MathMeanMathLogic042100

Then a follows from s algebraically iff a follows

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Unformatted text preview: ols appearing in S. Then a follows from S* algebraically iff a follows from S algebraically. We can compare the least size of a Herbrand proof of a from S* and from S. There is a necessary and sufficient iterated exponential blowup in passing from S* to S. This corresponds to the situa-tion with cut elimination in mathematical logic. 2. THE INCOMPLETENESS THEOREMS. Gödel’s first incompleteness theorem asserts that in any consistent recursively axiomatized formal system whose axioms contain a certain minimal amount of arithmetic, there exist sentences that are neither provable nor refutable. (This is actually a sharpening of Gödel’s original theorem due to Rosser). This can be made more friendly by 1) looking only at systems with finitely many axioms; 2) making the “minimal amount of arithmetic” very friendly. However, formal systems still remain. So we wish to go further and capture the mathematical essence without using formal systems. To maximize friendliness, we incorporate work of Matiyasevich/Robinson/Davis/Putnam on Hilbert’s 10th problem. THEOREM 7. Let S be a finite set of statements in free variable logic, including the ring axioms, that hold universally in some algebra. There is a ring inequation that holds universally in the ring of integers but does not follow from S algebraically. Gödel’s second incompleteness theorem is more delicate than his first incompleteness theorem. It asserts that for any consistent recursively axiomatized formal system whose axioms contain a certain minimal amount of arithmetic, that system cannot prove its own consistency. 7 (This is actually a sharpening of Gödel’s ori-ginal theorem due to several people). Again, this can be made more friendly as before by 1) looking only at systems with finitely many axioms; 2) making the “minimal amount of arithmetic” very friendly. However, formal systems still remain, as well as issues concerning appropriate formalizations of consistency. So we go further and capture the mathematical essence without using formal systems. For this purpose, we introduce the concept of an interpretation. This concept, formalized by Tarski, is normally presented in terms of the first order predicate calculus with equality. Here we only use inte...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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