GoedelBlessCurse020811

GoedelBlessCurse020811 - ADVENTURES IN LOGIC FOR...

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ADVENTURES IN LOGIC FOR UNDERGRADUATES by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Lecture 4. Gödel’s Blessing and Gödel’s Curse
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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
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KURT G Ö DEL Kurt Gödel (1906-1978) is responsible for the most celebrated results in mathematical logic of the 20th century. He had a singular ability to identify and focus on the central issues in logic. He combined powerful mathematical thinking with philosophical insights to transform logic into a highly sophisticated mathematical subject of great general intellectual interest. His doctoral dissertation (Habilitationsschrift) was accepted by the University of Vienna in 1932. He came to the USA in 1933 as a Visiting Professor at the Institute for Advanced Studies in Princeton, New Jersey, where he remained as a Professor until his death in 1978.
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GÖDEL’S BLESSING: COMPLETENESS Gödel’s Blessing is Completeness. Gödel’s Curse is Incompleteness. We have already encountered completeness theorems in the first two lectures. We won’t be relying on the first two lectures. The most relevant completeness theorem goes back to Gödel in his dissertation, and was discussed in general terms in the second lecture. We do need to give a high level review of Gödel’s famous COMPLETENESS THEOREM FOR PREDICATE CALCULUS in order to set the stage for the main business of this lecture.
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REVIEW OF PREDICATE CALCULUS - COMPLETENESS THEOREM In predicate calculus, we have assertions (formulas) and mathematical structures (models). The assertions and the structures are required to be of a specific kind which we will review in the next two slides. Formulas are true or false only relative to structures. Thus we speak of a given formula as being true in a given structure. PREDICATE CALCULUS COMPLETENESS THEOREM (Gödel 1928). There is a basic finite set of axioms and rules of inference, operating on the formulas, such that the following holds. A formula is true in all structures iff it is provable in this system. You DON’T NEED TO UNDERSTAND this brief review of predicate calculus to take advantage of the remainder of this lecture.
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REVIEW OF PREDICATE CALCULUS - FORMULAS From Lecture 2: Formulas in predicate calculus use i. Variables. v 1 ,v 2 ,... ii. Connectives. ¬, , , , iii. Quantifiers. , iv. Constant symbols. c 1 ,c 2 ,... v. Relation symbols. R n m , n,m 1 vi. Function symbols. F n m , n,m 1 vii. Equality. = Example from Lecture 2 of a formula of predicate calculus: ( x)(¬R(x,y) ( z)(F(z,H(y),c) = d ¬S(z,c))) x,y,z, variables; c,d, constant symbols; R,S, 2-ary relation symbols; H, 1-ary function symbol, F, 3-ary function symbol.
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REVIEW OF PREDICATE CALCULUS - STRUCTURES
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