Fri Jan 26, 2007
60 Evans Hall, 4:10–6 PM
W. Hugh Woodin (UC-Berkeley)
The Continuum Hypothesis, the Generic Multiverse, and the Ω Conjecture
This is a version of a talk given at the Cohen72 meeting at Stanford last September; this version will be directed more toward logicians. Godel’s axiom of constructibility is the canonical example of an axiom which provides a “semi-complete” theory for the universe of sets. But this axiom is not compatible with the known large cardinal axioms. Further, the independence of the Continuum Hypothesis challenges the idea that there is a meaningful conception of the universe of sets. The developments over the last 40 years have mitigated this challenge somewhat but the fact remains: there is no known extension of the axioms of set theory which is both compatible with large cardinal axioms and “semi-complete”. Finally, the credibility of any claim that the Continuum Hypothesis has an answer is severely challenged by the complete lack of any “semi-complete” theory for the universe of sets (again, which is compatible with large cardinals). Does this argue for a multiverse conception of truth? While this may look like an issue solely for philosophers, there are some intriguing mathematical questions and surprises which arise from the consideration of this question.