ConcCalcINF050309

ConcCalcINF050309 - 1 MUCH BETTER THAN Harvey M. Friedman

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Unformatted text preview: 1 MUCH BETTER THAN Harvey M. Friedman friedman@math.ohio-state.edu http://www.math.ohio-state.edu/~friedman/ May 3, 2009 ABSTRACT. This is the initial publication on Concept Calculus, which establishes mutual interpretability between formal systems based on informal commonsense concepts and formal systems for mathematics through abstract set theory. Here we work with axioms for "better than" and "much better than", and the Zermelo and Zermelo Frankel axioms for set theory. 1. Introduction. 2. Interpretation Power. 3. Basic Facts About Interpretation Power. 4. Better Than, Much Better Than. 5. Interpretation in ZF. 6. Interpretation in Z. 7. An Interpretation of Z. 8. An interpretation of ZF. 9. Some Further Results. 1. INTRODUCTION. We have discovered an unexpectedly close connection between the logic of mathematical concepts and the logic of informal concepts from common sense thinking. Our results indicate that they are, in a certain precise sense, equivalent. This connection is new and there is the promise of establishing similar connections involving a very wide range of informal concepts. We call this development the Concept Calculus. In this paper, we focus on just one context for Concept Calculus. We use two particular informal concepts from common sense thinking. These are the informal binary relations BETTER THAN. MUCH BETTER THAN. 2 As discussed in section 9, these relations can be looked at mereologically, using the part/whole and the infinitesimal part/whole relation. Sections 2,3 contain background information about interpretability between theories, which should be informative for readers not familiar with this fundamentally important concept credited to Alfred Tarski. We are now preparing a book on this topic (see [FVxx]). In section 4, we present some basic axioms involving "better than", "much better than", and identity. These axioms are of a simple character, and range from obvious to intriguingly plausible. In section 5, we give an interpretation of the system MBT (much better than) presented in section 4, in ZF. In section 6, we give an interpretation of ZF in MBT. It is well known that ZF and ZFC (ZF with the axiom of choice) are mutually interpretable. Thus we can replace ZF with ZFC in the statement of our results. The same remark applies to Z and ZC (Z with the axiom of choice). A Corollary of the results in sections 4,5 is a proof of the equivalence of the consistency of MBT and the consistency of ZF(C), within a weak fragment of arithmetic such as EFA = exponential function arithmetic. In particular, this provides a proof of the consistency of mathematics (as formalized by ZFC), assuming the consistency of MBT. We have also obtained a number of results in Concept Calculus involving a variety of other informal concepts, and a variety of formal systems including ZF and beyond. We are planning a comprehensive book on Concept Calculus....
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ConcCalcINF050309 - 1 MUCH BETTER THAN Harvey M. Friedman

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