Philosophy of Mathematics (Monday - Week 5, HT19)
Occasionally in the literature one can find the claim that the easiest way to define a finite axiomatization of an extension Th of PA is to consider CT--(Th) (extension of Th with the Tarski-style compositional axioms for the newly added truth predicate) and add the sentence saying "All axioms of Th are true". We show that this claim is false as there are very natural arithmetical theories Th for which CT--(Th)+ "All axioms of Th are true." is non-conservative over Th. Then we show that for every r.e. theory Th there exists a recursive axiomatization A such that CT--(Th)+ "All sentences from A are true." is conservative over Th. Finally, we discuss our main problem: which theories Th are the arithmetical parts of a theory of the form CT--(PA)+ "All sentences from PA' are true.", where PA' is an axiomatization of PA which is proof-theoretically reducible to (the standard schematic axiomatization of) PA.
Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Volker Halbach