The generalized Kurepa hypothesis at singular cardinals
Mohammad Golshani
备注：
In personal communication, Stevo Todorcevic informed the author that Theorem 3.1 has been obtained by him before; for example can be found on page 231 of his book Walks on ordinals and their characteristics. However our proof is different from him
We discuss the generalized Kurepa hypothesis $KH_λ$ at singular cardinals $λ$. In particular, we answer questions of Erdös-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $KH_{\aleph_ω}$ nor the existence of a family $ \mathcal{F} \subseteq [\aleph_ω]^{\aleph_0}$ of size $\aleph_{ω+1}$ such that $\mathcal{F} \restriction X$ has size $\aleph_0$ for every $X \subseteq S, |X|=\aleph_0$.
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 use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathc…
This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the fi…
In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $κ\ge ω_2$, $\textsf{ISP}(κ)$ implies that $\textsf{SCH}$ holds above $κ$, and (3) forcing posets which have the $ω_1$-approximation property also have the countable covering property. These results solve open problems of Viale and Hachtman-Sinapova.
A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.
We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.
We argue against Foreman's proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
In this note, we relax the hypothesis of the main results in Kellner-Shelah-Tǎnasie's "Another ordering of the ten cardinal characteristics in Cichoń's diagram".
The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $κ$ is singular strong limit, then $2^κ=κ^+$. We prove that given a singular cardinal $κ$ of {\em cofinality} $η$ in the ground model, which is a limit of suitable large cardinals, and $η^+=\aleph_γ$, then there is a forcing extension which preserves cardinals and cofinalities up to and including $η$, such that $κ$ becomes $\aleph_{γ+η}$, and SCH fails at $κ$. Furthermore, if $η$ is not an $\aleph$-fixed point, then in our model, SCH fails at $\aleph_η$. Our large c…
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…
Motivated by the goal of constructing a model in which there are no $κ$-Aronszajn trees for any regular $κ>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.
We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbolΠ^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} | κ_\infty, Λ),$ where $\mathcal{M}_\infty$ is a direct limit of iterates of $M_{n+1}$, $δ_\infty$ is the least Woodin cardinal in $\mathcal{M}_\infty$…
Let $M$ denote the Merimovich's model in which for each infinite cardinal $λ, 2^λ=λ^{+3}$. We show that in $M$ the following hold: (1) Shelah's strong hypothesis fails at all singular cardinals, indeed, $\forall λ(λ$ is a singular cardinal $\Rightarrow pp(λ)=λ^{+3}).$ (2) For each singular cardinal $λ$ there is an inner model $N$ of $M$ such that $M$ and $N$ have the same bounded subsets of $λ,$ $λ$ is a singular cardinal in $N$, $(λ^{+i})^N=(λ^{+i})^M$, for $i=1,2,3,$ and $N \models 2^λ=λ^{+}$. Thus it is possible to add many new fresh subsets to $λ$ without adding any new bounded subsets to …
The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $κ$ is supercompact and the cardinals $θ< κ$, $λ> κ$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(κ) = θ$, the Singular Cardinal Hypothesis fails at $κ$ and there exists a universal graph in cardinality $λ\in (κ,2^κ)$.
We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.
We consider the following dichotomy for $Σ^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $κ$: either all $R$-independent sets are of size at most $κ$, or there is a $κ$-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in (W. Kubiś, Proc. Amer. Math. Soc. 131 (2003), no 2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We prove that the above statement holds assuming $\Diamond_κ$ and the set theoretical hypothesis $I^-(κ)$, which is the modification of the hypothesis $I(κ)$ suitable for limit cardin…
This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic structures such as the Baire topology are introduced, and the Baire Category Theorem is extended to a statement that a $\aleph$-sequentially complete binary tree representation of a Hausdorff topological space that has a clopen base of cardinality $\aleph$ and no isolated or discrete points is not the union of $<\aleph+1$-many nowhere dense subsets for cardinal …
By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_σ$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^ωω$. We consider the analogous statement (which we call Hurewicz dichotomy) for $Σ^1_1$ subsets of the generalized Baire space ${}^κκ$ for a given uncountable cardinal $κ$ with $κ=κ^{<κ}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) hol…
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
Given a weakly compact cardinal $κ$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{κ^+, κ}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel for intuitionistic logic over $\mathcal{L}_{ω_1, ω}$. When $κ$ is a regular cardinal such that $κ^{<κ}=κ$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $\mathcal{L}_{κ^+, κ, κ}$, the language with disjunctions of at most…
