MathMeanMathLogic042100

Om oriented listener but more like painful long

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Unformatted text preview: . 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. For many of these presenta-tions, in order to be a.a.g. friendly, I do a certain amount of cheating. For instance, these presentations may be substantially less general than usual, even to the point of focusing on only a few illustrative examples. 1. THE COMPLETENESS THEOREM. The Gödel completeness theorem for first order predicate calculus with equality (1928) has a very simple formulation: every (set of) sentence(s) that is true in all structures has a proof (in fopce). Of course, this simplicity hides the fact that there is an elaborate system of defini-tions underneath that are not a.a.g. friendly. I start our treatment with equational logic. Let us consider systems of the form (D,f1,...,fn), where D is a nonempty set, n ≥ 0, and each fi is a multivariate function from D into D. We allow the arity of the various fi to be various nonnegative integers. The significance of arity 0 is that of a constant. 3 At the risk of offending most people in the audience, I call such a system D = (D,f1,...,fn) an algebra. The type of D is (k1,...,kn), where ki is the arity of fi. Ex: Groups. These are certain algebras of type (0,2,1). Here f1 is the identity element, f2 is the group operation, and f3 is the additive inverse operation. We build terms using the variables xj, j ≥ 1, and the functions f1,...,fn. One is normally pedantic, actually using function symbols F1,...,Fn standing for unknown actual functions f1,...,fn. Ex: In the type (0,2,1) of groups, the terms are just the words. In the type (0,0,2,1,2) of rings, the terms are just the polynomials with integer coefficien...
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