|Date: Tuesday, November 02, 2010
Title: Ziwet Lecture I: The search for Ultimate L (II, III Wednesday and Thursday at 4PM).
1) Title: Ã‚Â¬Ã¢â‚¬Â The search for Ultimate L.
Many of the basic questions of Set Theory, such as that of the Continuum Hypothesis, are not formally solvable on the basis of the ZFC axioms. But this does not imply that such questions cannot be answered. I shall review how strong axioms of infinity (these are large cardinal axioms) have been used to answer questions which, like that of the Continuum Hypothesis, cannot be answered on the basis of the ZFC axioms. Ã‚Â¬Ã¢â‚¬Â These questions, which concern simple subsets of Euclidean space, also date back to the beginnings of Set Theory.
This investigation has led to the identification, for the first time, of a candidate for an ultimate version of Goedel's constructible universe L. The axiom which asserts that this is the Universe of Sets resolves essentially all the questions of modern Set Theory which have been shown (using Cohen's method of forcing) to be unsolvable on the basis of the ZFC axioms.
2) Title: Ã‚Â¬Ã¢â‚¬Â Ã‚Â¬Ã¢â‚¬Â Forcing Axioms and unsolvable problems
An alternative approach to isolating new axioms for Set Theory involves Forcing Axioms which are simply generalizations of the Baire Category Theorem. This program has led to a provably maximum version of these axioms, Martin's Maximum, and remarkably, this axiom implies that the cardinal of the continuum is aleph_2. Recently there have been a second generation of results proved from Martin's Maximum. This includes the solution of the Basis Problem for Uncountable Orders by Justin Moore and the solution by Ilijas Farah to the problem of Brown-Douglas-Fillmore on automorphisms of the Calkin algebra.
Is Martin's Maximum the missing axiom?
3) Title: Multiverse views and the $\Omega$ Conjecture
Perhaps Ã‚Â¬Ã¢â‚¬Â the ubiquity of unsolvable problems in Set Theory is simply evidence for a multiverse conception of the universe of sets. Cohen's method of forcing, which is the basic technique for establishing unsolvability of problems with Set Theory, leads naturally to the generic-multiverse of sets. The corresponding conception of set theoretic truth in turn leads directly to the Omega Conjecture which if true arguably shows that the generic-multiverse conception is in essence a version of formalism. This compels one to reject the generic-multiverse conception as a basis for truth in Set Theory unless one can refute the Omega Conjecture, or otherwise argue it has no positive answer. Ã‚Â¬Ã¢â‚¬Â But the Omega Conjecture is invariant across the generic-multiverse and so it cannot be declared meaningless on the basis of the generic-multiverse conception of truth.
The Omega Conjecture thus emerges as a key problem and the investigation of it leads back to ultimate L.
Speaker: Hugh Woodin
Institution: UC Berkeley