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Unformatted text preview: 1 CONCEPT CALCULUS APA PANEL on LOGIC IN PHILOSOPHY by Harvey M. Friedman http://www.math.ohio-state.edu/~friedman/ Distinguished University Professor of Mathematics, Philosophy, and Computer Science Ohio State University Delivered December 28, 2007 Revised January 1, 2008 Revised January 2, 2008 1. Introduction. 2. Interpretation Power. 3. Basic Facts About Interpretation Power. 4. Initial Development of Concept Calculus. 5. Single Varying Quantity. 6. Single Varying Bit. 7. Persistently Varying Bit. 8. An Interpretation In Set Theory. 1. INTRODUCTION. We focus on an unexpectedly close connection between the logic of mathematical concepts and the logic of informal concepts from common sense thinking. 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. It is in a very early stage of development. The results in Concept Calculus are rather specific. For the initial result, we identify the “logic of mathematical concepts” to be the usual axioms of set theory that provide the usual foundations of mathematics - the ZFC axioms (Zermelo Frankel with the axiom of choice). And for the initial result, we focus on a two particular informal concepts from common sense thinking. These are the binary relations 2 BETTER THAN. MUCH BETTER THAN. In the initial result, we present some axioms involving only “better than”, “much better than”, and identity between objects. These axioms are of a simple basic character, and range from obvious to plausible. The initial result asserts the following. Let T be the system of axioms involving better than and much better than, presented below. THEOREM. ZFC and T are mutually interpretable. I.e., there is an interpretation of ZFC in T, and there is an interpretation of T in ZFC. COROLLARY. ZFC is consistent if and only if T is consistent. It should be noted that both of these results are proved in an extremely weak fragment of ordinary mathematics. If we use Gödel numberings throughout, then these results are proved in a very weak fragment of PRA = primitive recursive arithmetic, which is itself a very weak fragment of PA = Peano Arithmetic. In particular, EFA = exponential function arithmetic = I ∑ (exp), suffices. With some care, PFA = polynomial function arithmetic = bounded arithmetic = I ∑ , suffices. There are corresponding weak theories of finite strings that suffice if we treat formal systems more directly, without Gödel numberings. We also show that certain extensions of ZFC via so called “large cardinal hypotheses” correspond in the same way to certain very natural extensions of T, still using only better than and much better than....
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