Set Theory

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Omega-logic is the natural logic given by forcing:a sentence is Omega-valid if it holds in all rankinitial segments of all forcing extensions of V. Ifthere exists a proper class of Woodin cardinals thenwhether or not a given sentence is Omega-valid is itselfgenerically invariant. There is a natural candidateof the notion of proof for Omega-logic and the Omega-Conjectureis simply the conjecture that if there is a proper classof Woodin cardinals then for any sentence, the sentenceis Omega-provable if and only if it is Omega-valid.The significance of the Omega-Conjecture isthat if it is true then one has a complete analysis ofwhat can be achieved by forcing and moreover the setof Omega-valid sentences is definable in third ordernumber theory. This places a limit on the possibilitiesfor generic absoluteness.The Omega-Conjecture is also generically invariant andso it is unlikely to be independent is the same fashionthat CH is. Therefore any refutation of the Omega-Conjecture must come from large cardinal axioms. The Omega-Conjecture holds in the current generation of inner models (extender models). The focus of the proposal is to investigate the possibilitythat some large cardinal hypothesis refutes theOmega-Conjecture by examining extender models for large cardinals beyond superstrong.These extender models differ from the current familyin that long extenders are allowed on the sequence.Preliminary results have established several key points.First, for the standard generalization of extender modelsto the case of long extenders, comparison fails as soon asone reaches sequences of extenders for which the movingspaces problem arises (this problem was first identified by Steel).Nevertheless there is family of extender models which effectively exhaust all known large cardinal axioms and for theseextender sequences the moving spaces problem does not arise (the sequences are suitably short). If comparisoncan be established for these models then Omega-Conjectureholds in these inner models. As a corollary one would obtainthat no known large cardinal hypothesis can refutethe Omega-Conjecture. Finally if a suitable iterationhypothesis holds in these inner models then assumingappropriate large cardinals in V (extendible cardinals)the Omega-Conjecture must hold in V. Finally if, as onemight naturally expect, there are definable versions of these extendermodels then there are a number of truly profound corollaries.These include the following: if there is an extendible cardinal then HOD correctly computes the successor for a proper class of singular cardinals.Cantor's Continuum Hypothesis is arguably the most famousunsolvable problem in Mathematics, it was the first problemon Hilbert's list of 23 problems from 1900. The Continuum Hypothesissimply asserts that any infinite collection of real numbersis either equinumerous with the integers or equinumerouswith the collection of all real numbers.Around 40 years ago this problem was shown to be formallyunsolvable from the axioms of Set Theory. This seminal workof Cohen introduced a new technique to Set Theory, the method of forcing. However the fact that the Continuum Hypothesisis formally unsolvable does not necessarily imply thatit cannot be solved. Indeed the experience in Set Theoryover the last 40 years has demonstrated that some questionswhich are formally unsolvable can be answered. But theprecise methodology used in these cases cannot workfor the problem of the Continuum Hypothesis.The Omega-Conjecture (proposed around 10 years ago) arisesnaturally from an abstract analysis of the method of forcingbut in the context of so called large cardinal axioms. Ifthe Omega-Conjecture is true then there is an argument, or atleast strong evidence, that the Continuum Hypothesis is false. But independent of this, whether or not the Omega-Conjecture is true has profound consequences for the foundations of Set Theory. Cohen's method of forcing cannot be used to establish thatthe Omega-Conjecture is unsolvable so it is reasonable toexpect that whether or not the Omega-Conjecture is truecan be resolved. Any refutation of the Omega-Conjecturemust come from large cardinal axioms. This research projectconcerns trying to prove the Omega-Conjecture by a detailedanalysis of the hierarchy of large cardinal axioms. Preliminaryresults have been obtained which offer strong evidence forplausibility of this approach.