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ConceptCalc072606 - 1 CONCEPT CALCULUS by Harvey M Friedman...

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1 CONCEPT CALCULUS by Harvey M. Friedman Ohio State University July 27, 2006 PREFACE. We present a variety of basic theories involving fundamental concepts of naive thinking, of the sort that were common in "natural philosophy" before the dawn of physical science. The most extreme forms of infinity ever formulated are embodied in the branch of mathematics known as abstract set theory, which forms the accepted foundation for all of mathematics. Each of these theories embodies the most extreme forms of infinity ever formulated, in the following sense. ZFC, and even extensions of ZFC with the so called large cardinal axioms, are mutually interpretable with these theories. This is an extended abstract. Proofs of the claims will appear elsewhere. INTRODUCTION. 1. BETTER THAN. 1.1. Better Than, Much Better Than. 1.2. Better Than, Real. 1.3. Better Than, Real, Conceivable. 2. VARYING QUANTITIES. 2.1. Single Varying Quantity. 2.2. Two Varying Quantities, Three Separate Scales. 2.3. Varying Bit. 2.4. Persistently Varying Bit. 2.5. Naive Time. 3. BINARY RELATIONS. 3.1. Binary Relation, Single Scale. 3.2. Binary Relation, Two Separate Scales. 4. MULTIPLE AGENTS, TWO STATES. 5. POINT MASSES. 5.1. Discrete Point Masses in One Dimension. 5.2. Discrete Point Masses with End Expansion. 5.3. Discrete Point Masses with Inner Expansion. 5.4. Point Masses with Inner Expansion. 5.5. Discrete Point Masses with Inner Expansion Revisited. 6. TOWARDS THE MEREOLOGICAL.
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2 INTRODUCTION We present a variety of basic theories involving fundamental concepts of naive thinking, of the sort that were common in "natural philosophy" before the dawn of physical science. The most extreme forms of infinity ever formulated are embodied in the branch of mathematics known as abstract set theory, which forms the accepted foundation for all of mathematics. Each of these theories embodies the most extreme forms of infinity ever formulated, in the following sense. ZFC, and even extensions of ZFC with the so called large cardinal axioms, are mutually interpretable with these theories. Physical science, as we know it today, is based on measuring quantities via the usual real number system, and counting objects via the usual natural number system. Here we instead use abstract linear orderings (sometimes without linearity), and formulate principles that can be seen to be incompatible with the identification of these linear orderings with the real numbers, or segments of the real numbers. The principles have a clear conceptual meaning to any naive thinker. At this point, we are not suggesting the overthrow of the real number orthodoxy in physical science. That would be grossly premature, and may never be appropriate. Our aims are quite different and more philosophical. i. We show how a complex of very simple intuitive ideas that can be absorbed and reasoned with by naive thinkers, leads surprisingly quickly and naturally to a consistency proof for mathematics (ZFC, even with large cardinals).
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