We prove Łos conjecture = Morley theorem in ZF, with the same characterization, i.e., of first order countable theories categorical in $N_\alpha $ for some (equivalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality $N_\alpha $ is either ≥ |α| for every α or it has a small upper bound (independent of α close to Ð₂).

We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.

We investigate the monadic logic of trees with ω + 1 levels, the monadic topology of the product space ω λ and a strengthening of monadic logic for trees with ω levels.

We construct Boolean algebras with prescribed behaviour concerning depth for the free product of two Boolean algebras over a third, in ZFC using pcf; assuming squares we get results on ultraproducts. We also deal with the family of cardinalities and topological density of homomorphic images of Boolean algebras (you can translate it to topology - on the cardinalities of closed subspaces); and lastly we deal with inequalities between cardinal invariants, mainly.

Answering a problem of Fuhrken we prove that for every κ, 1 ≤ κ ≤ ℵ 0 , there is a (countable) complete theory T, with no prime model, and exactly κ minimal models (up to isomorphism).

We prove there are theories (stable or countable) for which over every A there is a prime model but it is not necessarily unique. We also give a simplified proof of the uniqueness theorem for countable stable theories.

We associate Hanf numbers to triples where T and T1 are theories and p is a type. We show that the Hanf number for the property: “there is a model M1 of which omits p, but is saturated” is larger than the Hanf number of but smaller than the Hanf number of when T is stable with. In fact, surprisingly, we even characterise the Hanf number of when we fix where T is a first order complete (and stable), and demand.

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context (...) of negation-less logics, _positive logics_, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. (shrink)

We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the o Conjecture for these classes. Further (...) results are in preparation. (shrink)

Assuming 0 ♯ does not exist, we present a combinatorial approach to Jensen's method of coding by a real. The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.

Under certain cardinal arithmetic assumptions, we prove that for every large enough regular λ cardinal, for many regular κ < λ, many stationary subsets of λ concentrating on cofinality κ has the “middle diamond”. In particular, we have the middle diamond on {δ < λ: cf(δ) = κ}. This is a strong negation of uniformization.

We shall prove that if a model of cardinality κ can be expanded to a model of a sentence ψ of Lλ+,ω by adding a suitable predicate in more than κ ways, then, it has a submodel of power μ which can be expanded to a model of ψ in $> \mu$ ways provided that λ,κ,μ satisfy suitable conditions.

We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or λ=λℵ0. We use versions of being reduced—replacing Q by a subring —and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with the oak property, a property of (...) complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups and deals more with the existence of universals. (shrink)

Let be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS. We prove that for a suitable Hanf number gc0 if χ0 < λ0 λ1, and is categorical inλ1+ then it is categorical in λ0.

We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA + or even PFA + (using the consistency of a large cardinal).

We present two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ,ω + 1) 2 , although λ is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, [7], that, for example, consistently, $\aleph_{\omega_1} \nrightarrow (\aleph_{\omega_1}, \omega + 1)^2$ and consistently, 2 $^{\aleph_0} \nrightarrow (2^{\aleph_0},\omega + 1)^2$.

A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n ≤ ω, there is a natural number m (n) such that, the following holds. Assume GCH: If $\lambda are regular, there is a cofinality preserving forcing extension in which 2 λ = μ and, for all $\sigma such that η +m(n)-1) ≤ μ, ((η +m(n)-1) ) σ ) → ((κ) σ ) η (1)n . This generalizes results of [3], Section 1, and (...) the forcing is a "many cardinals" version of the forcing there. (shrink)

We lay the combinatorial foundations for [5] by setting up and proving the essential properties of the coding apparatus for singular cardinals. We also prove another result concerning the coding apparatus for inaccessible cardinals.

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...) of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares. (shrink)

We prove that colouring of pairs from 2 with strong properties exists. The easiest to state problem it solves is: there are two topological spaces with cellularity 1 whose product has cellularity 2; equivalently, we can speak of cellularity of Boolean algebras or of Boolean algebras satisfying the 2-c.c. whose product fails the 2-c.c. We also deal more with guessing of clubs.

Let T be the family of open subsets of a topological space . We prove that if T has a base of cardinality μ, λμ<2λ, λ strong limit of cofinality 0, then T has cardinality μ or 2λ. This is our main conclusion . In Theorem 2 we prove it under some set-theoretic assumption, which is clear when λ = μ; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove (...) that the simplest examples are the basic ones; they occur in every example . The main result for the case λ = 0 was proved in [Sh 454]. (shrink)

We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two 1-separable abelian groups of cardinality 1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems from Eklof and Mekler [2].

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.

We prove here the consistency of u>i where: u=Min{|X|:X⫅P(ω) generates a non-principle ultrafilter}, i=Min{|A|:A is a maximal independent family of subsets of ω}In this we continue Goldstern and Shelah [G1Sh388] where Con(r>u) was proved using a similar but different forcing. We were motivated by Vaughan [V] (which consists of a survey and a list of open problems). For more information on the subject see [V] and [G1Sh388].

The main results in the paper are the following. Theorem A. Suppose that T is superstable and M ⊂ N are distinct models of T eq . Then there is a c ϵ N⧹M such that t is regular. For M ⊂ N two models we say that M ⊂ na N if for all a ϵ M and θ such that θ ≠ θ , there is a b ∈ θ ⧹ acl . Theorem B Suppose that T is (...) superstable , M ⊂ na N are models of T eq , and p is a regular type non-orthogonal to t . Then there is a c ϵ N such that t is regular and non-orthogonal to p. Furthermore, there is a formula θ ∈ t such that a ∈ θ and t ⊥ ̷ p ⇒ t is regular. We used these results to obtain ‘good’ tree decompositions of models in superstable theories with NDOP. See Definition 5.1 for the undefined terms. Theorem C Suppose that T is superstable with NDOP and M ⊨ T eq . Then every ⊂ na - decomposition inside M extends to a ⊂ na -decomposition of M. Furthermore, if 〈N η , a η : η ∈ I〉 is any ⊂ na -decomposition of M, then M is minimal over ∪N η and for all η ∈ I. M is dominated by ∪N η over N η . Using some stable group theory we show that when Th is superstable with NDOP and 〈 N η : η ∈ I 〉 is a tree decomposition of M, then M is constructible over ∪ n η with respect to a very strong isolation relation. (shrink)

We get consistency results on I(λ, T 1 , T) under the assumption that D(T) has cardinality $>|T|$ . We get positive results and consistency results on IE(λ, T 1 , T). The interest is model-theoretic, but the content is mostly set-theoretic: in Theorems 1-3, combinatorial; in Theorems 4-7 and 11(2), to prove consistency of counterexamples we concentrate on forcing arguments; and in Theorems 8-10 and 11(1), combinatorics for counterexamples; the rest are discussion and problems. In particular: (A) By Theorems (...) 1 and 2, if $T \subseteq T_1$ are first order countable, T complete stable but ℵ 0 -unstable, $\lambda > \aleph_0$ , and $|D(T)| > \aleph_0$ , then $IE(\lambda, T_1, T) \geq \operatorname{Min}\{2^\lambda, \beth_2\}$ . (B) By Theorems 4, 5, 6 of this paper, if e.g. V = L, then in some generic extension of V not collapsing cardinals, for some first order $T \subseteq T_1, |T| = \aleph_0, |T_1| = \aleph_1, |D(T)| = \aleph_2$ and IE(ℵ 2 , T 1 , T) = 1. This paper (specifically the ZFC results) is continued in the very interesting work of Baldwin on diversity classes [B1]. Some more advances can be found in the new version of [Sh300] (see Chapter III, mainly \S7); they confirm 0.1, 0.2 and 14(1), 14(2). (shrink)

We succeed to say something on the identities of when μ > θ > cf with μ strong limit θ-compact or even μ is limit of compact cardinals.