Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $δ$, $δ$ is strong limit, $2^δ=δ^{+3}$ and the tree property at $δ^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].
We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of $ZFC$ $+$ $CH$. In this paper we present a sufficient condition (for sets of cardinals) to be consistently the set of cardinalities of branches of Kurepa trees.
We prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.
Starting from the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
Assuming the existence of a strong cardinal $κ$ and a measurable cardinal above it, we force a generic extension in which $κ$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $κ^{++}$.
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
Assume $λ$ is a singular limit of $η$ supercompact cardinals, where $η\leq λ$ is a limit ordinal. We present two forcing methods for making $λ^+$ the successor of the limit of the first $η$ measurable cardinals while the tree property holding at $λ^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{η^2+1}$ with the failure of $SCH$ at $\aleph_{η^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and S…
An inaccessible cardinal $κ$ is supercompact when $(κ, λ)$-ITP holds for all $λ\geq κ.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where simultaneously $(\aleph_2, μ)$-ITP and $(\aleph_3, μ')$-ITP hold, for all $μ\geq \aleph_2$ and $μ'\geq \aleph_3.$
By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it has kappa many branches for some kappa strictly between omega_1 and 2^{omega_1}. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist Kurepa trees and there are no Jech--Kunen trees, (2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees with omega_3 many branches exist.
An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every natural number n greater than 1 and for every ordinal mu greater than or equal to aleph_n, we have (aleph_n, mu)-ITP.
Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For $κ$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite $d \ge 2$, we prove the consistency of the Halpern-Läuchli Theorem on $d$ many $κ$-trees at a measurable cardinal $κ$, given the consistency of a $κ+d$-strong cardinal. This follows from a more general consistency result at measurable $κ$, which includes the possibility of infinitely many trees, assuming partition relation…
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…
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining…
Let $\mathrm{cof}(μ)=μ$ and $κ$ be a supercompact cardinal with $μ<κ$. Assume that there is an increasing and continuous sequence of cardinals $\langleκ_ξ\mid ξ<μ\rangle$ with $κ_0:=κ$ and such that, for each $ξ<μ$, $κ_{ξ+1}$ is supercompact. Besides, assume that $λ$ is a weakly compact cardinal with $\sup_{ξ<μ}κ_ξ<λ$. Let $Θ\geqλ$ be a cardinal with $\mathrm{cof}(Θ)>κ$. Assuming the $\mathrm{GCH}_{\geqκ}$, we construct a generic extension where $κ$ is strong limit, $\mathrm{cof}(κ)=μ$, $2^κ= Θ$ and both $\mathrm{TP}(κ^+)$ and $\mathrm{TP}(κ^{++})$ hold. Further, in this model there is a very …
Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $κ$, there are $κ^+$-Aronszajn trees and all such trees are special.
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.
We investigate the consistency strength of the statement: $κ$ is weakly compact and there is no tree on $κ$ with exactly $κ^{+}$ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly $κ^{+}$ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal $κ$ the nonexistence of a tree on $κ$ with exactly $κ^{+}$ many branches and, in particular, the Perfect Subtree Property for $κ$, implies the consistency of $AD_{\mathbb{R}} + DC$.
We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notions of semi-proper and weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We prove the existence of semi-proper trees under certain cardinal arithmetic assumptions. W…
Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_κ$ consisting of perfect trees of height $λ$ and width $κ$ where $κ$ is a singular $ω$-strong limit of cofinality $λ$. They showed that if $κ$ is singular of countable cofinality, then $\mathbb{P}_κ$ is minimal for $ω$-sequences assuming that $κ$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_κ$ is $(ω,ν)$-distributive for all $ν<κ$ given a singular $ω$-strong limit cardinal $κ$ of countable cofinality, and Dobrinen et al$.$ asked whether …
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…
In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{la…
Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $λ,$ there is no universal graph of size $λ$.
Starting from a stationary set of supercompact cardinals we find a generic extension in which the tree property holds at every regular cardinal between $\aleph_2$ and $\aleph_{ω^2}$.
If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. …
Let $κ$,$λ$ be regular uncountable cardinals such that $λ> κ^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(κ) = λ$ starting from a ground model in which $o(κ) = λ$ and prove that assuming $\neg 0^¶$, $s(κ) = λ$ implies that $o(κ) \geq λ$ in the core model.
We investigate the unbalanced ordinary partition relations of the form $λ\rightarrow {(λ, α)}^{2}$ for various values of the cardinal $λ$ and the ordinal $α$. For example, we show that for every infinite cardinal $κ,$ the existence of a $κ^{+}-$Suslin tree implies $κ^{+} \not\rightarrow {\left( κ^{+}, {\log}_κ(κ^{+}) + 2 \right)}^{2}$. The consistency of the positive partition relation $\mathfrak{b} \rightarrow {(\mathfrak{b}, α)}^{2}$ for all $α< ω_{1}$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.
We prove that for regular $λ$ above a strong limit singular $μ$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $λ$ and $μ$ there are coboundedly many regular $κ<μ$ such that $\clubsuit^-(S^λ_κ)$ holds whenever $λ=λ^{<κ}$.
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
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.
For g < f in omega^omega we define c(f,g) be the least number of uniform trees with g-splitting needed to cover a uniform tree with f-splitting. We show that we can simultaneously force aleph_1 many different values for different functions (f,g). In the language of Blass: There may be aleph_1 many distinct uniform Pi^0_1 characteristics.
We provide a model theoretical and tree property like characterization of $λ$-$Π^1_1$-subcompactness and supercompactness. We explore the behaviour of those combinatorial principles at accessible cardinals.
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <ω$, is consistent with an arbitrary continuum function below $\aleph_ω$ which satisfies $2^{\aleph_n} > \aleph_{n+1}$, $n<ω$. Thus the tree property has no provable effect on the continuum function below $\aleph_ω$ except for the restriction that the tree property at $κ^{++}$ implies $2^κ>κ^+$ for every infinite $κ$.
We prove the consistency, assuming an ineffable cardinal, that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham-Shelah that any two normal $ω_1$-Aronszajn trees are club isomorphic, which follows from $\textsf{PFA}$. The statement that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic implies that there are no $ω_2$-Suslin trees, so our proof also expands on the method of Laver-Shelah for obtaining the $ω_2$-Suslin hypothesis.
In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of $\diamondsuit$. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path,…
We identify a premouse inner model $L[\mathbb{E}]$, such that for any coarsely iterable background universe $R$ modelling $\mathrm{ZFC}$, $L[\mathbb{E}]^R$ is a proper class premouse of $R$ inheriting all strong and Woodin cardinals from $R$. Moreover, for each $α\in\mathrm{OR}$, $L[\mathbb{E}]^R|α$ is $(ω,α)$-iterable, via iteration trees which lift to coarse iteration trees on $R$. We prove that $(k+1)$-condensation follows from $(k+1)$-solidity together with $(k,ω_1+1)$-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of $(k+1)$…
We investigate the problem of when $\leqλ$--support iterations of $<λ$--complete notions of forcing preserve $λ^+$. We isolate a property -- {\em properness over diamonds} -- that implies $λ^+$ is preserved and show that this property is preserved by $λ$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.
We use generalizations of concepts from descriptive set theory to study combinatorial objects of uncountable regular cardinality, focussing on higher Kurepa trees and the representation of the sets of cofinal branches through such trees as continuous images of function spaces. For different types of uncountable regular cardinals $κ$, our results provide a complete picture of all consistent scenarios for the representation of sets of cofinal branches through $κ$-Kurepa trees as retracts of the generalized Baire space ${}^κκ$ of $κ$. In addition, these results can be used to determine the consis…
It is shown that if there is a measurable cardinal above n Woodin cardinals and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but only iterable with respect to stacks of certain trees living between the Woodin cardinals. However, it is still true that if M is an omega-closed iterate of V then K^M is an iterate of K.
In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey theorem on aleph_1 which is again equiconsistent with a Pi^1_1 reflecting cardinal. We also prove that the addition of $MA$ to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal $κ$ has a homogeneous set of size $κ$ provided that the number of colors, $μ$ satisfies $μ^+<κ$. Another result is that an uncountable cardinal $κ$ is weakly compact if and only if $κ$ is regular, has the tree property and for each $λ,μ<κ$ there exists $κ^*<κ$ such that every tree of height $μ$ with $λ$ nodes has less than $κ^*$ branches.
Given an uncountable regular cardinal $κ$, a partial order is $κ$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $κ$ is stationary in $\mathcal{P}_κ(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $κ$ is weakly compact if and only if every partial order satisfying the $κ$-chain condition is $κ$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $κ$-Knaster partial…
Given a Woodin cardinal $δ$, I show that if $F$ is any Easton function with $F"δ\subseteqδ$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $λ$-supercompact cardinal $κ$ such that $\GCH$…
Under the assumption that $δ$ is a Woodin cardinal and $\GCH$ holds, I show that if $F$ is any class function from the regular cardinals to the cardinals such that (1) $κ<\cf(F(κ))$, (2) $κ<λ$ implies $F(κ)\leq F(λ)$, and (3) $δ$ is closed under $F$, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. Unlike the analogous results for supercompact cardinals [Men76] and strong cardinals [FH08], there is no requirement that the function $F$ be locally definable.
