Special Session

October 22, 2018

Two pictures from last weekend’s Session.

We rescued this one from Instagram a couple of minutes after it happened (Paul noticed that #determinacy was missing from the list):


And a decent group picture. Missing: Maryanthe Malliaris (who took it), Harry Altman and Jeffrey Bergfalk:


Many thanks to all the participants, and to the AMS for making it possible.


Large cardinals and combinatorial set theory

August 28, 2018

Paul Larson and I are organizing a special session at the Fall Central Sectional Meeting 2018 in Ann Arbor, on Large cardinals and combinatorial set theory. The session will take place Saturday October 20 and Sunday October 21. See here for the schedule and additional details. Paul and I are organizing dinner for the speakers for Saturday, at 7:30 p.m.

I transcribe the schedule below:

  • Saturday October 20, 2018, 8:30-11:20 a.m.

Room 2336, Mason Hall

  • Saturday October 20, 2018, 12:30-2:00 p.m. Open house at Mathematical Reviews.

416 Fourth Street

  • Saturday October 20, 2018, 2:00-4:20 p.m.

Room 2336, Mason Hall

Atrium, East Hall.

  • Sunday October 21, 2018, 8:00-10:20 a.m.

Room 2336, Mason Hall

  • Sunday October 21, 2018, 1:00-3:50 p.m.

Room 2336, Mason Hall

Professorship in Logic at the University of Vienna

February 4, 2018

I have been asked to help spread the word. Sy Friedman, the head of the Kurt Gödel Research Center at the University of Vienna (where I had my first postdoc), is retiring, and there is a search underway for a replacement. The KGRC has been significant in fostering a multitude of early career researchers in set theory, and it would be a shame if the position ended up in another discipline.

If you are interested, please consider applying. If you know of someone who may be interested, please help spread the word. The deadline is April 15.


From the link, the position is for a “University Professor of Mathematical Logic Taking into Account the Foundations of Computer Science”. Applications should be submitted by e-mail to the Dean of the Faculty of Mathematics of the University of Vienna, Univ.-Prof. Dr. Christian Krattenthaler, Oskar-Morgenstern-Platz 1, 1090 Vienna (dekanat.mathematik@univie.ac.at). Further details are at the link.

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:

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.


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.


July 5, 2016


Help us identify all mathematicians in this picture (click on it for a larger version). Please post comments here, on G+, or email me or Paul Larson.

The picture will appear in the book of proceedings of the Woodin conference, http://logic.harvard.edu/woodin_meeting.html. (Thanks to David Schrittesser for allowing us to use it.)