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MathMeanMathLogic042100

MathMeanMathLogic042100 - 1 THE MATHEMATICAL MEANING OF...

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1 THE MATHEMATICAL MEANING OF MATHEMATICAL LOGIC by Harvey M. Friedman [email protected] http://www.math.ohio-state.edu/~friedman/ April 15, 2000 Rev. April 21, 2000 I am going to discuss the mathematical meaning of 1. the completeness theorem. 2. the incompleteness theorems. 3. recursively enumerable sets of integers. 4. constructivity. 5. the Ackerman hierarchy. 6. Peano arithmetic. 7. predicativity. 8. Zermelo set theory. 9. ZFC and beyond. Each of these theorems and concepts arose from very specific considerations of great general interest in the foundations of mathematics (f.o.m.). They each serve well defined purposes in f.o.m. Naturally, the preferred way to formulate them for mathe-matical logicians is in terms that are close to their roots in f.o.m. However, the core mathematician does not come out of the f.o.m. tradition as does the mathematical logician. Instead, he/she comes out of the much older arithmetic/ algebraic/geometric (a.a.g.) tradition. The significance of these theorems and concepts are not readily apparent from the a.a.g. point of view. In fact, a full formulation of these theorems and concepts requires the introduction of rather elaborate structures which can only be properly appreciated from a distinctly f.o.m. perspective. In fact, the a.a.g. perspective is of little help in gaining facil-ity with these elaborate structures. So the core mathematician, steeped in a.a.g, is very unlikely to spend the considerable effort required to understand the meaning of such theorems and concepts. In wading through these develop-ments, he/she will not be putting the a.a.g. perspective to effective use, and will not anticipate any corresponding proportionate a.a.g. payoff.
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2 Of course, there is nothing to prevent a core mathematician from becoming familiar with and being perfectly comfort-able with the f.o.m. tradition. But for various reasons, this has become quite rare. To give a prime example of what I have in mind, most of these theorems and concepts depend on the syntax and semantics of so called first order predicate calculus with equality. This is a rather elaborate structure which, with proper substantial and detailed discussion, sounds like beautiful music to the ears of an f.o.m. oriented listener - but more like painful, long winded noise to many others. So in this talk, I want to give relatively a.a.g. friendly presentations of these fundamental theorems and concepts from f.o.m. I say relatively because I do not attempt to go all the way here. One can go much further. But I do go far enough in the direction of a.a.g. friendliness that the mathematical meaning of these presentations should be apparent to this audience. Bear in mind that the project of systematically giving such a.a.g. friendly treatments is, as far I know, quite new, and raises substantial issues - both technical and conceptual - about which I know very little at the present time.
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