ConIncompAmst051010

ConIncompAmst051010 - CONCRETE INCOMPLETENESS FROM EFA...

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CONCRETE INCOMPLETENESS FROM EFA THROUGH LARGE CARDINALS by Harvey M. Friedman Ohio State University www.math.ohio-state.edu/~friedman/ Institute for Logic, Language and Computation University of Amsterdam 1-3PM May 10, 2010 Monday, May 10, 2010
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THE ABSTRACT/CONCRETE DIVIDE There is a deep and growing conceptual divide between abstract set theory and normal mathematical culture. Normal mathematical culture is overwhelmingly concerned with finite structures, finitely generated structures, discrete structures (countably infinite), continuous and piecewise continuous functions between complete separable metric spaces, with lesser consideration of pointwise limits of sequences of such functions, and Borel measurable functions between complete separable metric spaces. More abstract mathematical objects are normally considered for two reasons: 1. They simplify presentations of material by avoiding the need for extra hypotheses. 2. They are used in proofs of normal statements. For 1, if great generality causes technical difficulties unrelated to the material, they are avoided in favor of less abstract formulations. This leaves 2 for a substantial role of abstract set theory in normal mathematical culture. Monday, May 10, 2010
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THE ABSTRACT/CONCRETE DIVIDE We will take the working definition of the Concrete as the universe of Borel measurable functions between complete separable metric spaces. Borel measurable functions of finite Borel rank between complete separable metric spaces. We have mentioned both as we like to keep the divide somewhat flexible. Obviously, this includes mathematical objects of far greater concreteness - particularly, countable structures. There are examples of uses of the Abstract for proving theorems in the Concrete - by normal mathematicians. Monday, May 10, 2010
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USES OF ABSTRACT FOR THE CONCRETE IN NORMAL MATHEMATICS The examples of normal mathematicians using the Abstract for the Concrete fall into these categories: 1. Convenience. It simplifies matters, and does not cause any special irrelevant difficulties. But it can obviously be removed, although actually removing it is judged not to be worth the effort. 2. Apparent necessity. There is no obvious way to remove it. Some ideas are needed to remove it. Interest varies concerning the issue. Subsequently, the Abstract is removed. 3. Necessity. It is known or believed that there is no way to remove it. Using “every field has an algebraic closure” is typical of category 1. We mention two particularly interesting cases of Apparent Necessity. Monday, May 10, 2010
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REMOVAL OF THE ABSTRACT FOR THE CONCRETE: TWO PARTICULARLY INTERESTING CASES FROM NORMAL MATHEMATICS We mention two cases. Both situations are fluid and ongoing. 1. Laver proved a number of new results about the free left
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ConIncompAmst051010 - CONCRETE INCOMPLETENESS FROM EFA...

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