Set theory, logic and Ramsey theory

October 16, 2017

José G. Mijares and I are organizing a special session at the 2018 Joint Mathematics Meetings in San Diego, cosponsored by the AMS and the ASL, on Set theory, logic and Ramsey theory. The session will take place the morning of Wednesday January 10 and the morning and afternoon of Thursday January 11. See here for the schedule and additional details. I transcribe the schedule below:


Monica Agana – The classical theory of rearrangements

July 16, 2017

Monica was my last masters student at Boise State, completing her thesis in December 2015, while I was already at Mathematical Reviews. My colleague at Boise State Zach Teitler coadvised her for the last few months. The following picture is from graduation day.

Monica Agana


Monica’s thesis, The classical theory of rearrangements, discusses Riemann’s famous rearrangement theorem for conditionally convergent series of real numbers, prior results, and some extensions and related topics. This previous blog post discusses some of these.

Bogotá, September 17-20, 2014, 50 años carrera de matemáticas, Universidad de los Andes

September 17, 2016

I ran last night into the Flickr page for this conference essentially by accident. Couldn’t find who to credit for this picture.



The Dalek invasion of MR

September 12, 2016

Maybe they are chasing me, see here.


On anti-foundation and coding the hereditarily finite sets

August 27, 2016

I would like to highlight a cute question in a recent paper,

Giovanna D’Agostino, Alberto Policriti, Eugenio G. Omodeo, and Alexandru I. Tomescu.
Mapping sets and hypersets into numbers.
Fund. Inform. 140 (2015), no. 3-4, 307–328.

Recall that W. Ackermann verified what in modern terms we call the bi-interpretability of \mathsf{ZFfin} and \mathsf{PA}, where the latter is (first-order) Peano arithmetic, and the former is finite set theory, the result of replacing in \mathsf{ZF} the axiom of infinity with its negation (and with foundation formulated as the schema of \in-induction). The reference is

Wilhelm Ackermann.
Die Widerspruchsfreiheit der allgemeinen Mengenlehre.
Math. Ann. 114 (1937), no. 1, 305–315.

I have written about this before. Briefly, one exhibits (definable) translations between the collection \mathsf{HF} of hereditarily finite sets and \mathbb{N}, and verifies that the translation extends to a definable translation of the relations, functions and constants of the language of each structure in a way that \mathsf{PA} verifies that \mathsf{ZFfin} holds in the translation of (\mathsf{HF},\in), and \mathsf{ZFfin} verifies that \mathsf{PA} holds in the translation of {\mathbb N}=(\omega,+,\times,<,0,1). Recall that \mathsf{HF} consists of those sets a whose transitive closure is finite, that is, a is finite, and all its elements are finite, and all the elements of its elements are finite, and so on. Using foundation, one easily verifies that \mathsf{HF}=V_\omega=\bigcup_{n\in\omega}V_n, that is, it is the collection of sets resulting from iterating the power-set operation (any finite number of times) starting from the empty set.

In the direction relevant here, one defines a map h:\mathsf{HF}\to\mathbb{N} by

h(a)=\sum_{b\in a}2^{h(b)}.

One easily verifies, using induction on the set-theoretic rank of the sets involved, that this recursive definition makes sense and is injective (and, indeed, bijective).

Of course this argument uses foundation. In the D’Agostino-Policriti-Omodeo-Tomescu paper they consider instead the theory resulting from replacing foundation with the  anti-foundation axiom, and proceed to describe a suitable replacement for h that injects (codes) \mathsf{HF} into the real numbers. They do quite a bit more in the paper but, for the coding itself, I highly recommend the nice review by Randall Holmes in MathSciNet, linked to above.

The anti-foundation axiom \mathsf{AFA} became known thanks to the work of Peter Aczel, and it is his formulation that I recall below, although it was originally introduced in work of Forti and Honsell from 1983, where they call it X_1. Aczel’s presentation appears in the excellent book

MR0940014 (89j:03039)
Peter Aczel.
Non-well-founded sets. With a foreword by Jon Barwise.
CSLI Lecture Notes, 14. Stanford University, Center for the Study of Language and Information, Stanford, CA, 1988. xx+137 pp.
ISBN: 0-937073-22-9.

The original paper is

MR0739920 (85f:03054)
Marco Forti, Furio Honsell.
Set theory with free construction principles.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 3, 493–522.

Given a binary relation R, its field \mathrm{fld}(R) is the union of its domain and codomain. A decoration of R is a function d:\mathrm{fld}(R)\to V satisfying

d(x)=\{d(y)\mid y\mathrel{R}x\}

for all x,y\in\mathrm{fld}(R). When R is \in and the sets in question are well-founded, the only decoration is the identity. Similarly, any well-founded relation R admits a unique decoration. Define \mathsf{AFA} as the statement that any binary R (whether well-founded or not) admits a unique decoration.

In \mathsf{ZF} with foundation replaced with \mathsf{AFA} one can prove the existence of many non-well-founded sets. One of the appealing aspects of \mathsf{AFA} is that the resulting univere is actually quite structured: Other anti-foundation axioms allow the existence of infinitely many Quine atoms, sets x such that x=\{x\}, for instance. Under \mathsf{AFA}, there is exactly one such x, usually called \Omega. The axiom is sometimes described as saying that it provides solutions to many “equations” among sets. For instance, consider the system of equations x=\{y\} and y=\{x\}. Under \mathsf{AFA} the system has x=y=\Omega as its unique solution. Note that assuming \mathsf{AFA}, \Omega is in \mathsf{HF}, as are many other non-well-founded sets.

Here is the open question from the D’Agostino-Policriti-Omodeo-Tomescu paper: Work in set theory with \mathsf{AFA} instead of foundation. Is there a unique, injective, function h:\mathsf{HF}\to \mathbb{R}  satisfying

h(x)=\sum_{y\in x}2^{-h(y)}

for all x,y\in\mathsf{HF}?

Note that there is a unique such h on the well-founded hereditarily finite sets, and it is in fact injective. In general, existence, uniqueness and injectivity of h appear to be open. The claim that there is such a function h is a statement about solutions of certain equations on the reals, and the claim that h is unique requires moreover uniqueness of such solutions. The expectation is that h(x) is transcendental for all non-well-founded hereditarily finite x but, even assuming this, the injectivity of h seems to require additional work.

For example, consider x=\Omega. The function h must satisfy


and, indeed h(\Omega)=0.6411857\dots is the unique solution x of the equation x=2^{-x}

I would be curious to hear of any progress regarding this problem.

Monochromatic colorings

August 20, 2016

Caïus Wojcik and Luca Zamboni recently posted a paper on the arXiv solving an interesting problem in combinatorics on words.
Monochromatic factorisations of words and periodicity.
Caïus Wojcik, Luca Q. Zamboni.

I had recently learned of the problem through another paper by Zamboni and a collaborator,

Aldo de Luca, Luca Q. Zamboni
On prefixal factorizations of words.
European J. Combin. 52 (2016), part A, 59–73.

It is a nice result and I think it may be enjoyable to work through the argument here. Everything that follows is either straightforward, standard, or comes from these papers.

1. The problem

To make the post reasonably self-contained, I begin by recalling some conventions, not all of which we need here.

By an alphabet we simply mean a set A, whose elements we refer to as letters. A word w is a sequence w:N\to A of letters from A where N is a (not necessarily non-empty, not necessarily proper) initial segment of \mathbb N. If we denote w_i=w(i) for all i\in N, it is customary to write the word simply as


and we will follow the convention. The empty word is typically denoted by \Lambda or \varepsilon. By A^* we denote the collection of all finite words from A, and A^+=A^*\setminus \{\varepsilon\}. By |x| we denote the length of the word x (that is, the size of the domain of the corresponding function).

We define concatenation of words in the obvious way, and denote by x_0x_1 the word resulting from concatenating the words x_0 and x_1, where x_0\in A^*. This operation is associative, and we extend it as well to infinite concatenations.

If a word w can be written as the concatenation of words x_0,x_1,\dots,


we refer to the right-hand side as a factorization of w. If w=xy and x is non-empty, we say that x is a prefix of w. Similarly, if y is non-empty, it is a suffix of w. By x^n for n\in\mathbb N we denote the word resulting form concatenating n copies of x. Similarly, x^{\mathbb N} is the result of concatenating infinitely many copies.

By a coloring we mean here a function c:A^+\to C where C is a finite set of “colors”.

Apparently the problem I want to discuss was first considered by T.C. Brown around 2006 and, independently, by Zamboni around 2010. It is a question about monochromatic factorizations of infinite words. To motivate it, let me begin with a cute observation.

Fact. Suppose w=w_0w_1\dots is an infinite word, and c is a coloring. There is then a factorization 


where all the x_i\in A^+ have the same color.

Proof. The proof is a straightforward application of Ramsey’s theorem: Assign to c the coloring of the set [\mathbb N]^2 of 2-sized subsets of \mathbb N given by d(\{i,j\})=c(w_iw_{i+1}\dots w_{j-1}) whenever i<j. Ramsey’s theorem ensures that there is an infinite set I=\{n_0<n_1<\dots\} such that all w_{n_i}w_{n_i+1}\dots w_{n_j-1} with i<j have the same color. We can then take p=w_0\dots w_{n_0-1} and x_i=w_{n_i}\dots w_{n_{i+1}-1} for all i. \Box

In the fact above, the word w was arbitrary, and we obtained a monochromatic factorization of a suffix of w. However, without additional assumptions, it is not possible to improve this to a monochromatic factorization of w itself. For example, consider the word w=01^{\mathbb N} and the coloring

c(x)=\left\{\begin{array}{cl}0&\mbox{if }0\mbox{ appears in }x,\\ 1&\mbox{otherwise.}\end{array}\right.

If nothing else, it follows that if w is an infinite word that admits a monochromatic factorization for any coloring, then the first letter of w must appear infinitely often. The same idea shows that each letter in w must appear infinitely often.

Actually, significantly more should be true. For example, consider the word

w=010110111\dots 01^n0 1^{n+1}\dots,

and the coloring

c(x)=\left\{\begin{array}{cl}0&\mbox{if }x\mbox{ is a prefix of }w,\\1&\mbox{otherwise.}\end{array}\right.

This example shows that in fact any such w must admit a prefixal factorization, a factorization


where each x_i is a prefix of w.

Problem. Characterize those infinite words w with the property P that given any coloring, there is a monochromatic factorization of w.

The above shows that any word with property P admits a prefixal factorization. But it is easy to see that this is not enough. For a simple example, consider


Consider the coloring c where c(x)=0 if x is not a prefix of w, c(0)=1, and c(x)=2 otherwise. If


is a monochromatic factorization of w, then x_0=01\dots so c(x_0)=2 and each x_i must be a prefix of w of length at least 2. But it is easy to see that w admits no such factorization: For any n>2, consider the first appearance in w of 0^{n+1} and note that none of the first n zeros can be the beginning of an x_i, so for some j we must have x_j=01\dots 10^n and since n>2, in fact x_j=01\dots 10^n10^n, but this string only appears once in w, so actually j=0. Since n was arbitrary, we are done.

Here is a more interesting example: The Thue-Morse word


was defined by Axel Thue in 1906 and became known through the work of Marston Morse in the 1920s. It is defined as the limit (in the natural sense) of the sequence x_0,x_1,\dots of finite words given by x_0=0 and x_{n+1}=x_n\bar{x_n} where, for x\in\{0,1\}^*, \bar x is the result of replacing each letter i in x with 1-i.

This word admits a prefixal factorization, namely


To see this, note that the sequence of letters of t can be defined recursively by t_0=0, t_{2n}=t_n and t_{2n+1}=1-t_n. To see this, note in turn that the sequence given by this recursive definition actually satisfies that t_n is the parity of the number of 1s in the binary expansion of n, from which the recursive description above as the limit of the x_n should be clear. The relevance of this observation is that no three consecutive letters in t can be the same (since t_{2n+1}=1-t_{2n} for all n), and from this it is clear that t can be factored using only the words 0, 01, and 011.

But it is not so straightforward as in the previous example to check whether t admits a factorization into prefixes of length larger than 1.

Instead, I recall a basic property of t and use it to exhibit an explicit coloring for which t admits no monochromatic factorization.

Read the rest of this entry »


August 11, 2016

I have just posted on my papers page a preprint of a review of

Smullyan, Raymond
Reflections—the magic, music and mathematics of Raymond Smullyan.
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. x+213 pp.
ISBN: 978-981-4644-58-7; 978-981-4663-19-9

that I have submitted to Mathematical Reviews.