Interactions Among Principles of Choice, Forcing, and Measure Theory

Abstract

In this thesis, we analyze the relationship between various set theoretic principles, motivated by individually plausible intuitions yet inconsistent with one another. Fragments of the Axiom of Choice (also known as choice principles) are motivated by the intuition that one can make infinitely many arbitrary choices, and the full Axiom of Choice is included in the standard mathematical foundation ZFC. We will examine some surprising consequences of Choice, in particular box game paradoxes and weak symmetry paradoxes. These theorems conflict with intuitions of randomness, in particular the belief that one cannot reliably predict the outcome of a random experiment with no relevant information. Measure-theoretic principles such as LM, the assertion “all sets of real numbers are Lebesgue measurable” and even the weaker assertion “there is a total-translation-invariant extension of Lebesgue measure” provide a natural alternative to Choice. A strong measure theory is appealing under a philosophical belief in truly random events, and refutes many but not all of the paradoxical consequences of Choice. We will formulate principles from the negation of these paradoxes and identify nontrivial combinatorial consequences, particularly the existence of ideals on the continuum with closure properties that cannot be achieved under Choice. One well-known principle of this form is Freiling’s axiom of symmetry. We identify a cardinal characterization of this axiom in the absence of choice and then consider strengthenings of this principle (equivalently, negations of weaker paradoxes) which refute moderate choice principles.