Bounded forcing axioms as principles of generic absoluteness
The abstract of the paper is quite descriptive: ``We show that the Bounded Forcing Axioms (for instance Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a \(\Sigma_1\) sentence of the language of set theory with parameters in a small transitive size is forceable, then it is true. We also show that the Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a \(\Sigma_3^1\) sentence with parameters is forceable, then it is true. Further, if for every real \(x\), \(x^\sharp\) exists, and the second uniform indiscernible is less than \(\omega_2\), then the same holds for \(\Sigma_4^1\) sentences.'' A forcing axiom for a poset \(\mathbb P\), \(\mathbf{FA}_\kappa(\mathbb P)\), asserts that for any family of at most \(\kappa\) maximal antichains of \(\mathbb P\), there is a filter which meets each of the antichains. Of course, the familiar Martin's Axiom is the assertion that \(\mathbf{FA}_\kappa(\mathbb P)\) holds for each ccc poset \(\mathbb P\) and each \(\kappa<\mathfrak c\). To introduce the notion of a \textbf{bounded} forcing axiom the reader must recall that each poset has a (unique) completion, let us denote the completion of \(\mathbb P\) as \(\overline{\mathbb P}\). The poset \(\mathbb P\) is a dense subset of \(\overline{ \mathbb P}\) and every subset of \(\mathbb P\) has a least upper bound in \(\overline{\mathbb P}\). Then a bounded forcing axiom for \(\mathbb P\), \(\mathbf{BFA}_\kappa(\mathbb P)\), asserts the same as \(\mathbf{FA}_\kappa(\mathbb P)\) except that the maximal antichains are \(\leq \kappa\) sized maximal antichains of \(\overline{\mathbb P}\). The notion of \(\Sigma_1(A)\)-absoluteness for a poset \(\mathbb P\) is defined as follows: if \(\varphi(x_1,\ldots, x_n)\) is a \(\Sigma_1\)-formula (only one quantifier, an existential, which does not bound its argument as in \(\exists x\in y\)), then for each \(\mathbb P\)-generic filter \(G\) and each \(a_1,\ldots, a_n\in A\), \(\varphi(a_1,\ldots,a_n)\) holds if and only if it holds in \(V[G]\). The interesting main result of the paper shows that for a poset \(\mathbb P\) and an infinite cardinal \(\kappa\), \(\mathbf{BFA}_\kappa(\mathbb P)\) is equivalent to \(\Sigma_1(\mathcal P(\kappa))\).
scientific article; zbMATH DE number 1523600