Ω-logic
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In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, Ω-logic is an
infinitary logic An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s. Some infinitary logics may have different properties from those of standard first-order lo ...
and
deductive system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
proposed by as part of an attempt to generalize the theory of
determinacy Determinacy is a subfield of game theory and set theory that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "dete ...
of
pointclass In the mathematical field of descriptive set theory, a pointclass is a collection of Set (mathematics), sets of point (mathematics), points, where a ''point'' is ordinarily understood to be an element of some perfect set, perfect Polish space. In ...
es to cover the structure H_. Just as the axiom of projective determinacy yields a canonical theory of H_, 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 In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
is false.


Analysis

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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over H_ (in Ω-logic), it must imply that the continuum is not \aleph_1. Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent \Pi_2 (over H_) sentence is true; this axiom implies that the continuum is \aleph_2. Woodin also related his Ω-conjecture to a proposed abstract definition of large cardinals: he took a "large cardinal property" to be a \Sigma_2 property P(\alpha) of ordinals which implies that α is a strong inaccessible, and which is invariant under forcing by sets of cardinal less than α. Then the Ω-conjecture implies that if there are arbitrarily large models containing a large cardinal, this fact will be provable in Ω-logic. 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 V^\mathbb_\alpha for some ordinal \alpha and some forcing notion \mathbb. This notion is clearly preserved under forcing, and in the presence of a proper class of Woodin cardinals it will also be invariant under forcing (in other words, Ω-satisfiability is preserved under forcing as well). There is also a notion of Ω-provability; 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. For a proof-set ''A'' the condition to be checked here is called "''A''-closed". A complexity measure can be given on the proofs by their ranks in the
Wadge hierarchy In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. W ...
. Woodin showed that this notion of "provability" implies Ω-validity for sentences which are \Pi_2 over ''V''. The Ω-conjecture states that the converse of this result also holds. 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.


Notes


References

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External links

*W. H. Woodin
Slides for 3 talks
{{DEFAULTSORT:Omega Logic Set theory Systems of formal logic