Souslin Hypothesis Research Articles

A Simple Characterization of Useful Topologies in Mathematical Utility Theory

In this paper, we present a simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. We introduce the concept of weak open and closed countable chain condition (WOCCC) relative to a topology, and we then show that a useful topology always satisfies this condition. The most important result in the paper shows that a completely regular topology is useful if and only if it is separable and it satisfies WFOCCC (a stricter version of WOCCC). In this way, we generalize all the previous results concerning useful topologies. We finish the paper by presenting a simple axiomatization of useful topologies under the well-known Souslin Hypothesis.

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.

The basis problem for subspaces of monotonically normal compacta

We prove, assuming Souslin's Hypothesis, that each uncountable subspace of each zero-dimensional monotonically normal compact space contains an uncountable subset of the real line with either the metric, the Sorgenfrey, or the discrete topology.

An variation for one souslin tree

AbstractWe present a variation of the forcing as presented in Woodin [4], Our forcing is a ℙmax-style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree TG which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree,” with TG being this minimal tree. In particular, in the extension this Souslin tree has the property that forcing with it gives a model of Souslin's Hypothesis.

The Souslin Hypothesis and continuous utility functions: A remark

The Souslin Hypothesis of Axiomatic Set Theory is shown to provide a basis for a simple characterization of continuous utility function representation of preferences.

Universality of the automorphism group of the real line

We discuss the question, whether each automorphism group (of cardinality at most\(2^{\aleph _0 } \)) of a linear order is embeddable into the automorphism group of the real line. We show that the answer to this question is independent of the axioms of ZFC: the answer is positive, if we assume\(2^{\aleph _0 } \)<\(2^{\aleph _1 } \) and Souslin's hypothesis; the answer is negative, if we assume\(2^{\aleph _0 } = 2^{\aleph _1 } \) orV=L.

John Gregory. Higher Souslin trees and the generalized continuum hypothesis. The journal of symbolic logic, vol. 41 (1976), pp. 663–671. - Richard Laver and Saharon Shelah. The ℵ2-Souslin hypothesis. Transactions of the American Mathematical Society, vol. 264(1981), pp. 411–417. - S. Shelah and L. Stanley. S-forcing, I. A “black-box” theorem for morasses, with applications to super-Souslin trees. Israel journal of

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Generalized Martin’s Axiom and Souslin’s hypothesis for higher cardinals

We consider different generalizations of Martin’s Axiom to higher cardinals. For ℵ1, assuming CH+2ℵ 1>ℵ2+□ℵ1 we show that a generalized Martin’s Axiom considered by Baumgartner settles the ℵ2 Souslin Hypothesis ... the wrong way. We further show that, assuming CH+2ℵ 1>ℵ2, a strengthening of this axiom implies □ℵ 1. Finally, we show that a seemingly innocuous further strengthening is inconsistent with CH+2ℵ 1>ℵ2.

Free limits of forcing and more on Aronszajn trees

We prove that the Souslin Hypothesis does not imply “every Aron. (=Aronszajn) tree is special”. For this end we introduce variants of the notion “special Aron. tree”. We also introduce a limit of forcings bigger than the inverse limit, and prove it preserves properness and related notions not less than inverse limit, and the proof is easier in some respects.

Concerning the consistency of the Souslin hypothesis with the continuum hypothesis

Concerning the consistency of the Souslin hypothesis with the continuum hypothesis

Between Martin's Axiom and Souslin's Hypothesis

Between Martin's Axiom and Souslin's Hypothesis

R. M. Solovay and S. Tennenbaum. Iterated Cohen extensions and Souslin's problem. Annals of mathematics, ser. 2 vol. 94 (1971), pp. 201–245.

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

Topos theory and souslin's hypothesis

Topos theory and souslin's hypothesis

Iterated Cohen Extensions and Souslin's Problem

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2