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

The real numbers in inner models of set theory

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Joan BagariaWe study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, Gödel's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their generation process is linked to the properties of $L$ and its levels, mainly referring to [18]. We provide detailed proofs for the results of that paper, generalize them in some directions hinted at by the authors, and present a generalization of our own by introducing the concept of an infinite order gap, which is natural and yields some new insights. On the other hand, we present and prove some well-known results that build pathological sets of reals. We generalize this study to $L\left[\#_1\right]$ (the smallest inner model closed under the sharp operation for reals) and $L[\#]$ (the smallest inner model closed under all sharps), for which we provide some introduction and basic facts which are not easily available in the literature. We also discuss some relevant modern results for bigger inner models

On Projective Ordinals

We study in this paper the projective ordinals δ^1_n, where δ^1_n = sup{ξ: ξ is the length of ɑ Δ^1_n prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the "definable length" of the continuum. We prove first in §2 that projective determinacy implies δ^1_n 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in §3) the classical fact that δ^1_1 ℵ_l and the result of Martin that δ^1_3 = ℵ_(ω + 1) by proving that δ^1_(n2+1) = λ^+_(2n+1), where λ_(2n+1) is a cardinal of cofinality ω. Finally we discuss in §4 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that ∀α (α^# exists) implies that every δ^1_n with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles