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Foundations021611

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ADVENTURES IN LOGIC FOR UNDERGRADUATES by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Lecture 5. Foundations of Mathematics

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LECTURE 1. LOGICAL CONNECTIVES. Jan. 18, 2011 LECTURE 2. LOGICAL QUANTIFIERS. Jan. 25, 2011 LECTURE 3. TURING MACHINES. Feb. 1, 2011 LECTURE 4. GÖDEL’S BLESSING AND GÖDEL’S CURSE. Feb. 8, 2011 LECTURE 5. FOUNDATIONS OF MATHEMATICS Feb. 15, 2011 SAME TIME - 10:30AM SAME ROOM - Room 355 Jennings Hall WARNING: CHALLENGES RANGE FROM EASY, TO MAJOR PARTS OF COURSES
FOUNDATIONS OF MATHEMATICS In mathematics, we find a huge variety of concepts. We also find “proofs” that we believe are beyond dispute. But what exactly are the rules of the game? I.e., what are the allowable methods of creating new concepts, and what are the allowable methods of reasoning in proofs? Some very precise (mathematical!) structures have evolved to address this question. These structures evolved through the efforts and insights of many mathematicians and mathematical philosophers, such as Cauchy, Dedekind, Boole, Cantor, Frege, Russell, Zermelo, Frankel, and others. The culmination of these efforts has led to the current foundations of mathematics called the ZFC axiom system.

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VARIETY OF MATHEMATICAL CONCEPTS We find a huge variety of concepts in mathematics. For example: integers, rationals, reals, complexes, addition, subtraction, multiplication, exponentiation, sequence, series, less than, greater than, sets, functions, graphs, relations, member, semigroups, groups, rings, fields, integral domains, vector spaces, topological spaces, continuous functions, differentiable functions, analytic functions, and so forth. We also have logical notions that allow us to make assertions and do reasoning. We have already talked about the logical notions in the first two Lectures: the variables v 1 ,v 2 ,...; the connectives ¬, , , , ; the quantifiers , , and =. These nine items can be reduced, but we gain very little by this, and it is not generally done.
REDUCTION OF MATHEMATICAL CONCEPTS integers, rationals, reals, complexes, addition, subtraction, multiplication, exponentiation, sequence, series, less than, greater than, sets, functions, graphs, relations, member, semigroups, groups, rings, fields, integral domains, vector spaces, topological spaces, continuous functions, differentiable functions, analytic functions, and so forth. We keep v 1 ,v 2 ,...,¬, , , , , , ,=. We definitely benefit from sharply reducing the mathematical concepts. Experience has shown that the most advantageous reduction is to SETS WITH THE MEMBERSHIP RELATION. Alternative reductions have been proposed, but none have overthrown this classic standard choice of primitive concepts.

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SET THEORETIC FOUNDATIONS OF MATHEMATICS Everything is a set. The only mathematical relation is membership between sets, written . I.e., x y. The logical symbols are v 1 ,v 2 ,...,¬, , , , , , ,=.
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