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

132 concept calculus: much better thanWe put primary emphasis on the system MBT (much better than) = Basic + DiverseExactness + Strong Unlimited Improvement. We prove that MBT is mutually interpretablewith ZF (and, hence, ZFC, as ZF and ZFC are mutually interpretable).However, there are other meritorious combinations that we show are mutually interpretablewith ZF. In fact, we show that if we choose one axiom from each of the threegroups (thus, Basic must be included), then we get a system mutually interpretablewith ZF, with exactly one exception: Basic + Diverse Exactness + Unlimited Improvementis interpretable in ZF/P (ZF without the power set) and may be much weakerstill.Zermelo set theory (Z) is a particularly important relatively strong fragment ofZF of substantial foundational significance. In particular, ZC (Z with the axiom ofchoice) forms a very smooth and workable foundation for mathematics that is nearly ascomprehensive, in practice, as ZFC. Of course, known exceptions to this are particularlyinteresting and noteworthy. See Friedman (in preparation) for a discussion.We show that Basic plus any of the last three forms of Diverse Exactness form asystem that is mutually interpretable with Z.We close with Section 6.10, in which we give a very brief discussion of some furtherdevelopments.A corollary of the results here is a proof of the equivalence of the consistency ofMBT and the consistency of ZF(C), within a weak fragment of arithmetic such asEFA = exponential function arithmetic. In particular, this provides a proof of theconsistency of mathematics (as formalized by ZFC), assuming the consistency ofMBT. The same holds for the variants discussed earlier that are mutually interpretablewith ZF.We have also obtained a number of results in concept calculus involving a varietyof other informal concepts and a variety of formal systems, including ZF and beyond.We are planning a comprehensive book on concept calculus.6.2 Interpretation PowerThe notion of interpretation plays a crucial role in concept calculus. Interpretabilitybetween formal systems was first precisely defined by Alfred Tarski. We work in theusual framework of first-order predicate calculus with equality.DefinitionAn interpretation of S in T consists of:i. A one-place relation defined in T that is meant to carve out the domain of objectsthat S is referring to, from the point of view of T.ii. A definition of the constants, relations, and functions in the language of S byformulas in the language of T, whose free variables are restricted to the domainof objects that S is referring to (in the sense of i).iii. It is required that every axiom of S, when translated into the language of T bymeans of i and ii, becomes a theorem of T.It is now standard to allow quite a lot of flexibility in i through iii. Specifically: