# Blog Archives

# Topic Archive: CH

# Freiling’s axiom of symmetry, or throwing darts at the real line

This will be a talk for the GC Math Program Graduate Student Colloquium. The talk will be aimed at a general audience of mathematics graduate students.

I shall give an elementary presentation of Freiling’s axiom of symmetry, which is the principle asserting that if we map every real $x$ to a countable set of reals $A_x$, then there are two reals $x$ and $y$ for which $x$ is not in $A_y$ and $y$ is not in $A_x$. To argue for the truth of this principle, Freiling imagined throwing two darts at the real number line, landing at $x$ and $y$ respectively: almost surely, the location $y$ of the second dart is not in the set $A_x$ arising from that of the first dart, since that set is countable; and by symmetry, it shouldn’t matter which dart we imagine as being first. So it may seem that almost every pair must fulfill the principle. Nevertheless, the principle is independent of the axioms of ZFC and in fact it is provably equivalent to the failure of the continuum hypothesis. I’ll introduce the continuum hypothesis in a general way and discuss these foundational matters, before providing a proof of the equivalence of the negation of CH with the axiom of symmetry. The axiom of symmetry admits natural higher dimensional analogues, such as the case of maps from pairs $(x,y)$ to countable sets $A_{x,y}$, where one seeks a triple $(x,y,z)$ for which no member is in the set arising from the other two, and these principles also have an equivalent formulation in terms of the size of the continuum.

Questions and commentary can be made at: jdh.hamkins.org/freilings-axiom-of-symmetry-graduate-student-colloquium-april-2016/.

# The continuum hypothesis and other set-theoretic ideas for non-set-theorists

This is a talk for the Einstein Chair Mathematics Seminar at the CUNY Graduate Center, a talk on set theory for non-set-theorists, in two parts:

- An introductory background talk at 11 am
- The main talk at 2 – 4 pm

I shall present several set-theoretic ideas for a non-set-theoretic mathematical audience, focusing particularly on the continuum hypothesis and related issues.

At the morning talk, I shall discuss and prove the Cantor-Bendixson theorem, which asserts that every closed set of reals is the union of a countable set and a perfect set (a closed set with no isolated points), and explain how it led to Cantor’s development of the ordinal numbers and how it establishes that the continuum hypothesis holds for closed sets of reals. We’ll see that there are closed sets of arbitrarily large countable Cantor-Bendixson rank. We’ll talk about the ordinals, about $omega_1$, the long line, and, time permitting, we’ll discuss Suslin’s hypothesis. Dennis has requested that at some point the discussion turn to the role of set theory in the foundation for mathematics, compared for example to that of category theory, and I would look forward to that. I would be prepared also to discuss the Feferman theory in comparison to Grothendieck’s axiom of universes, and other issues relating set theory to category theory.

At the main talk in the afternoon, I’ll begin with a discussion of the continuum hypothesis, including an explanation of the history and logical status of this axiom with respect to the other axioms of set theory, and establish the connection between the continuum hypothesis and Freiling’s axiom of symmetry. I’ll explain the axiom of determinacy and some of its applications and its rich logical situation, connected with large cardinals. I’ll prove the determinacy of open sets and show that AD implies that every set of reals is Lebesgue measurable. I’ll briefly mention the themes and goals of the subjects of cardinal characteristics of the continuum and of Borel equivalence relation theory. If time permits, I’d like to explain some fun geometric decompositions of space that proceed in a transfinite recursion using the axiom of choice, mentioning the open questions concerning whether there can be such decompositions that are Borel.

See also the profile of this talk on my blog.

# Pluralism in set theory: does every mathematical statement have a definite truth value?

I shall give a summary account of some current issues in the philosophy of set theory, specifically, the debate on pluralism and the question of the determinateness of set-theoretical and mathematical truth. The traditional Platonist view in set theory, what I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. What I would like to do is to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, which I call the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.