The singular world of singular cardinals

The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones

Square compactness and the filter extension property

We show that the consistency strength of κ being 2κ-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juhasz

András Hajnal, life and work

Life and Work of Andras Hajnal, honorary President of the European Set Theory Society. Published in News Bulletin of the Iranian Association for Logic

Paragraded structures inspired by mathematical logic

We use methods from mathematical logic to give new examples of paragraded structures, showing that at certain cardinals all first order structures are paragraduaded. We introduce the notion of bi-embeddability to measure when two paragraduaded structures are basically the same. We prove that the bi-embeddability of the paragraduating system gives rise to the bi-embeddability of the limiting structures. Under certain circumstances the converse is also true, as we show here. Finally, we show that one paragraduaded structure can have many graduaded substructures, to the extent that the number of the same is not always decidable by the axioms of set theory

Universal infinite clique-omitting graphs

The main result of the paper is that when κ is a cardinal of cofi- nality ω and λ ≥ κ, the class of graphs of size λ omitting cliques of size κ has no universal element under graph homomorphisms (or the weak and strong embeddings). This theorem only requires ZF

Memories of Mary Ellen Rudin

An invited collective remembrance celebrating Mary Ellen Rudin's lif

Some strong logics within combinatorial set theory and the logic of chains

In this note we report on a project in progress, where we study compactness of infinitary logics, including the logic of chains. The motivation of this project is to find logical reasons for the set-theoretic phenomenon of com- pactness at singular cardinals

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