In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure
Just as the axiom of projective determinacy yields a canonical theory of
, he sought to find axioms that would give a canonical theory for the larger structure.
The theory he developed involves a controversial argument that the continuum hypothesis is false.
Woodin's Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most results in the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completeness theorem.
Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent
The theory involves a definition of Ω-validity: a statement is an Ω-valid consequence of a set theory T if it holds in every model of T having the form
There is also a notion of Ω-provability;[1] here the "proofs" consist of universally Baire sets and are checked by verifying that for every countable transitive model of the theory, and every forcing notion in the model, the generic extension of the model (as calculated in V) contains the "proof", restricted its own reals.
A complexity measure can be given on the proofs by their ranks in the Wadge hierarchy.
Woodin showed that this notion of "provability" implies Ω-validity for sentences which are
In all currently known core models, it is known to be true; moreover the consistency strength of the large cardinals corresponds to the least proof-rank required to "prove" the existence of the cardinals.