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1 INTERPRETING SET THEORY IN ORDINARY THINKING: CONCEPT CALCULUS Harvey M. Friedman* Nineteenth Annual Tarski Lectures Interpretations of Set Theory in Discrete Mathematics and Informal Thinking Lecture 3 Delivered, April 13, 2007 Expanded, May 24, 2007 1. Interpretation hierarchy (lecture #1). 2. The general nature of concept calculus. 3. Better than, much better than. 4. Single varying quantity. 5. Single varying bit. 6. Persistently varying bit. 7. Some models. 1. INTERPRETATION HIERARCHY (Lecture #1). The first of these Tarski lectures was devoted to a discussion of Tarski interpretability. This bears repeating. STARTLING OBSERVATION. Any two natural theories S,T, known to interpret EFA, are known (with small numbers of exceptions) to have: S is interpretable in T or T is interpretable in S. The exceptions are believed to also have comparability. Exceptions to linearity are known, which have clearly identifiable elements of artificiality. It would be interesting to explore just how natural one can (artificially!) make the incomparability. Because of this observation, there has emerged a rather large linearly ordered table of “interpretation powers” represented by natural formal systems. Generally, several natural formal systems may occupy the same position. We call this growing table, the Interpretation Hierarchy . 2. THE GENERAL NATURE OF CONCEPT CALCULUS.
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2 Concept Calculus is a new mathematical/philosophical program of wide scope. The development of Concept Calculus began in Summer, 2006. There is a report in [1]. Concept Calculus promises to connect mathematics, philosophy, and commonsense thinking in a radically new way. Advances in Concept Calculus are made through rigorous mathematical findings, and promise to be of immediate and growing interest to philosophers. Developments in Concept Calculus generally consist of the following. a. An identification of a few related concepts from informal thinking. In the various developments, the choice of these concepts will vary greatly. In fact, all concepts from ordinary language are prime targets. b. Formulation of a variety of fundamental principles involving these concepts. These various principles may have various degrees of plausibility, and may even be incompatible with each other. There may be no agreement among philosophers as to just which principles to accept. Concept Calculus is concerned only with logical structure. c. Formulation of a variety of systems of such fundamental principles in b. These systems generally combine several such fundamental principles in some attractive way. d. An identification of the location of the theory in the interpretation hierarchy. e. In particular, for each of the resulting systems, a determination of whether they interpret mathematics – as formalized by ZFC.
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