# Blog Archives

# Topic Archive: set theory of the reals

# 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.

# On the independence of Borel’s Conjecture

In 1919 Borel introduced the notion of a strong measure zero set and stated what has become known as Borel’s conjecture (BC): every strong measure zero set of reals is countable. Sierpiński soon proved (1928) that CH implies the failure of BC. However, a proof for the consistency of BC with ZFC would have to wait for the development of more powerful tools. In 1976, Laver used an iterated forcing argument to produce a model of ZFC + BC. I will present an exposition of these classical results. Time permitting, I will sketch some analogous results for the dual Borel Conjecture, the category analogue of BC.