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)

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)

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

We give a construction of the square principle by means of forcing with finite conditions.

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. (shrink)

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, ♣*− 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 ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

Let κ be a cardinal which is measurable after generically adding ${\beth_{\kappa+\omega}}$ many Cohen subsets to κ and let ${\mathcal G= ( \kappa,E )}$ be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value ${r_m^+}$ such that the set [κ] m can be partitioned into classes ${\langle{C_i:i (...) G}$ in ${\mathcal G}$ such that ${[\mathcal{G} ^\ast ] ^m\cap C_i}$ is monochromatic. It follows that ${\mathcal{G}\rightarrow (\mathcal{G} ) ^m_{<\kappa/r_m^+}}$ , that is, for any coloring of ${[ \mathcal {G} ] ^m}$ with fewer than κ colors there is a copy ${\mathcal{G} ^{\prime}}$ of ${\mathcal{G}}$ such that ${[\mathcal{G} ^{\prime} ] ^{m}}$ has at most ${r_m^+}$ colors. On the other hand, we show that there are colorings of ${\mathcal{G}}$ such that if ${\mathcal{G} ^{\prime}}$ is any copy of ${\mathcal{G}}$ then ${C_i\cap [\mathcal{G} ^{\prime} ] ^m\not=\emptyset}$ for all ${i 2 we have ${r_m^+ > r_m}$ where r m is the corresponding number of types for the countable Rado graph. (shrink)

The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author asked the first author about the development of forcing, the use of new axioms and set-theoretic intuition on independence. Parts of these (...) conversations are directly presented in this article. They are supplemented by important mathematical results as well as discussion sections. Finally, we present three hypotheses about set-theoretic practice: First, that most set theorists were surprised by the introduction of the forcing method. Second, that most set theorists think that forcing is a natural part of contemporary set theory. Third, that most set theorists prefer an answer to a problem with the help of a new axiom of lowest possible consistency strength, and that for most set theorists, a difference in consistency strength weighs much more than the difference between Forcing Axiom and Large Cardinal Axiom. (shrink)

We consider various versions of the ♣ principle. This principle is a known consequence of $\lozenge$ . It is well known that $\lozenge$ is not sensitive to minor changes in its definition, e.g., changing the guessing requirement form "guessing exactly" to "guessing modulo a finite set". We show however, that this is not true for ♣. We consider some other variants of ♣ as well.

This book, published in 2019 as an open-access edition of the Fordham University Press, attracts by its title. Imagination, as we mathematicians know only too w.

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 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 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. (shrink)

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$ , of cofinality $\omega$ , such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ many colours has a g-colour class on which c takes every value. The paper also contains some generalisations of the above statement in which $\mu^{+}$ is replaced (...) by other cardinals < $\mu$. (shrink)

Let κ be a singular cardinal. Karp's 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. With a notion of satisfaction and -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 -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 = 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. 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. (shrink)

We show the relative consistency of ℵ1 satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω1 satisfying a certain density condition necessarily contain all (...) finite subsets of an infinite subset of ω1, and specifically if this and a stronger property hold under MA + ¬CH. Towards this we show that if MA + ¬CH holds, then for every family ? of ℵ1 many infinite subsets of ω1, one can find a family ? of finite subsets of ω1 which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?.We then pose some open problems related to the question. (shrink)

A wide Aronszajn tree is a tree of size and height $\omega _{1}$ with no uncountable branches. We prove that under $MA$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under $MA$, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under $MA$ every wide Aronszajn tree embeds into a (...) Lipschitz tree or a coherent tree. We also prove that under $MA$ there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todorčević from 2007 who proved that under $MA$ there are no universal Aronszajn trees. (shrink)

We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable separable positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its (...) space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA+⇁ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA+⇁ CH every atomless ccc Boolean algebra satisfies this stronger chain condition. (shrink)

The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author asked the first author about the development of forcing, the use of new axioms and set-theoretic intuition on independence. Parts of these (...) conversations are directly presented in this article. They are supplemented by important mathematical results as well as discussion sections. Finally, we present three hypotheses about set-theoretic practice: First, that most set theorists were surprised by the introduction of the forcing method. Second, that most set theorists think that forcing is a natural part of contemporary set theory. Third, that most set theorists prefer an answer to a problem with the help of a new axiom of lowest possible consistency strength, and that for most set theorists, a difference in consistency strength weighs much more than the difference between Forcing Axiom and Large Cardinal Axiom. (shrink)