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Ω-logic
In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses 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 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. 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_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
ω-consistent Theory
In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) W. V. O. Quine (1971), ''Set Theory and Its Logic''. theory is a theory (collection of sentences) that is not only (syntactically) consistentS. C. Kleene, ''Introduction to Metamathematics'' (1971), p.207. Bibliotheca Mathematica: ''A Series of Monographs on Pure and Applied Mathematics'' Vol. I, Wolters-Noordhoff, North-Holland 0-7204-2103-9, Elsevier 0-444-10088-1. (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem. Definition A theory ''T'' is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of ''T'' so that ''T'' is able to prove the basic axioms of the natural numbers under this translation. A ''T'' tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |