The notion of distributivity comes from the Latin word distribut-, meaning “divided up”, and has since evolved into how mathematics deals with things that are divided up. This starts back in school when we learn that . This property can be generalised in the language of Boolean algebras, still maintaining the intent of dealing with divided stuff, leading to the axiom of choice being a notion of distributivity as well!
When dealing with games in general, we can vary different parameters. We could vary (1) how big the payoff set is, (2) which objects we’re playing and (3) for how many rounds we’re playing. In a ZFC context, which is what I’ll be working with here as well, I’ve previously written about what limitations we’re facing. In particular, when we restrict ourselves to definable games then we can’t have determinacy of games on integers of length . Restricting ourselves to definable games of countable length on the integers, what large cardinal strength do we obtain?
Looking at a map of the large cardinal hierarchy for the first time can be a dizzying experience. What are the differences between them, and which ones are similar? Some of them are defined using partition properties and some of them are defined using elementary embeddings, and others have a whole myriad of equivalent characterisations! What’s the intuition about the different sections of the hierarchy, and what type of set theorists are working in each section?
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.
There’s a very neat way of encoding any set as a set of ordinals, which has the somewhat peculiar feature of it being hard (which here meaning that it requires the axiom of choice) to encode sets, but easy to decode them. Like some kind of a very ineffective crypto-system.
Mentioning the core model induction to a fellow set theorist is akin to mentioning that you’re a mathematician to the layman — you receive a reaction which is struck by a delightful mix of terror and awe. My humble goal with this blog post is not to offer a “fix-all” solution to this problem, but rather to give a vague (but correct) explanation of what’s actually going on in a core model induction, without getting too bogged down on the details.
D. S. Nielsen and P. Welch, Games and Ramsey-like cardinals, 2018, manuscript under review — arXiv.
Abstract. We generalise the -Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals to arbitrary ordinals , and answer several questions posed in that paper. In particular, we show that -Ramseys are downwards absolute to the core model for all $\alpha$ of uncountable cofinality, that strategic -Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic -Ramsey cardinals are equiconsistent with measurable cardinals. We also show that the -Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing connections between the -Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).