The PI proposes to continue work on several problems in set theory, using the methods of set and class forcing, infinite combinatorics, infinitary logic, and fine structure. Previous work has shown that certain combinatorial characterization problems do not have first-order solutions. For example, in general there is no first-order definition of the set of subsets of omega-2 that, in some omega-1 and omega-2 preserving outer model, have a closed unbounded subset. Other examples regarding subsets of other cardinals, branches through trees of certain sorts, and large homogeneous subsets for certain partitions are known. One line of work concerns settling some further cases. Another line of work concerns a more fundamental question. Can this phenomenon can be mitigated by working relative to some reasonable extension of ZFC, for example, one that is consistent with all large cardinal axioms. The logically most simple case of incompleteness of the third kind lies (necessarily) just beyond the scope of Woodin's celebrated generic absoluteness theorem: Assume CH. Consider Sigma-2-1 sentences of analysis with sets of reals as parameters. In general the set of such sentences that are satisfiable in some outer model having the same reals is not first-order definable. Does there exist an extension of ZFC that is consistent with all large cardinal axioms and relative to which this set is (lightface) Delta-2-2 definable? Finally, the PI is interested in several questions regarding class forcing.
The proposed work centers on incompleteness of a "third kind" in set theory. Incompleteness in set theory is important because all mathematics can be formalized in set theory. Propositions that are neither provable nor refutable from the axioms of set theory cannot be settled mathematically, at least in our current understanding. Goedel's famous Incompleteness Theorems show that such propositions exist. Sentences demonstrating this first kind of incompleteness formalize metamathematical statements. For example, the formalization of "ZFC is consistent" is neither provable nor refutable from the axioms of set theory (ZFC), provided those axioms are, in fact, consistent. Even though "ZFC is consistent" is not provable from ZFC, there is an obvious reason to favor it over its negation---studying mathematics within ZFC presupposes that ZFC is consistent. Incompleteness results proved using Cohen's method of (set) forcing represent a second kind of incompleteness. Typically, given a "standard" model of ZFC, one constructs an outer model in which a given statement is true and one in which it is false. In the case of this second kind of incompleteness, there is often no reason to favor a statement or its negation. Deep work by Woodin, Steel, Martin, Foreman, and a number of others has suggested reasons to favor certain statements up through a certain level of logical complexity. Just beyond this level of logical complexity lies incompleteness of a third kind. Here it is not possible even to say which statements are satisfiable in some outer model. "Characterization problems" are the combinatorial form of this phenomenon. Past work of the PI has highlighted that, in general, it is not possible to characterize in set theory the "satiable objects" of certain sorts. Precise statements are technical, but an analogy gives the general idea. In this analogy, the "objects" correspond to equations. An object is "sated" if the corresponding equation is solvable. An object is "satiable" if the corresponding equation is potentially solvable, that is, either solvable or solvable in some larger number system. In elementary mathematics, there is no reason to distinguish solvable and potentially solvable equations because typically there exist maximal number systems in which every potentially solvable equation of a particular sort is actually solvable. Such "maximal standard models" do not exist in set theory. The analog of an anticharacterization result in set theory would be a type of equation for which there cannot be a good criterion for potential solvability. Anticharacterization is troubling for two reasons. First, in mathematics one expects that anything that is true is true for a good reason---so there ought to be a criterion for insatiability. Secondly, in their strongest form, these anticharacterization results hold only if the universe fails to be "sufficiently non-minimal". To some extent this threatens the well established thesis that all of mathematics is formalizable in first-order set theory, because non-minimality cannot be expressed in this language. The PI seeks to explore three aspects of anticharacterization. First, he seeks to determine whether some specific cases of characterization problems are solvable. Secondly, he seeks to discover whether adding auxiliary axioms to the usual axioms of set theory might allow satiable objects to be characterized. This could be construed as evidence in favor of these axioms. Finally, he seeks to continue work on abstract class forcing with an eye towards understanding general outer models, at least in the presence of conditions that render models highly non-minimal.
