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 a\cdot (b+c)=ab+ac. 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!

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Mice and long games

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 \omega_1+\omega. Restricting ourselves to definable games of countable length on the integers, what large cardinal strength do we obtain?

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A travel guide to the large cardinals

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?

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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.

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Core Model Induction 101


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.

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Paper: Games and Ramsey-like cardinals

D. S. Nielsen and P. Welch, Games and Ramsey-like cardinals, 2018, manuscript under review — arXiv.

Abstract. We generalise the \alpha-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals \alpha to arbitrary ordinals \alpha, and answer several questions posed in that paper. In particular, we show that \alpha-Ramseys are downwards absolute to the core model K for all \alpha of uncountable cofinality, that strategic \omega-Ramsey cardinals are equiconsistent with virtually measurable cardinals and that strategic \omega_1-Ramsey cardinals are equiconsistent with measurable cardinals. Further, if 2^\kappa=\kappa^+ then strategic \omega_1-Ramseys carry precipitous ideals. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing connections between the \alpha-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).

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