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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
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
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
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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