Ω-logic

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 ${\displaystyle H_{\aleph _{2}}}$. Just as the axiom of projective determinacy yields a canonical theory of ${\displaystyle H_{\aleph _{1}}}$, 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. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over ${\displaystyle H_{\aleph _{2}}}$ (in Ω-logic), it must imply that the continuum is not ${\displaystyle \aleph _{1}}$. Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent ${\displaystyle \Pi _{2}}$ (over ${\displaystyle H_{\aleph _{2}}}$) sentence is true; this axiom implies that the continuum is ${\displaystyle \aleph _{2}}$.

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