My work intends to bring some understanding about some instances of very large cardinal axioms by using tools from quite a new field of research called Generalised Descriptive Set Theory. In particular, I work in Generalised Descriptive Set Theory for singular cardinals of countable cofinality and its connections with I0.
I0 states that there exists and elementary embedding j from L(Vλ+1) into L(Vλ+1) with critical point less than λ, where L(Vλ+1) is the class of sets constructible relative to Vλ+1. It is a very large cardinal axiom, the closest to inconsistency (with ZF plus Choice) so far conceived and it has deep connections with Determinacy and Descriptive Set Theory. For instance, Woodin showed that the Axiom of Determinacy holds in L(R) under the assumption of I0 and that Vλ+1 curiously exhibits interesting descriptive set theoretic properties.
While Classical Descriptive Set Theory studies definable sets in Polish spaces, GDST of singular cardinals of countable cofinality focus on definable sets in λ-Polish spaces with λ of countable cofinality. A λ-Polish space is a topological space which is completely metrizable and has weigth less than or equal to λ. Now, if λ is a limit cardinal, Vλ+1 endowed with the Woodin topology has a basis of cardinality |Vλ|. Moreover, if λ is assumed to be a strong limit, then wt(Vλ+1)=|Vλ| and, if cf(λ)=ω, we can define a complete metric on Vλ+1. Therefore, if λ is of countable cofinality and |Vλ|=λ, then Vλ+1 is a λ-Polish space. It turns out that if λ is the supremum of the critical sequence of j witnessed by I0, it satisfies all the previous requirements.
λ-Polish spaces admit generalised forms of those regularity properties that had historically motivated the development of Classical Descriptive Set Theory such as the Baire and Perfect Set Properties. We know already that λ-Polish spaces, under suitable assumptions, satisfy the λ-Baire property and that if λ witnesses I0, then every subset of every λ-Polish space X ∈ L(Vλ+1) has the λ-PSP. We wonder whether the same can be done for a generalised version of Lebesgue measurability, which is what articulates my research project. Right now I am working on a theory for λ-additive measures. Being my ultimate aim to show that I0 implies that every set is λ-Lebesgue measurable.
As an accessory project, I am trying to provide a “correct” generalisation of the intuitive notion of the continuum line into the transfinite. This has led me to the study of the potential role that singular cardinals of countable cofinality may play as generalisations of ω: although it is a quite common thing to say that the motto of large cardinals is to generalise feautures of ω into the transfinite, there are situations in which ω behaves more like a cardinal of countable cofinality than like a regular cardinal.