Souslin Tree Research Articles

Full Souslin trees at small cardinals

AbstractA ‐tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full ‐Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal . Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be many full ‐trees such that the product of any countably many of them is an ‐Souslin tree.

Locally compact, ω1-compact spaces

An ω1-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ω1-compact space is σ-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, ω1-compact space of cardinality ℵ1 is σ-countably compact. Whether ℵ1 can be replaced with ℵ2 is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, ω1-compact space is σ-countably compact.As a result, it is also ZFC-independent whether there is a locally compact, ω1-compact Dowker space of cardinality ℵ1, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space ω1.Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as “MM(S)[S]”. Most of the work is done by the P-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality ℵ1, as it is in several theorems.

A guessing principle from a Souslin tree, with applications to topology

We introduce a new combinatorial principle which we call ♣AD. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces.Our main result states that strong instances of ♣AD follow from the existence of a Souslin tree. It is also shown that the weakest instance of ♣AD does not follow from the existence of an almost Souslin tree.As an application, we obtain a simple, de Caux type proof of Rudin's result that if there is a Souslin tree, then there is an S-space which is Dowker.

On the rigidity of Souslin trees and their generic branches

We show it is consistent that there is a Souslin tree S such that after forcing with S, S is Kurepa and for all clubs C subset omega _1, Supharpoonright C is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with diamondsuit that for every Souslin tree T there is a dense X subseteq T which does not contain a copy of T. This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982).

Ramsey theory over partitions III: Strongly Luzin sets and partition relations

The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size ℵ 1 \aleph _1 . In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpiński a hundred years ago.

On the ideal J[κ

Motivated by a question from a recent paper by Gilton, Levine and Stejskalová, we obtain a new characterization of the ideal J[κ], from which we confirm that κ-Souslin trees exist in various models of interest.As a corollary we get that for every integer n such that b<2ℵn=ℵn+1, if □(ℵn+1) holds, then there exists an ℵn+1-Souslin tree.

A microscopic approach to Souslin-tree construction, Part II

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-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 ⋄. 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, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal.Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved.

Souslin trees at successors of regular cardinals

AbstractWe 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.

More Notions of Forcing Add a Souslin Tree

An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree. In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.

A Remark on Schimmerling’s Question

Schimmerling asked whether \(\square ^{\ast }_{\lambda }+\mathsf {GCH}\) entails the existence of a λ+-Souslin tree, for a singular cardinal λ. We provide an affirmative answer under the additional assumption that there exists a non-reflecting stationary subset of \({E}^{{\lambda }^{+}}_{{\neq } \text {cf}(\lambda )}\). As a bonus, the outcome λ+-Souslin tree is moreover free.

PFA(S) and countable tightness

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we consider status under PFA(S) of two important consequences of PFA concerning spaces of countable tightness. We prove that the existence of a Souslin tree does not imply the existence of a compact non-sequential space of countable tightness. We contrast this with M.E. Rudin's result that the existence of a Souslin tree does imply the existence of an S-space (and the later improvement by Dahrough to a compact S-space). On the other hand, PFA(S) implies there is a first-countable perfect pre-image of ω1 that contains no copies of ω1.

A Forcing Axiom Deciding the Generalized Souslin Hypothesis

AbstractWe 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$\unicode[STIX]{x1D706}$, if$\unicode[STIX]{x1D706}^{++}$is not a Mahlo cardinal in Gödel’s constructible universe, then$2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$entails the existence of a$\unicode[STIX]{x1D706}^{+}$-complete$\unicode[STIX]{x1D706}^{++}$-Souslin tree.

PFA(S) and automorphisms of [formula omitted

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we show that forcing with the Souslin tree S, as postulated by PFA(S), will preserve that all automorphisms of the Boolean algebra P(N)/fin are trivial.

THE EIGHTFOLD WAY

AbstractThree central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ &lt; \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$.

Normality Versus Paracompactness in Locally Compact Spaces

AbstractThis note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ω1, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ω1.

PFA(S)[S] for the masses

We present S. Todorcevic's method of forcing with a coherent Souslin tree over restricted iteration axioms as a black box usable by those who wish to avoid its complexities but still access its power.

SQUARE WITH BUILT-IN DIAMOND-PLUS

AbstractWe formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds in L for every infinite cardinal.As an application, we prove that the following two hold in L:1.For every infinite regular cardinal λ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;2.For every infinite cardinal λ, there exists a respecting λ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed in L.

Higher Souslin trees and the GCH, revisited

It is proved that for every uncountable cardinal λ, GCH+□(λ+) entails the existence of a cf(λ)-complete λ+-Souslin tree. In particular, if GCH holds and there are no ℵ2-Souslin trees, then ℵ2 is weakly compact in Gödel's constructible universe, improving Gregory's 1976 lower bound. Furthermore, it follows that if GCH holds and there are no ℵ2 and ℵ3 Souslin trees, then the Axiom of Determinacy holds in L(R).

REDUCED POWERS OF SOUSLIN TREES

We study the relationship between a $\unicode[STIX]{x1D705}$-Souslin tree $T$ and its reduced powers $T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$.Previous works addressed this problem from the viewpoint of a single power $\unicode[STIX]{x1D703}$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph _{6}$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle {\mathcal{U}}_{n}\mid n&lt;6\rangle$ such that $T^{\aleph _{n}}/{\mathcal{U}}_{n}$ is $\aleph _{6}$-Aronszajn if and only if $n&lt;6$ is not a prime number.This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces

We apply a method of Todorcevic in showing that PFA(S) implies that the coherent Souslin tree S forces various strengthenings of the assertion that every locally countable space of size ℵ1 that has a countably tight compactification must be σ-discrete.