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Hilb2panel020107

# Hilb2panel020107 - 1 CONTEMPORARY PERSPECTIVES ON HILBERT'S...

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1 CONTEMPORARY PERSPECTIVES ON HILBERT'S SECOND PROBLEM AND THE GÖDEL INCOMPLETENESS THEOREMS AMS/ASL PANEL DISUCSSION Moderator: Akihiro Kanamori Panelists: Harvey M. Friedman, David E. Marker, Michael Rathjen January 6, 2007 1. First Incompleteness Theorem. 2. First Incompleteness Theorem in ( ,<,0,1,+,•). (!!!) 3. Second Incompleteness Theorem. 4. Is there any real logical strength? 5. Strict Reverse Mathematics. It is not yet clear just what the most illuminating ways of rigorously stating the Incompleteness Theorems are. This is particularly true of the Second. Also I believe that there are more illuminating proofs of the Second that have yet to be uncovered. NOTE: See “Formal Statements of Godel’s Second Incompleteness Theorem”, http://www.math.ohio- state.edu/%7Efriedman/ There is also a very interesting viscously anti foundational argument which suggests that mathematics can be developed in a way that can be proved to be free of contradiction in Peano Arithmetic, or even in weak fragments such as Exponential Function Arithmetic = EFA = I S 0 (exp) – thereby suggesting that the Incompleteness Theorems are an irrelevant and misleading distraction. Refutation of this mind numbing heresy is ongoing and leads to some very interesting formal work, called Strict Reverse Mathematics. NOTE: See “The Inevitability of Logical Strength”, February, 2007, recently submitted for publication. http://www.math.ohio-state.edu/%7Efriedman/ 1. FIRST INCOMPLETENESS THEOREM. R.M. Robinson’s Q. L(Q) = 0,S,+,•, with =, and

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2 1. Sx 0. 2. Sx = Sy x = y. 3. x 0 ( \$ y)(x = Sy). 4. x+0 = x. 5. x+Sy = S(x+y). 6. x•0 = 0. 7. x•Sy = (x•y)+x. THEOREM 1.1. Let T be a consistent many sorted theory with finitely many axioms, and p be an interpretation of Q in T. Then there is a sentence j of L(Q) such that p ( j ) is neither provable nor refutable in T. Note that Theorem 1.1 is extremely clean and fully rigorously stated. There is an important extension that is not so clean. THEOREM 1.2. Let T be a consistent many sorted theory, and p be an interpretation of Q into T. Assume that the set of axioms of T is recursively enumerable. Then there is a sentence j of L(Q) such that p ( j ) is neither provable nor refutable in T. Now the statement involves Gödel numberings of syntax. Questions of robustness occur, which are nowadays considered trivial.
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