We relate Popper functions to regular and perfectly additive such non-Archimedean probability functions by means of a representation theorem: every such non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that regular and perfectly additive non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions triangulate to the same place: they are in a good sense interchangeable.

In his seminal paper introducing the fine structure of L, Jensen proved that under \ any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen’s result that is straight-forward and accessible to those without a knowledge of Jensen’s fine structure theory. The proof here instead uses hyperfine structure, a very natural and simpler alternative to fine structure theory introduced by Friedman and Koepke.

In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets, or, in other words, to have a specific probability of having a given set-theoretic property.