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ConcCalcInf082709 - 1 CONCEPT CALCULUS MUCH BETTER THAN...

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1 CONCEPT CALCULUS: MUCH BETTER THAN Harvey M. Friedman* [email protected] http://www.math.ohio-state.edu/~friedman/ May 4, 2009 revised August 28, 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. Some Implications. 6. Interpretation of MBT in ZF. 7. Interpretation of B + VSDE + VSDE + SSDE in Z. 8. Interpretation of Z in B + SDE. 9. Interpretation of ZF in MBT. 10. 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
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2 BETTER THAN. MUCH BETTER THAN. As discussed briefly in section 10, 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 our basic axioms involving "better than", "much better than", and identity. These axioms are of a simple character, and range from obvious to intriguingly plausible. We anticipate that Concept Calculus is extremely flexible, so that axioms can be chosen to accommodate many diverse points of view - while still maintaining the mutual interpretability with systems such as Z and ZF that we establish here. For instance, the axioms investigated here preclude there being a best object. There are important viewpoints where a best object is an essential component. We anticipate a formulation accommodating a best object that stands in relation to the system here as does class theory to set theory. In section 4, you will find three groups of axioms. BASIC. DIVERSE EXACTNESS. STRONG DIVERSE EXACTNESS. VERY STRONG DIVERSE EXACTNESS. SUPER STRONG DIVERSE EXACTNESS. UNLIMITED IMPROVEMENT. STRONG UNLIMITED IMPROVEMENT.
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