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 26, 2016

Announcement of the plan to revise the Mathematics Subject Classification

Mathematical Reviews (MR) and zbMATH cooperate in maintaining the Mathematics Subject Classification (MSC), which is used by these reviewing services, publishers, and others to categorize items in the mathematical sciences literature. The current version, MSC2010, consists of 63 areas classified with two digits refined into over 5000 three- and five-digit classifications. Details of MSC2010 can be found at www.msc2010.org or www.ams.org/msc/msc2010.html and zbmath.org/classification/.

MSC2010 was a revision of the 2000 subject classification scheme developed through the collaborative efforts of the editors of zbMATH and MR with considerable input from the community. zbMATH and MR have initiated the process of revising MSC2010 with an expectation that the revision will be used beginning in 2020. From the perspective of MR and zbMATH, the five-digit classification scheme MSC is an extremely important device that allows editors and reviewers to process the literature. Users of the publications of zbMATH and MR employ the MSC to search the literature by subject area. In the decade since the last revision, keyword searching has become increasingly prevalent, with remarkable improvements in searchable databases. Yet, the classification scheme remains important. Many publishers use the subject classes at either the time of submission of an article, as an aid to the editors, or at the time of publication, as an aid to readers. The arXiv uses author-supplied MSC codes to classify submissions, and as an option in creating alerts for the daily listings. Browsing the MR or zbMATH database using a two- or three-digit classification search is an effective method of keeping up with research in specific areas.

Based in part on some thoughtful suggestions from members of the community, the editors of MR and zbMATH have given preliminary consideration to the scope of the revision of the MSC. We do not foresee any changes at the two-digit level; however, it is anticipated that there will be refinement of the three- and five-digit levels.

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Version of 2016.07.25


April 22, 2015

I will be taking a leave from BSU this coming academic year, and moving to Ann Arbor, to work as an Associate Editor at MathReviews.