Maximal Methods for Small Sets

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We intend to study maximal models for the powersets of the firsttwo uncountable cardinals as realized by the forcing method in thecontext of large cardinals and determinacy, and the applicationsof these models to other areas, especially topology. Many of theseissues complement a new theory of the infinite developed by W. Hugh Woodin called Omega-logic. By now it is a well establishedtheme in set theory that large cardinals impose certain forms ofcanonicity and absoluteness on the universe of sets. Inparticular, the existence of certain large cardinals implies thatthe theories of certain definable inner models of the universe areinvariant under forcing. Furthermore, these large cardinals alsotend to give rise to a detailed structure theory for these innermodels. The prototypical results of this type are results ofWoodin, building on work of Foreman, Magidor, Shelah, Martin andSteel, showing that a proper class of Woodin cardinals impliesthat the theory of the least inner model of set theory containingthe reals and the ordinals (L(R)) cannot be changed by setforcing, and that this fixed theory includes the Axiom ofDeterminacy. One natural program in the wake of these results isto identify and study larger models for which similar resultshold. Another direction, noting that the Axiom of Determinacycontradicts the Axiom of Choice, is to find similar forms ofabsoluteness compatible with AC. One way of doing this is toconsider statements to the effect that the universe of sets isclosed under certain forcing operations. Such statements aretypically called forcing axioms. Another approach is to considerforcing extensions of these inner models of determinacy. One majoradvance in this direction is Woodin's forcing Pmax. Heuristically,every natural question about the subsets of the first uncountablecardinal should have an answer in the Pmax extension of L(R).Nonetheless, there are several important questions about the Pmaxextension which remain open. Some of these questions concern theproperties of the nonstationary ideal on the first uncountablecardinal. One goal in pursuing these questions is to develop afiner analysis of the Pmax extension. In the other direction thereis the issue of whether results obtained by Pmax can be obtainedby other methods. Furthermore, the Pmax method has a number ofvariations, some of which have found application in topology.Cohen's method of forcing is a way of taking models of themathematical universe and producing larger, often very differentmodels. We intend to study properties of the first two uncountablecardinals as realized by the forcing method in the context of theregularity imposed by assuming the existence of large infiniteobjects (large cardinals) and certain regularity properties forset of real numbers (determinacy), and the applications of thesemodels to other areas, especially topology. Many of these issuescomplement a new theory of the infinite developed by W. HughWoodin. By now it is a well established theme in set theory thatlarge cardinals impose certain forms of canonicity andabsoluteness on the universe of sets. In particular, the existenceof certain large cardinals implies that the theories of certaindefinable inner models of the universe are invariant underforcing. Furthermore, these large cardinals also tend to give riseto a detailed structure theory for these inner models. One naturalprogram in the wake of these results is to identify and study larger models for which similar results hold. One way of doingthis is to consider statements to the effect that the universe ofsets is closed under certain forcing operations. Another approachis to consider forcing extensions of canonical inner models ofdeterminacy. One major advance in this direction is Woodin'sforcing Pmax. Heuristically, every natural question about thesubsets of the first uncountable cardinal should have an answer inthe Pmax extension. Nonetheless, there are several importantquestions about the Pmax extension which remain open. One goal inpursuing these questions is to develop a finer analysis of thePmax extension. In the other direction there is the issue ofwhether results obtained by Pmax can be obtained by other methods.Furthermore, the Pmax method has a number of variations, some ofwhich have found application in other areas of mathematics.