Tuesday, September 7, 2004 Overcoming Goedel's Incompleteness Theorem |
Abstract: (I will review the basic notions of a 'first order formal theory', so that the talk should be accessible to all. ) Goedel's Incompleteness theorem implies that a complete axiomatic formal "foundation" of mathematics a la Hilbert is not possible---or so it seems. Although this hardly affects the work of any mathematician, one does feel a sense of apprehension regarding the problem with the touchstone of mathematical truth---deducibility from first principles. We adopt a different viewpoint which lets us by-pass Goedel's theorem and provides a means to better understand the deductive structure of mathematics. If we call a 'nice' (formal) theory an aristotelian system, then Goedel says that the deductive structure of mathematics is not aristotelian. However, it is trivially a LOCALLY aristotelian, meaning it can be covered by more than one separate aristotelian systems, which agree wherever they overlap. (This is somewhat similar to saying that a sphere is not euclidean, but is locally euclidean.) The crucial result of our investigation is a 'compactness theorem'
which says that FINITELY MANY such aristotelian systems are sufficient
to 'cover' mathematics. Thus, Hilbert's purely finitary formal foundation
is, in principle, possible. Finally, it can be shown that TWO such
patches are sufficient to cover mathematics----first order theory
of sets (say ZFC), and a first order theory of categories. This is
very satisfactory, because set-theoretic and categoric viewpoints
are also conceptually complementary---the former being introverted
and the latter being extroverted viewpoint on mathematical structures. |