I’m giving a contributed talk at the inner model theory conference in Girona and at the Set Theory Today conference in Vienna. Both talks are going to be on results from my recent paper, where the Girona talk will be more specialised in a game-theoretic direction and the Set Theory Today talk will be more of an overview of the Ramsey-like cardinals and some of our results. Here are a couple of abstracts.
Set Theory Today talk: Games and Ramsey-like cardinals (slides)
Most “large” large cardinals can be characterised in terms of elementary embeddings from V into some inner model M. As for the “smaller” large cardinals, for instance weakly compacts, ineffables, Erdős cardinals and Ramsey cardinals, these are mostly defined combinatorially using partition properties. Recently however, Gitman, Holy and Schlicht have introduced several so-called Ramsey-like cardinals, all lying in this smaller region of the large cardinals and which do characterise some of the “partition-type” cardinals. These Ramsey-like cardinals are all defined in terms of elementary embeddings between small structures of the same size and can thus be seen as “mini-versions” of the larger large cardinals. Holy and Schlicht introduced a hierarchy of these cardinals, and we show that these cardinals range from weakly compacts and all the way up to (and including) the measurable cardinals. A curious aspect of these cardinals is that they admit a game-theoretic definition, which leads us to certain determinacy results as well. This is joint work with Philip Welch.
Girona conference talk: Game-theoretic Ramsey-like cardinals
Holy and Schlicht recently introduced a new hierarchy of Ramsey-like cardinals, the (strategic) α-Ramsey cardinals, which can be seen as “mini-versions” of measurable cardinals and which also admit a game-theoretic definition. We show that strategic ω₁-Ramseys are measurable in K and that the α-length game associated to the (strategic) α-Ramseys is determined when α≤ω and not determined when α≥ω₁. This is joint work with Philip Welch.