Countably compact groups without non-trivial convergent sequences

We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.Comment: 21 pages, to be published in Transactions of the American Mathematical Societ

Chain logic and Shelah’s infinitary logic

For a cardinal of the form kappa = (sic)(kappa), Shelah's logic L-kappa(1) has a characterisation as the maximal logic above boolean OR(lambda We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah's [28], which asked whether for kappa singular of countable cofinality there was a logic strictly between L kappa+,omega and L kappa(+),kappa(+) having interpolation. We show that modulo accepting as the upper bound a model class of L-kappa,L-kappa, Karp's chain logic satisfies the required properties. In addition, we show that this chain logic is not kappa-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chainindependent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic L-kappa(1) in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.Peer reviewe

Some size and structure theorems for ultrapowers

In this thesis we study the mapping D → A[sup I]/D, between ultrafilters and models, given by the ultrapower construction. Under this mapping homomorphisms of ultrapowers induce elementary embeddings of ultrapowers. Using these embeddings we investigate the dependence of the structure of an ultrapower A[sup I]/D on the cardinality of the index set I. With each ultrafilter D we associate a set of cardinals a(D) which we term the shadow of D. We investigate the form of the sets a(D). It is shown that if σ(D) has "gaps" then isomorphisms arise between ultrapowers of different index sizes. In terms of σ(D) we prove new results on the properties of the set of homomorphic images of an ultrafilter. Finally we introduce a new class of "quasicomplete" ultrafilters and prove several results about ultrapowers constructed using these. Two results which can be mentioned here are the following: Let α be a regular cardinal. We establish necessary and sufficient conditions on D (i) for the cardinality of α to be raised in the passage to α[sup I]/D. (ii) for the confinality of α[sup I]/D (regarded as an ordered set) to be greater than α⁺. Some of the results of this thesis depend on assumption of the Generalised Continuum Hypothesis. The result (i) above is a case in point.Science, Faculty ofMathematics, Department ofGraduat