ShockOSU041610

ShockOSU041610 - SHOCKING(?) UNPROVABILITY by Harvey M....

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Unformatted text preview: SHOCKING(?) UNPROVABILITY by Harvey M. Friedman OSU Math Dept Columbus, Ohio April 16, 2010 1. WHAT KIND OF SHOCK? 2. GENERAL THESIS. 3. BASES AND KERNELS. 4. LOCAL BASES. 5. LOCAL BASIS CONSTRUCTIONS. 6. POLYNOMIALS AND LOWER YIELDS. 7. INFINITARY LOCAL BASIS CONSTRUCTION THEOREMS. 8. FINITARY LOCAL BASIS CONSTRUCTION THEOREMS. 9. ORDER INVARIANT LOCAL BASIS CONSTRUCTION THEOREMS. 10. UPPER SHIFT LOCAL BASIS THEOREM ON Q k . The work reported on here is ultimately aimed at the working mathematician who is not particularly concerned with issues in the foundations of mathematics. We have maintained a variety of contacts with working mathematicians inside and outside OSU. We wish to acknowledge the highly valuable feedback I have obtained in recent years from Ovidiu Costin (analysis) and Stephen Milne (combinatorics/number theory) from our Department. WHAT KIND OF SHOCK? Mathematical Logic had a glorious period in the 1930s, which was briefly rekindled in the 1960s. Any Shock Value, such as it is, has surrounded unprovability from ZFC. These unprovability results had some shock value: 1. ZFC is free of contradiction is not provable in ZFC. (Goedel 1930s). 2. Every uncountable set of reals is in one-one correspondence with the set of all real numbers is not refutable in ZFC (Goedel 1930s). 3. Every uncountable set of reals is in one-one correspondence with the set of all real numbers is not provable in ZFC (Cohen, 1960s). Since then, there have been unprovability results from ZFC more connected with mathematics, but entirely on the set theoretic side of mathematics. Old Shock Value has long since worn off. It is essential for logic to expand unprovability from ZFC into the finite - and ultimately to familiar, beautiful, and essential finite contexts. I have been trying to do this for about 100,000 hours over about 40 years. On occasion, I make a report. It is almost always quickly replaced by a major improvement. Today is going to be no exception. GENERAL THESIS There appears to be a general concept of an inductive construction of a structure satisfying constraints. The constraints are to be met at any stage. Each relevant object is processed only after all of the relevant objects have been previously processed. However, we can attempt to radically alter the construction so that objects are processed at a given stage well before relevant objects have been processed. The multiple objects processed at a given stage will include all objects built out of the previously processed objects, regardless of whether we are really ready to know what to do with them with any confidence. It is not at all clear how we can carry out such a construction meeting the constraints - except for one method....
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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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ShockOSU041610 - SHOCKING(?) UNPROVABILITY by Harvey M....

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