Perfect tree forcings for singular cardinals

Handbook of the History of Logic, 2012

2020

If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

Fundamenta Mathematicae, 2010

The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ ++ ; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If 0 # exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inaccessible cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.

We investigate some versions of amoeba for tree forcings in the generalized Cantor and Baire spaces. This answers [9, Question 3.20] and generalizes a line of research that in the standard case has been studied in [10], [11] and [7]. Moreover, we also answer questions posed in [4] by Friedman, Khomskii and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show Σ 1 1-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between Ramsey and Baire property , in slight contrast with the standard case.

Archive of Mathematical Logic 54: 961-984, 2015, 2015

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of L(R) is absolute for proper forcing (Schindler in Bull Symbolic Logic 6(2):176-184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all two-step iterations of the form Add(κ, θ) * Ṙ, where Ṙ is forced to be <κ-closed and ≤κdistributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the GCH.

The Journal of Symbolic Logic

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma $ : the hierarchy of bounded forcing axioms, of $\Sigma ^1_1$ -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level $\kappa $ says that whenever T is a tree of height $\omega _1$ and width $\kappa $ that does not have a branch of order type $\omega _1$ , and whenever ${\mathord {\mathbb P}}$ is a forcing notion in $\Gamma $ , then it is not the case that ${\mathord {\mathbb P}}$ forces that T has such a branch. $\Sigma ^1_1$ -absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don’t add reals, the three principles at level $2^\omega $ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don’t add reals.

Russian Mathematical Surveys, 2016

Archive for Mathematical Logic, 2016

We introduce and study the first-order Generic Vopěnka's Principle, which states that for every definable proper class of structures C of the same type, there exist B = A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka's Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for n ∈ ω, which we introduce here, is a natural generic analogue for the C (n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka's Principle in [1]. Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, wPFA, which states that for every transitive model M in the language of set theory with some ω 1-many additional relations, if it is forced by a proper forcing P that M satisfies some Σ 1-property, then V has a transitive modelM, which satisfies the same Σ 1property, and in some set-forcing extension there is an elementary embedding fromM into M. This is a weakening of a formulation of PFA due to Schindler and Claverie [2], which asserts that the embedding fromM to M exists in V. We show that wPFA is equiconsistent with a remarkable cardinal and that wPFA implies PFA ℵ 2 , the proper forcing axiom for antichains of size at most ω 2 , but it is consistent with κ for all κ ≥ ω 2 , and therefore does not imply PFA ℵ 3 .

Proceedings of the American Mathematical Society, 1974

It is proved that supercompact cardinals can be characterized by combinatorial properties which are generalizations of ineffability. 0. Introduction. A A B is the symmetric difference of A and B. Greek letters will denote ordinals. Pk(A) is the set of all nonempty subsets of A of cardinality less than k. The cardinality of A is \A\. [A]' is the set of all subsets of A of cardinality a. An ultrafilter U on Pk(A) is normal if (a) U is k complete ; (b) for every a e A {P\P e Pk(A), aePjeU; (c) any choice function on Pk(A) is almost everywhere constant. The definition of supercompact cardinal is due to Solovay [6]. A cardinal k is A supercompact if there is a normal ultrafilter on Pk(A). k is supercompact if it is A supercompact for all A. R(a.) is the set of all sets of rank < a. The purpose of this paper is to show that being a supercompact cardinal can be characterized by partition properties which are natural generalizations of those defining ineffable cardinal (Jensen-Kunen [3]). By this we partially settle problem 4 of T. J. Jech [2]. We assume the reader knows the definition of closed subset of an ordinal, unbounded subset of an ordinal, and stationary set, as well as the basic properties of such sets (cf. Fodor [1]). A cardinal k is ineffable iff for any sequence {Ax}x<k, such that Ax^a. for all <t<k, there is A^k and {a|oc</c, An<x=Ax} is stationary in k. An equivalent definition is the following partition property : For any function/: [fc]2-»-2 there is a homogeneous stationary set for/ (that is, a stationary set A such that |/"[/l]2| = l). An ineffable cardinal is weakly compact and therefore inaccessible. 1. Basic facts. A natural generalization of closed, unbounded and stationary sets are the following:

2015