Blog Archives

The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading

Posted in Compactness | Tagged , , , , | 1 Comment

Ordinal definable subsets of singular cardinals

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading

Posted in Publications, Singular Cardinals Combinatorics | Tagged , , | 2 Comments

Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading

Posted in Blog, Expository | Tagged , , , | 13 Comments

Prikry Forcing

Recall that the chromatic number of a (symmetric) graph $(G,E)$, denoted $\text{Chr}(G,E)$, is the least (possible finite) cardinal $\kappa$, for which there exists a coloring $c:G\rightarrow\kappa$ such that $gEh$ entails $c(g)\neq c(h)$. Given a forcing notion $\mathbb P$, it is … Continue reading

Posted in Blog, Expository | Tagged | 16 Comments