Approach to Self-Referential Truth".  (I don't
have the reference handy. The truth theories they discuss avoid semantic
paradoxes by weakening the classical axioms on 'true'; they discuss a number
of different ways of doing this.) Take 'All the axioms of ZFC+ are true' to
be part of the supplemented theory.  You can carry out the intuitive
reasoning for Godel sentences of ZFC+ in the supplemented theory, but the
intuitive reasoning for Godel sentences of the supplemented theory turns on
intuitive principles of truth not accepted in the truth theory.  (The
details depend on which of their axiomatized truth theories you pick.) --
Hartry Field