Strong Omega Consistency

Abstract

This dissertation attempts to formalize the informal observation that large cardinal axioms are consistent with each other and are well-ordered in terms of consistency strength. In Chapter 1, we review the more general observation that natural'' mathematical theories tend to be well-ordered in terms of consistency strength, and we survey several unnatural'' counterexamples to well-ordering in the consistency strength hierarchy. In Chapter 2, we observe that the hierarchy of beta-consistent sentences is well-ordered and generally agrees with the ordering of large cardinal axioms in terms of consistency strength, but this hierarchy admits sentences that are inconsistent with each other. In Chapter 3, we use Hugh Woodin's Omega-logic to define a hierarchy of strongly Omega-consistent large cardinal axioms. We show, under certain strong hypotheses, that the strongly Omega-consistent large cardinal axioms are well-ordered and jointly strongly Omega-consistent with each other.