Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

134 concept calculus: much better thanWe now discuss a much more sophisticated example. Let P = Peano arithmetic bethe theory in 0, S, +, • with successor axioms, defining equations for +, •, and thescheme of induction for all formulas in this language.Now consider “finite set theory.” By this, we mean ZF with the axiom of infinityreplaced by its negation, that is, ZF\I +¬I.Theorem (well known)P, ZF\I +¬I are mutually interpretable.The theorem is usually attribution to Tarski.To interpret P in ZF\I +¬I, nonnegative integers are interpreted as the finite vonNeumann ordinals in ZF\I +¬I. 0, S, +, •, = are interpreted in the normal way on thefinite von Neumann ordinals in ZF\I +¬I.To interpret ZF\I +¬I in P, sets are coded by the natural numbers in P. A commonmethod writes n = 2 m 1+···+2 m kand has n coding the set of sets coded by the m’s.This uses all of the natural numbers in P, with = interpreted as =.In many examples of mutual interpretability, the considerably stronger relation ofsynonymy holds. The strongest notion of synonymy normally considered is that ofhaving a common definitional extension. There are some important weaker notions.Notions of synonymy and other topics concerning interpretability are treated systematicallyand extensively in a forthcoming book (Friedman and Visser in preparation).Synonymy and its natural variants exhibit many delicate phenomena. It is obviousthat S, T cited previously are synonymous. It is proved in Kaye and Wong (2007) thatP and ZF\I +¬I are synonymous, if we formulate the axiom of foundation in ZF as ascheme.However, it is proved in Enayat, Schmerl, and Visser (2008) that P and ZF\I +¬Iare not synonymous, if foundation is formulated in the more usual way as a singlesentence.6.3 Basic Facts about Interpretation PowerEvery theory is interpretable in any inconsistent theory. Thus, the most powerfullevel of interpretation power is inconsistency. The following fundamental fact thereis no maximal interpretation power, short of inconsistency. From Feferman (1960)demonstrates thatTheorem 3.1 (in ordinary predicate calculus with equality) Let S be a consistentrecursively axiomatized theory. There exists a consistent finitely axiomatizedsystem T such that S is interpretable in T but T is not interpretable in S.This is proved using Gödel’s second incompleteness theorem. Consider T = EFA +Con(S), where EFA is exponential function arithmetic. If T is interpretable in S, thenEFA proves Con(S) → Con(EFA + Con(S)). By Gödel’s second incompleteness theorem,EFA + Con(S) is inconsistent, which is a contradiction.Comparability(?). Let S, T be recursively axiomatized theories. Then S is interpretablein T or T is interpretable in S?There are plenty of natural and interesting examples of incomparability for finitelyaxiomatized theories that are rather weak.