Abstract
As we all know, in consequence of the Axiom of Choice many so-called "monstrous" sets exist, such as Hamel bases, a subset of the plane which meets every line in exactly two points, MAD families, and so on... For many of these sets, it is known that they must be complicated, in the sense that their construction cannot proceed in few, simple, effective steps (this can be made precise). Interestingly, the usual standard axioms of set theory leave open how complex a monstrous set must be: This depends, roughly speaking, on what large cardinals exist in the set theoretical universe. A classical result of Arnie Miller shows that in the smallest universe (Gödel's constructible universe) there are co-analytic monstrous sets (that is, they are as simple as they can possibly be; meaning this universe is itself somewhat "monstrous"). In this talk I will discuss a recent joint result with Vera Fischer and Thilo Weinert, stating that a similar same conclusion can be proved in a specific, well-known extension of the axioms of set theory: Namely, under the Bounded Proper Forcing Axiom (BPFA) plus an assumption which limits the size of the large cardinals in our universe (an anti-large cardinal assumption), there are Π¹₂ Hamel bases, mad families, etc. This is to say, provided there are almost no large cardinals, models of BPFA are "almost as monstrous" as the minimal model of set theory, Gödel's constructible universe.