RUI: Incompleteness of the Third Kind in Set Theory

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The PI proposes to continue work on several problems in set theory, using themethods of set and class forcing, infinite combinatorics, infinitary logic,and fine structure. Previous work has shown that certain combinatorialcharacterization problems do not have first-order solutions. For example, ingeneral there is no first-order definition of the set of subsets of omega-2that, in some omega-1 and omega-2 preserving outer model, have a closedunbounded subset. Other examples regarding subsets of other cardinals,branches through trees of certain sorts, and large homogeneous subsets forcertain partitions are known. One line of work concerns settling some furthercases. Another line of work concerns a more fundamental question. Can thisphenomenon can be mitigated by working relative to some reasonable extensionof 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 genericabsoluteness theorem: Assume CH. Consider Sigma-2-1 sentences of analysiswith sets of reals as parameters. In general the set of such sentences thatare satisfiable in some outer model having the same reals is not first-orderdefinable. Does there exist an extension of ZFC that is consistent with alllarge cardinal axioms and relative to which this set is (lightface) Delta-2-2definable? Finally, the PI is interested in several questions regarding classforcing.The proposed work centers on incompleteness of a "third kind" in set theory.Incompleteness in set theory is important because all mathematics can beformalized in set theory. Propositions that are neither provable norrefutable from the axioms of set theory cannot be settledmathematically, at least in our current understanding. Goedel's famousIncompleteness Theorems show that such propositions exist. Sentencesdemonstrating this first kind of incompleteness formalize metamathematicalstatements. For example, the formalization of "ZFC is consistent" is neitherprovable nor refutable from the axioms of set theory (ZFC), provided thoseaxioms are, in fact, consistent. Even though "ZFC is consistent" is notprovable from ZFC, there is an obvious reason to favor it over itsnegation---studying mathematics within ZFC presupposes that ZFC is consistent.Incompleteness results proved using Cohen's method of (set) forcing representa 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 inwhich it is false. In the case of thissecond kind of incompleteness, there is often no reason to favor a statementor its negation. Deep work by Woodin, Steel, Martin, Foreman,and a number of others has suggested reasons to favor certain statements upthrough a certain level of logical complexity. Just beyond this level oflogical complexity lies incompleteness of a third kind. Here it is notpossible 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 notpossible to characterize in set theory the "satiable objects" of certainsorts. Precise statements are technical, but an analogy gives the generalidea. In this analogy, the "objects" correspond to equations. An object is"sated" if the corresponding equation is solvable. An object is "satiable" ifthe corresponding equation is potentially solvable, that is, either solvableor solvable in some larger number system. In elementary mathematics, there isno reason to distinguish solvable and potentially solvable equations becausetypically there exist maximal number systems in which every potentiallysolvable equation of a particular sort is actually solvable. Such "maximalstandard models" do not exist in set theory. The analog of ananticharacterization result in set theory would be a type of equation forwhich there cannot be a good criterion for potential solvability.Anticharacterization is troubling for two reasons. First, in mathematics oneexpects that anything that is true is true for a good reason---so there oughtto be a criterion for insatiability. Secondly, in their strongest form, theseanticharacterization results hold only if the universe fails to be"sufficiently non-minimal". To some extent this threatens the well establishedthesis that all of mathematics is formalizable in first-order set theory,because non-minimality cannot be expressed in this language. The PIseeks to explore three aspects of anticharacterization. First, he seeks todetermine whether some specific cases of characterization problems aresolvable. Secondly, he seeks to discover whether adding auxiliary axioms tothe usual axioms of set theory might allow satiable objects to becharacterized. This could be construed as evidence in favor of these axioms.Finally, he seeks to continue work on abstract class forcing with an eyetowards understanding general outer models, at least in the presence ofconditions that render models highly non-minimal.