Omega-logic is the natural logic given by forcing: a sentence is Omega-valid if it holds in all rank initial segments of all forcing extensions of V. If there exists a proper class of Woodin cardinals then whether or not a given sentence is Omega-valid is itself generically invariant. There is a natural candidate of the notion of proof for Omega-logic and the Omega-Conjecture is simply the conjecture that if there is a proper class of Woodin cardinals then for any sentence, the sentence is Omega-provable if and only if it is Omega-valid. The significance of the Omega-Conjecture is that if it is true then one has a complete analysis of what can be achieved by forcing and moreover the set of Omega-valid sentences is definable in third order number theory. This places a limit on the possibilities for generic absoluteness.
The Omega-Conjecture is also generically invariant and so it is unlikely to be independent is the same fashion that 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 possibility that some large cardinal hypothesis refutes the Omega-Conjecture by examining extender models for large cardinals beyond superstrong. These extender models differ from the current family in that long extenders are allowed on the sequence. Preliminary results have established several key points. First, for the standard generalization of extender models to the case of long extenders, comparison fails as soon as one reaches sequences of extenders for which the moving spaces 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 these extender sequences the moving spaces problem does not arise (the sequences are suitably short). If comparison can be established for these models then Omega-Conjecture holds in these inner models. As a corollary one would obtain that no known large cardinal hypothesis can refute the Omega-Conjecture. Finally if a suitable iteration hypothesis holds in these inner models then assuming appropriate large cardinals in V (extendible cardinals) the Omega-Conjecture must hold in V. Finally if, as one might naturally expect, there are definable versions of these extender models 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 famous unsolvable problem in Mathematics, it was the first problem on Hilbert's list of 23 problems from 1900. The Continuum Hypothesis simply asserts that any infinite collection of real numbers is either equinumerous with the integers or equinumerous with the collection of all real numbers.
Around 40 years ago this problem was shown to be formally unsolvable from the axioms of Set Theory. This seminal work of Cohen introduced a new technique to Set Theory, the method of forcing. However the fact that the Continuum Hypothesis is formally unsolvable does not necessarily imply that it cannot be solved. Indeed the experience in Set Theory over the last 40 years has demonstrated that some questions which are formally unsolvable can be answered. But the precise methodology used in these cases cannot work for the problem of the Continuum Hypothesis.
The Omega-Conjecture (proposed around 10 years ago) arises naturally from an abstract analysis of the method of forcing but in the context of so called large cardinal axioms. If the Omega-Conjecture is true then there is an argument, or at least 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 that the Omega-Conjecture is unsolvable so it is reasonable to expect that whether or not the Omega-Conjecture is true can be resolved. Any refutation of the Omega-Conjecture must come from large cardinal axioms. This research project concerns trying to prove the Omega-Conjecture by a detailed analysis of the hierarchy of large cardinal axioms. Preliminary results have been obtained which offer strong evidence for plausibility of this approach.
