Absoluteness for Universally Baire Sets and the Uncountable I

Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion of independence results in the following decades. Many other statements about infinite sets, such as the Borel Conjecture, Whitehead’s problem, and automatic continuity for Banach Algebras, were proved independent, perhaps leaving an impression that most nontrivial statements about infinite sets can be neither proved nor refuted in ZFC. Moreover, some classical statements imply the consistency of ZFC and stronger theories, and by Gödel’s incompleteness theorems the consistency of these statements with ZFC can be proved only by using strong axioms of infinity, so-called large cardinal axioms. A classical example is Banach’s ‘Lebesgue measure has a σ-additive extension to all sets of reals.’ While it is fairly easy to find a model in which this is false and there is no known ZFC-proof of its negation, proving the consistency of this statement requires assuming the existence of a measurable cardinal ([32]). A remarkable result was proved by Shoenfield ([31]): every statement of the form (∃x ∈ R)(∀y ∈ R)φ(x, y), where all quantification in φ is over the natural numbers and all of its parameters are real numbers, is absolute between models of ZFC that are transitive and contain all countable ordinals. In this form Shoenfield’s theorem is best possible, as it cannot even be improved by adding one more alternation of quantifiers ranging over R. However, a corollary that the truth of any Σ2 statement (i.e., one of the above syntactical form) cannot be changed by forcing turned out to be susceptible to far-reaching generalizations. One of the more striking results in modern set theory is that the existence of suitable large cardinals implies that the theory of the inner model L(R) (the smallest inner model of ZF, the usual axioms of Set Theory without the Axiom of Choice, containing all real numbers) cannot be changed by set forcing (see [13, 21]). In particular, a sentence with real parameters and any number of alterations of quantifiers ranging over R, has a fixed truth value that cannot be changed by forcing. The impact of large cardinals on sets of reals goes well beyond L(R) to imply absoluteness for certain canonical sets of reals, the universally Baire sets ([11]), as defined below. A remarkable consequence is that the existence of large cardinals outright implies that all sets of reals in L(R), and indeed all universally Baire sets, share all the classical regularity properties of Borel sets such as Lebesgue measurability.