Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.Comment: 3 figure

Stably Measurable Cardinals

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second uniform indiscernible for bounded subsets of $\kappa$: $u_2(\kappa)$, and secondly to give the consistency strength of a property of L\"ucke's. Theorem: The following are equiconsistent: (i) There exists $\kappa$ which is stably measurable; (ii) for some cardinal $\kappa$, $u_2(\kappa)=\sigma(\kappa)$; (iii) The {\boldmath $\Sigma_1$}-club property holds at a cardinal $\kappa$. Here $\sigma(\kappa)$ is the height of the smallest $M \prec_{\Sigma_1} H(\kappa^+)$ containing $\kappa+1$ and all of $H(\kappa)$

Canonical Truth

We introduce and study a notion of canonical set theoretical truth, which means truth in a `canonical model', i.e. a transitive class model that is uniquely characterized by some $\in$-formula. We show that this notion of truth is `informative', i.e. there are statements that hold in all canonical models but do not follow from ZFC, such as Reitz' ground model axiom or the nonexistence of measurable cardinals. We also show that ZF+$V=L[\mathbb{R}]$+AD has no canonical models. On the other hand, we show that there are canonical models for `every real has sharp'. Moreover, we consider `theory-canonical' statements that only fix a transitive class model of ZFC up to elementary equivalence and show that it is consistent relative to large cardinals that there are theory-canonical models with measurable cardinals and that theory-canonicity is still informative in the sense explained above

An undecidable extension of Morley's theorem on the number of countable models

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of $\sigma$-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.Comment: 31 page

Determinacy of refinements to the difference hierarchy of co-analytic sets

In this paper we develop a technique for proving determinacy of classes of the form ω²-Π¹₁+Γ (a refinement of the difference hierarchy on Π¹₁ lying between ω²-Π¹₁ and (ω²+1)-Π¹₁) from weak principles, establishing upper bounds for the determinacy- strength of the classes ω²-Π¹₁+Σ^0_α for all computable α and of ω²-Π¹₁+Δ¹₁. This bridges the gap between previously known hypotheses implying determinacy in this region