We study projective stationary sets. The Projective Stationary Reflection principle is the statement that every projective stationary set contains an increasing continuous $\in$--chain of length $ω_1$. We show that if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also it is equivalent to the Strong Reflection Principle. We show that the saturation of the nonstationary ideal on $ω_1$ is equivalent to a certain kind of reflection.
We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set r…
Bounded stationary reflection at a cardinal $λ$ is the assertion that every stationary subset of $λ$ reflects but there is a stationary subset of $λ$ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $μ> \aleph_ω$ and models in which bounded stationary reflection holds at $μ^+$ but the approachability property fails at $μ$.
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of ideals.
We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(κ)$ introduced by Brodsky and Rinot for the purpose of constructing $κ$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $κ$ but the stronger is not. We then prove that, if $μ$ is a singular cardinal, $\square_μ$ implies the existence of a special $μ^+$-tree with a $\mathrm{cf}(μ)$-ascent path, thus answering a question of Lücke.
A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a stationary subset of alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.
It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result is best possible.
We prove that there is a certain degree of independence between stationary reflection phenomena at different cofinalities; e.g. it is consistent that every stationary subset of S_1^3 reflects at a point of cofinality aleph_2 while every stationary subset of S^3_0 has a non-reflecting stationary subset
If $S,T$ are stationary subsets of a regular uncountable cardinal $κ$, we say that $S$ reflects fully in $T$, $S<T$, if for almost all $α\in T$ (except a nonstationary set) $S \cap α$ is stationary in $α.$ This relation is known to be a well founded partial ordering. We say that a given poset $P$ is realized by the reflection ordering if there is a maximal antichain $\langle X_p ; p \in P \rangle$ of stationary subsets of $Reg(κ)$ so that $$\forall p,q \in P \; \forall S\subseteq X_p, T\subseteq X_q \text{ stationary}:(S<T \leftrightarrow p<_P q ) .$$ We prove that if $κ$ is $\Cal P _2 κ-$stro…
It is well-known that the square principle $\square_λ$ entails the existence of a non-reflecting stationary subset of $λ^+$, whereas the weak square principle $\square^*_λ$ does not. Here we show that if $μ^{\mathrm{cf}(λ)} < λ$ for all $μ< λ$, then $\square^*_λ$ entails the existence of a non-reflecting stationary subset of $E^{λ^+}_{\mathrm{cf}(λ)}$ in the forcing extension for adding a single Cohen subset of $λ^+$. It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplet…
A stationary subset $S$ of a regular uncountable cardinal $κ$ {\it reflects fully} at regular cardinals if for every stationary set $T \subseteq κ$ of higher order consisting of regular cardinals there exists an $α\in T$ such that $S \cap α$ is a stationary subset of $α$. {\it Full Reflection} states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than $κ$ having Mitchell order $κ^{++}$ it is consistent that Full Reflection holds at every $λ\leq κ$ and $κ$ is measurable.
We prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every stationary family of countable subsets of mu^+ reflect in some subset of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad stationary set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four stationary sets.
We prove that it is consistent that Club Stationary Reflection and the Special Aronszajn Tree Property simultaneously hold on $ω_2$, thereby contributing to the study of the tension between compactness and incompactness in set theory. The poset which produces the final model follows the collapse of an ineffable cardinal first with an iteration of club adding (with anticipation) and second with an iteration specializing Aronszajn trees. In the first part of the paper, we prove a general theorem about specializing Aronszajn trees on $ω_2$ after forcing with what we call $\mathcal{F}$-Strongly Pr…
We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $κ$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$ (both with and without $\mathsf{AP}(κ^{++})$) and models of the conjunctions $\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$ and $\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using di…
We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by…
Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to $<\aleph_2$ is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters down to $<2^{\aleph_0}$ implies that the size of the continuum is $\aleph_2$. In contrast, an internal interpretation of the stationary logic can satisf…
section 2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of S_{< aleph_2}(lambda) = {a subseteq lambda : |a| < aleph_2}]. section 3 - NPT is not transitive. We prove NPT(lambda, mu) + NPT(mu, kappa) not => NPT(lambda, kappa)
Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the Löwenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the situations in the models obtained by mixed support iteration of a supercompact length and then collapsing another supercompact cardinal to make it $(2^{\aleph_0})^+$. We show, among other things, that the reflection down to $< 2^{\aleph_0}$ of the non-metrizability of topological spaces with small character is independent from the reflection properties studied in [Fu…
We show that the tree property, stationary reflection and the failure of approachability at $κ^{++}$ are consistent with $\mathfrak{u}(κ) = κ^+ < 2^κ$, where $κ$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $λ$ is a regular cardinal, then stationary reflection at $λ^+$ is indestructible under all $λ$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).
We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.
Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the Löwenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets ($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the restriction of the LST property to (downward) reflection of $Π^1_1$ formulas, which we call the $Π^1_1 \! \! \downarrow$-LST property, is equivalent to the \emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results from \cite{Cox_RP_IS}, this shows that the $Π^1_1 \! \! \downarrow$-LST Property for Stationary Lo…
The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/95042…
A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle are analyzed, and consequences are presented. Some of these consequences are ``exact'' versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.
Let kappa be a regular uncountable cardinal and lambda >=kappa^+ . The principle of stationary reflection for P_kappa lambda has been successful in settling problems of infinite combinatorics in the case kappa=omega_1. For a greater kappa the principle is known to fail at some lambda. This note shows that it fails at every lambda if kappa is the successor of a regular uncountable cardinal or kappa is countably closed.
For an arbitrary forcing class $Γ$, the $Γ$-fragment of Todorcevic's strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for $Γ$ implies the $Γ$-fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their correspond…
A cardinal $λ$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < λ$, $λ$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large cardinals can be satisfied robustly by small cardinals. We focus in particular on stationary reflection and the tree property, both of which can consistently hold but fail to be robust at small cardinals. We introduce natural strengthenings of these principles which are always robust and which hold at sufficiently large cardinals, consider the extent to which these strengthe…
