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GodelLegPanelNij080906

GodelLegPanelNij080906 - 1 GODELS LEGACY IN MATHEMATICAL...

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1 GODEL’S LEGACY IN MATHEMATICAL PHILOSOPHY by Harvey M. Friedman Ohio State University LC ’06 Gödel Legacy Panel Nijmegen, Netherlands Presented August 2, 2006 Revised August 9, 2006 NOTE: This is an edited version of my 20 minute lecture at the Gödel legacy panel of the LC ‘06. Gödel's definitive results and his essays leave us with a rich legacy of philosophical programs that promise to be subject to mathematical treatment. After surveying some of these, we focus attention on the program of circumventing his demonstrated impossibility of a consistency proof for mathematics by means of extramathematical concepts. This program has seen substantial progress through our new Concept Calculus. 1. INCOMPLETENESS THEOREMS. Gödel’s First Incompleteness theorem asserts that any consistent formal system obeying rather weak conditions, such as PA = Peano Arithmetic, is incomplete. Gödel’s Second Incompleteness theorem asserts that any consistent formal system obeying rather weak conditions, equipped with a “standard” formalized proof predicate with which to formulate its own consistency, cannot prove its own consistency. This suggests a number of obviously important programs, which are, to various degrees, implicit and explicit in Gödel’s writings, and certainly well known to him. 1a. What mathematics can be formalized in a way that is not subject to the First Incompleteness theorem, so that one has completeness? Perhaps the most well known examples of real depth are the complete axiomatizations of the ordered group of integers
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2 (Presburger) and the ordered field of reals (Tarski) and Euclidean geometry (Tarski). I think that there is room for new dramatic results where complete axiomatizations of natural significant portions of mathematics are given. However, many of these will likely involve rather imaginative delineations that are not simply the full first order theory of some fundamental structures, as was the case with Presburger and Tarski.
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