Three Days of Ω-logic
نویسنده
چکیده
The Zermelo-Fraenkel axioms for set theory with the Axiom of Choice (ZFC) form the most commonly accepted foundations for mathematical practice, yet it is well-known that many mathematical statements are neither proved nor refuted from these axioms. One example is given by Gödel’s Second Incompleteness Theorem, which says that no consistent recursive axiom system which is strong enough to describe arithmetic on the integers can prove its own consistency. This presents a problem for those interested in finding an axiomatic basis for mathematics, as the consistency of any such theory would seem to be as well justified as the theory itself. There are many views on how to resolve this situation. While some logicians propose weaker foundations for mathematics, some study the question of how to properly extend ZFC. Large cardinal axioms and their many consequences for small sets have been studied as a particularly attractive means to this goal. Large cardinals resolve the projective theory of the real line and banish the various measure-theoretic paradoxes derivable from the Axiom to Choice from the realm of the definable. Moreover, the large cardinal hierarchy itself appears to serve as universal measuring stick for consistency strength, in that the consistency strength of any natural statement of mathematics (over ZFC) can be located on this hierarchy (see [7] for the definitive reference on large cardinals). While there is no technical definition of “large cardinal,” Woodin’s Ω-logic is an attempt to give a formal definition for the set of consequences of large cardinals for rank initial segments of the universe. Two definitions are proposed. The first, a form of generic invariance, is called the semantic relation: a theory T is said to imply a statement φ in Ω-logic if φ holds in every rank initial segment satisfying T in every set forcing extension (we deal only with set forcing in this article). Another more elaborate notion, involving a correctness property with respect to universally Baire sets of reals, plays the role of proofs. Woodin’s Ω-conjecture is the statement that these two definitions are
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