Shelah's Theorem Research Articles

Stability, the NIP, and the NSOP: model theoretic properties of formulas via topological properties of function spaces

AbstractWe study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (NSOP) in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, Talagrand's stability, and explain the relationship between this property and the NIP in continuous logic. Using a result of Bourgain, Fremlin, and Talagrand, we prove almost definability and Baire 1 definability of coheirs assuming the NIP. We show that a formula has the strict order property if and only if there is a convergent sequence of continuous functions on the space of φ‐types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein‐Šmulian theorem.

A game‐theoretic proof of Shelah's theorem on labeled trees

AbstractWe give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality κ of the family is much larger (in the sense of large cardinals) than the cardinality λ of the set of labels, more precisely if the partition relation holds, then there is a homomorphism from one labeled tree in the family to another. Our proof uses a characterization of such homomorphisms in terms of games.

Corrigendum: The Keisler—Shelah theorem for QmbC through semantical atomization

Corrigendum: The Keisler—Shelah theorem for QmbC through semantical atomization

The Keisler–Shelah theorem for $\mathsf{QmbC}$ through semantical atomization

AbstractIn this paper, we consider some contributions to the model theory of the logic of formal inconsistency $\mathsf{QmbC}$ as a reply to Walter Carnielli, Marcelo Coniglio, Rodrigo Podiacki and Tarcísio Rodrigues’ call for a ‘wider model theory.’ This call demands that we align the practices and techniques of model theory for logics of formal inconsistency (LFIs) as closely as possible with those employed in classical model theory. The key result is a proof that the Keisler–Shelah isomorphism theorem holds for $\mathsf{QmbC}$, i.e. that the strong elementary equivalence of two $\mathsf{QmbC}$ models $\mathfrak{A}$ and $\mathfrak{B}$ is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between $\mathsf{QmbC}$ models, explicitly prove Łoś’ theorem and introduce a useful technique of model-theoretic ‘atomization’ in which the satisfaction sets of non-deterministically evaluated formulae are associated with new predicates. Finally, we consider some of the extensions of $\mathsf{QmbC}$, explicitly showing that Keisler–Shelah holds for $\mathsf{QCi}$ and suggesting that it holds of extensions like $\mathsf{QCila}$ and $\mathsf{QCia}$ as well.

Homotopy properties of subsets of Euclidean spaces

This paper is devoted to prove several results concerning the homotopy groups of separable metric spaces that generalize some of the main results of [4] and [5] to homotopy groups. In particular, we focus on subspaces of Euclidean spaces. Among the results, we proposed a partial generalization of Shelah's Theorem to higher homotopy groups for noncompact spaces. Also, we discuss n-homotopically Hausdorff property, a separation axiom for n-loops introduced in [12], and conclude that each subset of Rn+1 is n-homotopically Hausdorff. Moreover, the concept of a Hawaiian n-wild point will be introduced that illustrates the complexity of homotopy group at that point. We show that any (n−1)-connected locally (n−1)-connected subspaces of Rn+1 with uncountable nth homotopy group admit a Hawaiian n-wild point.Finally, we prove that nth homotopy group of any (n−1)-connected locally (n−1)-connected subspace of Rn+1 is free provided that it is n-semilocally simply connected, and then we study the free Abelian factor groups of the homotopy groups of these spaces.

Putting a diamond inside the square

By a 35-year-old theorem of Shelah, □ λ + ♦ ( λ + ) does not imply for regular uncountable cardinals λ. Here, it is proved that □ λ + ♦ ( λ + ) is equivalent to for every singular cardinal λ.

Independence, dimension and continuity in non-forking frames

AbstractThe notion J is independent in (M, M0, N) was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal λ and has a non-forking relation, satisfying the good λ-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.

The failure of diamond on a reflecting stationary set

1. It is shown that the failure of ♦S , for a set S ⊆ אω+1 that reflects stationarily often, is consistent with GCH and APאω , relatively to the existence of a supercompact cardinal. By a theorem of Shelah, GCH and λ entails ♦S for any S ⊆ λ that reflects stationarily often. 2. We establish the consistency of existence of a stationary subset of [אω+1] that cannot be thinned out to a stationary set on which the sup-function is injective. This answers a question of Konig, Larson and Yoshinobu, in the negative. 3. We prove that the failure of a diamond-like principle introduced by Džamonja-Shelah is equivalent to the failure of Shelah’s strong hypothesis.

CCC forcing and splitting reals

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.

A short proof of the preservation of the ωω‐bounding property

AbstractThere are two versions of the Proper Iteration Lemma. The stronger (but less well‐known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω‐bounding property. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Shelah's work on non-semi-proper iterations, II

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

Instances of dependent choice and the measurability of [formula omitted

Starting from cardinals κ < λ where κ is 2 λ supercompact and λ > κ is measurable, we construct a model for the theory “ ZF + ∀n < ω[DC ℵ n ] + ℵ ω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ℵ ω + 1 , and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.

Homogeneous models for unidimensional theories

A stable theory is called unidimensional if any two of its nonalgebraic types are nonorthogonal. In the paper we consider some problems concerning homogeneous models of unidimensional theories. For homogeneous models, in particular, we prove an analog of the theorem of Shelah which states that the concepts of unidimensionality and saturation are equivalent in any sufficiently saturated model of that theory.

Saharon Shelah, Can you take Solovay's inaccessible away?Israel journal of mathematics, vol. 48 (1984), pp. 1–47. - Jean Raisonnier, A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel journal of mathematics, vol. 48 (1984), pp. 48–56.

Saharon Shelah, Can you take Solovay's inaccessible away?Israel journal of mathematics, vol. 48 (1984), pp. 1–47. - Jean Raisonnier, A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel journal of mathematics, vol. 48 (1984), pp. 48–56.

On singular compactness

We present some applications of Shelah's singular compactness theorem to algebraic situations where the Shreier property fails. The principal application is to valuated vector spaces, where we make use of an alternate, unpublished, version of Shelah's theorem.

A new proof of a theorem of Shelah

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.