Mirna Džamonja

On the existence of universal models

Abstract.Suppose that λ=λ <λ ≥ℵ0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤λ+, and is meaningful when 2λ +>λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such λ, for any fixed μ>λ+ regular with μ=μλ+, it is consistent that 2λ=μ and there is no normed vector space over ℚ of size <μ which is universal for normed vector spaces over ℚ of dimension λ+ under the notion of embedding h which specifies (a,b) such that ||h(x)/||x∈(a,b) for all x.

Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣∗−, and show that if it holds on a stationary subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣∗−(S) is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣∗−(S) holds modulo an ideal obtained through the weak reflection.

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

Abstract If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which ♢ κ ( REG ) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

Does not imply the existence of a Suslin tree

We prove that ♣ does not imply the existence of a Suslin tree, so answering a question of I. Juhász.

On properties of theories which preclude the existence of universal models

Abstract We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH (and close to the failure of SCH) hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to non-universality, as was raised by the earlier work of Shelah. One of our examples is the theory T feq ∗ for which non-universality results similar to the ones we obtain are already known; hence we may view our results as an abstraction of the known results from a concrete theory to a class of theories. We show that no theory with the oak property is simple.

Small Universal Families of Graphs on ℵ ω+1

We prove that it is consistent that אω is strong limit, 2אω is large and the universality number for graphs on אω+1 is small. The proof uses Prikry forcing with interleaved collapsing.

Forcing □ ω1 with finite conditions.

Abstract We give a construction of the square principle □ ω 1 by means of forcing with finite conditions.

A poset hierarchy

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.

Chain models, trees of singular cardinality and dynamic EF-games

Let κ be a singular cardinal. Karps notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf(κ)κ. The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models.

WEAK REFLECTION AT THE SUCCESSOR OF A SINGULAR CARDINAL

The notion of stationary reflection is one of the most important notions of combinatorial set theory. Weak reflection, which is, as its name suggests, a weak version of stationary reflection, is investigated. The main result is that modulo a large cardinal assumption close to 2-hugeness, there can be a regular cardinal