Maximal Methods for Small Sets

Miami University, Oxford OH

Investigators

Abstract

We intend to study maximal models for the powersets of the first two uncountable cardinals as realized by the forcing method in the context of large cardinals and determinacy, and the applications of these models to other areas, especially topology. Many of these issues complement a new theory of the infinite developed by W. Hugh Woodin called Omega-logic. By now it is a well established theme in set theory that large cardinals impose certain forms of canonicity and absoluteness on the universe of sets. In particular, the existence of certain large cardinals implies that the theories of certain definable inner models of the universe are invariant under forcing. Furthermore, these large cardinals also tend to give rise to a detailed structure theory for these inner models. The prototypical results of this type are results of Woodin, building on work of Foreman, Magidor, Shelah, Martin and Steel, showing that a proper class of Woodin cardinals implies that the theory of the least inner model of set theory containing the reals and the ordinals (L(R)) cannot be changed by set forcing, and that this fixed theory includes the Axiom of Determinacy. One natural program in the wake of these results is to identify and study larger models for which similar results hold. Another direction, noting that the Axiom of Determinacy contradicts the Axiom of Choice, is to find similar forms of absoluteness compatible with AC. One way of doing this is to consider statements to the effect that the universe of sets is closed under certain forcing operations. Such statements are typically called forcing axioms. Another approach is to consider forcing extensions of these inner models of determinacy. One major advance in this direction is Woodin's forcing Pmax. Heuristically, every natural question about the subsets of the first uncountable cardinal should have an answer in the Pmax extension of L(R). Nonetheless, there are several important questions about the Pmax extension which remain open. Some of these questions concern the properties of the nonstationary ideal on the first uncountable cardinal. One goal in pursuing these questions is to develop a finer analysis of the Pmax extension. In the other direction there is the issue of whether results obtained by Pmax can be obtained by other methods. Furthermore, the Pmax method has a number of variations, some of which have found application in topology. Cohen's method of forcing is a way of taking models of the mathematical universe and producing larger, often very different models. We intend to study properties of the first two uncountable cardinals as realized by the forcing method in the context of the regularity imposed by assuming the existence of large infinite objects (large cardinals) and certain regularity properties for set of real numbers (determinacy), and the applications of these models to other areas, especially topology. Many of these issues complement a new theory of the infinite developed by W. Hugh Woodin. By now it is a well established theme in set theory that large cardinals impose certain forms of canonicity and absoluteness on the universe of sets. In particular, the existence of certain large cardinals implies that the theories of certain definable inner models of the universe are invariant under forcing. Furthermore, these large cardinals also tend to give rise to a detailed structure theory for these inner models. One natural program in the wake of these results is to identify and study larger models for which similar results hold. One way of doing this is to consider statements to the effect that the universe of sets is closed under certain forcing operations. Another approach is to consider forcing extensions of canonical inner models of determinacy. One major advance in this direction is Woodin's forcing Pmax. Heuristically, every natural question about the subsets of the first uncountable cardinal should have an answer in the Pmax extension. Nonetheless, there are several important questions about the Pmax extension which remain open. One goal in pursuing these questions is to develop a finer analysis of the Pmax extension. In the other direction there is the issue of whether results obtained by Pmax can be obtained by other methods. Furthermore, the Pmax method has a number of variations, some of which have found application in other areas of mathematics.