A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}.

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.

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)

Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.

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)

The stability theory of first order theories was initiated by Saharon Shelah in 1969. The classification of abstract elementary classes was initiated by Shelah, too. In several papers, he introduced non-forking relations. Later, Shelah [17, II] introduced the good non-forking frame, an axiomatization of the non-forking notion.We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost λ-stability hypothesis: The number of types over a model of cardinality (...) λ is at most λ+.We present conditions on Kλ, that imply the existence of a model in Kλ+n for all n. We do this by providing sufficiently strong conditions on Kλ, that they are inherited by a properly chosen subclass of Kλ+. What are these conditions? We assume that there is a ‘non-forking’ relation which satisfies the properties of the non-forking relation on superstable first order theories. Note that here we deal with models of a fixed cardinality, λ.While in Shelah [17, II] we assume stability in λ, so we can use brimmed models, here we assume almost stability only, but we add an assumption: The conjugation property.In the context of elementary classes, the superstability assumption gives the existence of types with well-defined dimension and the ω-stability assumption gives the existence and uniqueness of models prime over sets. In our context, the local character assumption is an analog to superstability and the density of the class of uniqueness triples with respect to the relation ≼bs is the analog to ω-stability. (shrink)

We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H , be the set of order types of ℵ 1 -dense homogeneous subsets of (...) R with the relation of embeddability. We prove that for every finite model L , : Con iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ℵ 1 , and { U 0 ,\h.;, U n−1 } is a cover of D ▪ X x X -} x,x> ¦ x ϵ X } consisting of symmetric open sets, then X can be partitioned into { X i \brvbar; i ϵ ω } such that for every i ϵ ω there is l such that D ⊇ U l ”. We also prove that MA+OCA [xrArr] 2 ℵ 0 = ℵ 2. (shrink)

We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.

This paper investigates a connection between the semantic notion provided by the ordering * among theories in model theory and the syntactic SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article 229) it was shown that SOP3 implies *-maximality and we prove here that *-maximality in a model of GCH implies a property called SOP2″. It has been (...) subsequently shown by Shelah and Usvyatsov that SOP2″ and SOP2 are equivalent, so obtaining an implication between *-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent. Together with the known results about the connection between the SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property. (shrink)

In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in Hyttinen 167–182). We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see Hyttinen.

We use a creature construction to show that consistently $$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}< {{\mathrm{non}}}< {{\mathrm{non}}}< {{\mathrm{cof}}} < 2^{\aleph _0}. \end{aligned}$$The same method shows the consistency of $$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}< {{\mathrm{non}}}< {{\mathrm{non}}}< {{\mathrm{cof}}} < 2^{\aleph _0}. \end{aligned}$$.

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded (...) proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure. (shrink)

We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals.

We prove some consistency results about and δ, which are natural generalisations of the cardinal invariants of the continuum and . We also define invariants cl and δcl, and prove that almost always = cl and = cl.

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...) prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$ . In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property " $F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$ . In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$. (shrink)

In this paper we study the question assuming MA+⌝CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals 0. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.

We use κ-free but not Whitehead Abelian groups to constructElementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is (...) an AEC with amalgamation which is not (N₀, N₁)-tame but is 2(N0, ∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ N₂, ≤ N₂)-compact. (shrink)

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)

Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove--again in ZFC--that for a large class of cardinals there is no universal linear order (e.g. in every regular $\aleph_1 < (...) \lambda < 2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ "resembles" ℵ1--a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on). (shrink)

A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

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Model Theory in Logic and Philosophy of Logic

Philosophy of Mathematics

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In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider $\operatorname{seq}^{1 - 1}(A)$ , the set of all sequences of A without repetition. We compare $|\operatorname{seq}^{1 - 1}(A)|$ , the cardinality of this set, to |P(A)|, the cardinality of (...) the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that $ZF \vdash \forall A(|\operatorname{seq}^{1 - 1}(A)| \neq|\mathscr{P}(\mathscr{A})|)$ , and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then $|\operatorname{fin}(B)| <|\mathscr{P}(B)|$ even though the existence for some infinite set B* of a function f from $\operatorname{fin}(B^\ast)$ onto P(B*) is consistent with ZF. (shrink)

We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi [12].

The theory of trees with additional unary predicates and quantification over nodes and branches embraces a rich branching time logic. This theory was reduced in the companion paper to the first-order theory of binary, bounded, well-founded trees with additional unary predicates. Here we prove the decidability of the latter theory.

This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 229–255] and [M. Džamonja, S. Shelah, On left triangle, open*-maximality, Annals of Pure and Applied Logic 125 119–158]. We present a rank function for NSOP1 theories and give an example of a theory which is NSOP1 but not simple. We also investigate the connection between maximality in the ordering left triangle, open* among complete first order theories and the SOP2 property. We (...) prove that left triangle, open*-maximality implies SOP2 and obtain certain results in the other direction. The paper provides a step toward the classification of unstable theories without the strict order property. (shrink)

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely MA(Γ + ℵ 0 ), and using the results on Souslin forcing we show that MA(Γ + ℵ 0 ) is consistent with the existence of a Souslin tree and with the splitting number s = ℵ 1 . We prove that MA(Γ + ℵ 0 ) proves the additivity of measure. Also we introduce (...) the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + Σ 1 2 -measurability. (shrink)

We deal with several pcf problems: we characterize another version of exponentiation: maximal number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further T D (f). The sections can be read independently, although there are some minor dependencies.

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at the (...) successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method. (shrink)

We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote \\). Here mif abbreviates maximal independent family. We show that:1.whenever \ are finitely many regular uncountable cardinals, it is consistent that \\); 2.whenever \ has uncountable cofinality, it is consistent that \=\{\aleph _1,\kappa =\mathfrak {c}\}\). Assuming large cardinals, in addition to above, we can provide that $$\begin{aligned} \cap \hbox {Spec}=\emptyset \end{aligned}$$for each i, \.

For f, g $ \in \omega ^\omega $ let $c_{f,g}^\forall $ be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. $c_{f,g}^\exists $ is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often. It is consistent that $c_{f \in ,g \in }^\exists = c_{f \in ,g \in }^\forall = k_ \in $ for N₁ many (...) pairwise different cardinals $k_ \in $ and suitable pairs $(f_{ \in ,g \in } ).$ For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision. (shrink)

We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...) measure and CATEGORY = 1. Several important new metamathematical patterns among classical statements of set theory are pointed out. (shrink)

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD(A), where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an∈ P and finite subset C ⊆ Q, but (...) [Formula: see text] for some independent a1, …, an, an+1∈ P and some finite subset C ⊆ Q.Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] interprets a non-classical group, or n = 1,2,3 and•If n = 1 then G is abelian and acts regularly on P′.•If n = 2 the action of G on P′ is isomorphic to the affine action of K ⋊ K* on the algebraically closed field K.•If n = 3 the action of G on P′ is isomorphic to the action of PGL2(K) on the projective line ℙ1(K) of the algebraically closed field K.We prove a similar result for excellent classes. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

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].

We prove independence results concerning the number of nonisomorphic models (using the S-chain condition and S-properness) and the consistency of "ZCF + 2 ℵ 0 = ℵ 2 + there is a universal linear order of power ℵ 1 ". Most of these results were announced in [Sh 4], [Sh 5]. In subsequent papers we shall prove an analog f MA for forcing which does not destroy stationary subsets of ω 1 , investigate D-properness for various filters and prove the (...) consistency with G.C.H. of an axiom implying SH (for ℵ 1 ), and connected results. (shrink)

We continue investigations of reasonable ultrafilters on uncountable cardinals defined in previous work by Shelah. We introduce stronger properties of ultrafilters and we show that those properties may be handled in λ-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal λ with generating systems of size less than $2^\lambda$ . We also show how ultrafilters generated by small systems can be killed by forcing notions which have enough (...) reasonable completeness to be iterated with λ-supports. (shrink)

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A For every natural number k, there is a class $K_k $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{k - 1} $ , but $K_k $ has the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_{k - 3} $ and has models of cardinality $\mathfrak{N}_{k (...) - 1} $ . More strongly, we can have disjoint amalgamation up to $\mathfrak{N}_\alpha $ for α > ω, but have a bound on size of models. Theorem B For every countable ordinal a, there is a class $K_\alpha $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{\omega 1} $ , but K does have the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_\alpha $ . Finally we show that we can extend the $\mathfrak{N}_\alpha $ to $ \supset _\alpha $ in the second theorem consistently with ZFC and while having $\mathfrak{N}_i \ll \supset _i $ for 0 > i < α. Similar results hold for arbitrary ordinals α with |α|= K and $L_{k + ,\omega } $. (shrink)