in Set Theory

The Model of Solovay

The objective of measure theory is to make rigorous the intuitive notions of length, surface area, volume, and measurement of quantity. A mathematical model of all of these can be found in the concept of measure on a set S, which in its simplest form is a function \mu : 2^S \rightarrow [0,1] such that the following hold:

  •  Monotonicity: X \subset Y \Rightarrow \mu(X) \le \mu(Y)
  • \sigma-additivity: \mu(\sqcup_{n\in\omega} X_n) = \Sigma_{n \in\omega} \mu(X_n)
  • Normalization: \mu(\emptyset) = 0 and \mu(S) = 1
  • Atomlessness: \forall a \in S: \mu(\{a\}) = 0

On the one hand, this notion replicates a number of facts about volume in the physical world. For instance, the first condition can be rephrased as “Bigger objects have bigger volume” and the second one translates to “The combined volume of a collection of disjoint objects is the sum of their respective volumes”. On the other hand, we can interpret a measure as an order-preserving map between partially ordered sets. Here the set 2^S is ordered by inclusion, and \R is ordered by \le. From this perspective, (i) just means that \mu is order-preserving, (ii) restricts the value of \mu on certain joins, and (iii) and (iv) are normalization conditions. In analogy of the representation theory of groups, the structure of 2^S as a poset is compared trough \mu to the poset structure of [0,1]. This interrelates the two structures and allows to analyze the structure of either one.With this definition in mind, there are a few questions that arise naturally. From the practical point of view, me might ask:

  • Does euclidean space \R^n admit a measure that is translation invariant and hence reflects our experience of volume?

From the set theoretic point of view, one might ask:

  • Which sets S admit a measure? In particular, since S admits a measure exactly when |S| does, which cardinals admit measures? This question is known as the Measure Problem.

As we will show in this semester project, the answer to the first question is no under \ac. This has peculiar consequences such as the possibility to decompose a solid ball into a small number of pieces, rearranging them, and ending up with two exact copies of the initial ball. This mathematical possibility of copying a ball without adding any mass is known as the Banach-Tarski paradox, and would be impossible if a measure existed on all of \R^3, for then mass would have to be conserved.

From here, there are essentially two possible directions to proceed. Either one can weaken the requirement that the measure be translation invariant, in which case there may exist a solution, or one can weaken \ac, which is the route we are taking. Robert M. Solovay showed in 1970  that there is a model of set theory in which every set of reals is Lebesgue measurable and where a such a weakening of \ac, the axiom of dependent choice \dc, still holds.

The main goal of this thesis is to construct this model and to discuss how the structure of the real line relates to strong assumptions, such as further axioms of set theory or large cardinal hypotheses.In the first chapter, we introduce essential concepts and tools from descriptive set theory, such as a representation of the reals used in set theory, the Borel sets and alongside Lebesgue measurability two other regularity properties of the reals, the property of Baire and the perfect set property. The second chapter constructs the Model of Solovay and proves that any subset of the real line in it is Lebesgue measurable, has the Property of Baire and the perfect set property. The proof requires the existence of an inaccessible cardinal to define the Lévy collapse, which is a forcing notion that allows to overwrite any bad behaviour the reals might display. With this a possible solution to the measure problem is presented if \ac is not assumed.In the last chapter we present the work of Shelah and Specker, illuminating when the assumption of the inaccessible cardinal is really needed, and in which cases it can be avoided. Finally, we will compare different alternatives to \ac and present their impact on the regularity of the reals.We generally follow the books by Jech and Kanamori.

The rest of this semester project can be found here.