# Choiceless large cardinals and set-theoretic potentialism

• R. Cutolo and J. D. Hamkins, “Choiceless large cardinals and set-theoretic potentialism,” , p. 10 pages, 2020. (Under review)
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Abstract. We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe $V$. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory S5, both for assertions in the language of ZF and for assertions in the full potentialist language.

# Forcing as a computational process

• J. D. Hamkins, R. Miller, and K. J. Williams, “Forcing as a computational process,” Mathematics arXiv, 2020. (Under review)
@ARTICLE{HamkinsMillerWilliams:Forcing-as-a-computational-process,
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Abstract. We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory $\langle M,\in^M\rangle$, we explain senses in which one may compute $M$-generic filters $G\subseteq\mathbb{P}\in M$ and the corresponding forcing extensions $M[G]$. Specifically, from the atomic diagram one may compute $G$, from the $\Delta_0$-diagram one may compute $M[G]$ and its $\Delta_0$-diagram, and from the elementary diagram one may compute the elementary diagram of $M[G]$. We also examine the information necessary to make the process functorial, and conclude that in the general case, no such computational process will be functorial. For any such process, it will always be possible to have different isomorphic presentations of a model of set theory $M$ that lead to different non-isomorphic forcing extensions $M[G]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

# Categorical cardinals, CUNY Set Theory Seminar, June 2020

This will be an online talk for the CUNY Set Theory Seminar, Friday 26 June 2020, 2 pm EST = 7 pm UK time. Contact Victoria Gitman for Zoom access.

Abstract: Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$ in his 1930 quasi-categoricity result asserting that the models of $\text{ZFC}_2$ are precisely those isomorphic to a rank-initial segment $V_\kappa$ of the cumulative set-theoretic universe $V$ cut off at an inaccessible cardinal $\kappa$. I shall discuss the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the $V_\kappa$ universes are categorically characterized by their sentences or theories. For example, if $\kappa$ is the smallest inaccessible cardinal, then up to isomorphism $V_\kappa$ is the unique model of $\text{ZFC}_2$ plus the sentence “there are no inaccessible cardinals.” This cardinal $\kappa$ is therefore an instance of what we call a first-order sententially categorical cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of $\text{ZFC}_2$ by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford).

# My view of Univ

Appearing in The Martlet, Issue 11, Spring 2020, University College, Oxford.

## My view of Univ

“I came to Oxford last year, leaving an established career in New York, and found a welcoming new home, an ideal environment for research and intellectual stimulation. Through the big wooden door to the Main Quad, I enter the College each day to find fascinating new conversations with historians, classicists, geologists, political scientists, medical scientists, mathematicians, philosophers, artists and even Egyptologists. What a life! I take on Oxford like a fine wool coat, enveloping me, suiting me perfectly.”

Professor Joel David Hamkins, Sir Peter Strawson Fellow in Philosophy at Univ and Professor of Logic at Oxford

# The theory of infinite games, including infinite chess, Talk Math With Your Friends, June 2020

This will be accessible online talk about infinite chess and other infinite games for the Talk Math With Your Friends seminar, June 18, 2020 4 pm EST (9 pm UK).  Zoom access information.  Please come talk math with me!

Abstract. I will give an introduction to the theory of infinite games, with examples drawn from infinite chess in order to illustrate various concepts, such as the transfinite game value of a position.

See more of my posts on infinite chess.

# Proof and the Art of Mathematics

A coming-of-age book for mathematicians aspiring to write proofs.

• J. D. Hamkins, Proof and the Art of Mathematics, MIT Press, 2020.
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title = {Proof and the {Art} of {Mathematics}},
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}

Now available!

## From the Preface:

This is a mathematical coming-of-age book, for students on the cusp, who are maturing into mathematicians, aspiring to communicate mathematical truths to other mathematicians in the currency of mathematics, which is: proof. This is a book for students who are learning—perhaps for the first time in a serious way—how to write a mathematical proof. I hope to show how a mathematician makes an argument establishing a mathematical truth.

Proofs tell us not only that a mathematical statement is true, but also why it is true, and they communicate this truth. The best proofs give us insight into the nature of mathematical reality. They lead us to those sublime yet elusive Aha! moments, a joyous experience for any mathematician, occurring when a previously opaque, confounding issue becomes transparent and our mathematical gaze suddenly penetrates completely through it, grasping it all in one take. So let us learn together how to write proofs well, producing clear and correct mathematical arguments that logically establish their conclusions, with whatever insight and elegance we can muster. We shall do so in the context of the diverse mathematical topics that I have gathered together here in this book for the purpose.

# Bi-interpretation of weak set theories, Oxford Set Theory Seminar, May 2020

This will be a talk for the newly founded Oxford Set Theory Seminar, May 20, 2020. Contact Sam Adam-Day (me@samadamday.com) for the Zoom access codes.

Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

This is a version of the talk that I had planned to give at the 2020 Set Theory meeting Oberwolfach, before that meeting was canceled on account of the Covid-19 situation.

Slides

• A. R. Freire and J. D. Hamkins, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020. (Under review)
@ARTICLE{FreireHamkins:Bi-interpretation-in-weak-set-theories,
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# Oxford Set Theory Seminar

Welcome to the Oxford Set Theory Seminar.

We focus on all aspects of set theory and the philosophy of set theory. Topics will include forcing, large cardinals, models of set theory, set theory as a foundation, set-theoretic potentialism, cardinal characteristics of the continuum, second-order set theory and class theory, and much more.

Technical topics are completely fine. Speakers are encouraged to pick set-theoretic topics having some philosophical angle or aspect, although it is expected that this might sometimes be a background consideration, while at other times it will be a primary focus.

The seminar will last 60-90 minutes, and are generally held on Wednesdays 4:00 – 5:30 UK time. Speakers are requested to prepare a one hour talk, and we expect a lively discussion with questions.

## Hilary Term 2021

The seminar this term is again held jointly with the University of Bristol, organized by myself, Samuel Adam-Day, and Philip Welch.

Talks are held Wednesdays 4:00 – 5:30 pm UK time.

## Iteration, reflection, and singular cardinals

Abstract. Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and
(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.
These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH.  Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.

## Stationary reflection at successors of singular cardinals

We survey some recent progress in understanding stationary reflection at successors of singular cardinals and its influence on cardinal arithmetic:
1) In joint work with Yair Hayut, we reduced the consistency strength of stationary reflection at aleph_{omega+1} to an assumption weaker than kappa is kappa+ supercompact.
2) In joint work with Yair Hayut and Omer Ben-Neria, we prove that from large cardinals it is consistent that there is a singular cardinal nu of uncountable cofinality where the singular cardinal hypothesis fails at nu and every collection of fewer than cf(nu) stationary subsets of nu+ reflects at a common point.
The statement in the second theorem was not previously known to be consistent.  These results make use of analysis of Prikry generic objects over iterated ultrapowers.

## Title to be announced

Abstract. Will be posted later.

## Michaelmas Term 2020

This term, we are coordinating the seminar in collaboration with Bristol, and so let me announce the joint meetings of the Oxford Set Theory Seminar and the Bristol Logic and Set Theory seminar. Organized by myself, Samuel Adam-Day, and Philip Welch.

For the Zoom access code (which is the same as last term), contact Samuel Adam-Day me@samadamday.com.

Talks are held on Wednesdays 4:00 – 5:30 UK time.

# Ultrafilters on omega versus forcing

Abstract. I plan to survey known facts and open questions about ultrafilters on omega generating (or not generating) ultrafilters in forcing extensions.

# On wide Aronszajn trees

Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).

In our upcoming JSL  paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree.

The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version doi: 10.1017/jsl.2020.42.

# Even ordinals and the Kunen inconsistency

Abstract. The Burali-Forti paradox suggests that the transfinite cardinals “go on forever,” surpassing any conceivable bound one might try to place on them. The traditional Zermelo-Frankel axioms for set theory fall into a hierarchy of axiomatic systems formulated by reasserting this intuition in increasingly elaborate ways: the large cardinal hierarchyOr so the story goes. A serious problem for this already naive account of large cardinal set theory is the Kunen inconsistency theorem, which seems to impose an upper bound on the extent of the large cardinal hierarchy itself. If one drops the Axiom of Choice, Kunen’s proof breaks down and a new hierarchy of choiceless large cardinal axioms emerges. These axioms, if consistent, represent a challenge for those “maximalist” foundational stances that take for granted both large cardinal axioms and the Axiom of Choice. This talk concerns some recent advances in our understanding of the weakest of the choiceless large cardinal axioms and the prospect, as yet unrealized, of establishing their consistency and reconciling them with the Axiom of Choice.

# The geology of inner mantles

An inner model is a ground if V is a set forcing extension of it. The intersection of the grounds is the mantle, an inner model of ZFC which enjoys many nice properties. Fuchs, Hamkins, and Reitz showed that the mantle is highly malleable. Namely, they showed that every model of set theory is the mantle of a bigger, better universe of sets. This then raises the possibility of iterating the definition of the mantle—the mantle, the mantle of the mantle, and so on, taking intersections at limit stages—to obtain even deeper inner models. Let’s call the inner models in this sequence the inner mantles.

In this talk I will present some results, both positive and negative, about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz, results which are analogues of classic results about the sequence of iterated HODs. On the positive side: (Joint with Reitz) Every model of set theory is the eta-th inner mantle of a class forcing extension for any ordinal eta in the model. On the negative side: The sequence of inner mantles may fail to carry through at limit stages. Specifically, it is consistent that the omega-th inner mantle not be a definable class and it is consistent that it be a definable inner model of ¬AC.

## Trinity Term 2020

In Trinity term 2020, the seminar is organized by myself and Samuel Adam-Day. In light of the corona virus situation, we will be meeting online via Zoom for the foreseeable future.

## 6 May 2020, 4 pm UK

Victoria Gitman, City University of New York

## Elementary embeddings and smaller large cardinals

Abstract  A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

## 20 May 2020, 4 pm

Joel David Hamkins, Oxford

## Bi-interpretation of weak set theories

Abstract. Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

## 27 May 2020, 4 pm

Ali Enayat, Gothenberg

## Leibnizian and anti-Leibnizian motifs in set theory

Abstract. Leibniz’s principle of identity of indiscernibles at first sight appears completely unrelated to set theory, but Mycielski (1995) formulated a set-theoretic axiom nowadays referred to as LM (for Leibniz-Mycielski) which captures the spirit of Leibniz’s dictum in the following sense:  LM holds in a model M of ZF iff M is elementarily equivalent to a model M* in which there is no pair of indiscernibles.  LM was further investigated in a 2004  paper of mine, which includes a proof that LM is equivalent to the global form of the Kinna-Wagner selection principle in set theory.  On the other hand, one can formulate a strong negation of Leibniz’s principle by first adding a unary predicate I(x) to the usual language of set theory, and then augmenting ZF with a scheme that ensures that I(x) describes a proper class of indiscernibles, thus giving rise to an extension ZFI of ZF that I showed (2005) to be intimately related to Mahlo cardinals of finite order. In this talk I will give an expository account of the above and related results that attest to a lively interaction between set theory and Leibniz’s principle of identity of indiscernibles.

## 17 June 2020, 4 pm

Corey Bacal Switzer, City University of New York

## Some Set Theory of Kaufmann Models

Abstract.
A Kaufmann model is an $\omega_1$-like, recursively saturated, rather classless model of PA. Such models were shown to exist by Kaufmann under the assumption that $\diamondsuit$ holds, and in ZFC by Shelah via an absoluteness argument involving strong logics. They are important in the theory of models of arithmetic notably because they show that many classic results about countable, recursively saturated models of arithmetic cannot be extended to uncountable models. They are also a particularly interesting example of set theoretic incompactness at $\omega_1$, similar to an Aronszajn tree.

In this talk we’ll look at several set theoretic issues relating to this class of models motivated by the seemingly naïve question of whether or not such models can be killed by forcing without collapsing $\omega_1$. Surprisingly the answer to this question turns out to be independent: under $\mathsf{MA}_{\aleph_1}$ no $\omega_1$-preserving forcing can destroy Kaufmann-ness whereas under $\diamondsuit$ there is a Kaufmann model $M$ and a Souslin tree $S$ so that forcing with $S$ adds a satisfaction class to $M$ (thus killing rather classlessness). The techniques involved in these proofs also yield another surprising side of Kaufmann models: it is independent of ZFC whether the class of Kaufmann models can be axiomatized in the logic $L_{\omega_1, \omega}(Q)$ where $Q$ is the quantifier “there exists uncountably many”. This is the logic used in Shelah’s aforementioned result, hence the interest in this level of expressive power.

The seminar talks appear in the compilation of math seminars at https://mathseminars.org/seminar/oxford-set-theory.

# Philosophy of Mathematics, graduate lecture seminar, Oxford, Trinity term 2020

This will be a graduate-level lecture seminar on the Philosophy of Mathematics held during Trinity term 2020 here at the University of Oxford, co-taught by Dr. Wesley Wrigley and myself.

The broad theme for the seminar will be incompleteness, referring both to the incompleteness of our mathematical theories, as exhibited in Gödel’s incompleteness theorems, and also to the incompleteness of our mathematical domains, as exhibited in mathematical potentialism.

All sessions will be held online using the Zoom meeting platform. Please contact Professor Wrigley for access to the seminar (wesley.wrigley@philosophy.ox.ac.uk). The Zoom meetings will not be recorded or posted online.

The basic plan will be that the first four sessions, in weeks 1-4, will be led by Dr. Wrigley and concentrate on his current research on the incompleteness of mathematics and the philosophy of Kurt Gödel, while weeks 5-8 will be led by Professor Hamkins, who will concentrate on topics in potentialism.

Weeks 1 & 2  (28 April, 5 May)
Kurt Gödel “Some basic theorems on the foundations of mathematics and their implications (*1951)”,  in: Feferman, S. et al.  (eds) Kurt Gödel: Collected Works Volume III, pp.304-323. OUP (1995). And Wrigley “Gödel’s Disjunctive Argument”. (Also available on Canvas).

Week 3 (12th May)
Donald Martin, “Gödel’s Conceptual Realism”, Bulletin of Symbolic Logic 11:2 (2005), 207- 224 https://www.jstor.org/stable/1556750. And Wrigley “Conceptual Platonism.”

Week 4 (19th May)
Bertrand Russell “The Regressive Method of Discovering the Premises of Mathematics (1907)”, in: Moore , G. (ed) The Collected Papers of Bertrand Russell, Volume 5, pp.571-580. Routledge (2014). And Wrigley “Quasi-Scientific Methods of Justification in Set Theory.”

Week 5 (26th May)
Øystein Linnebo & Stewart Shapiro, “Actual and potential infinity”, Noûs 53:1 (2019), 160-191, https://doi.org/10.1111/nous.12208. And Øystein Linnebo. “Putnam on Mathematics as Modal Logic,” In: Hellman G., Cook R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96274-0_14

Week 6 (2nd June)
The topic this week is: tools for analyzing the modal logic of a potentialist system. This seminar will be based around the slides for my talk “Potentialism and implicit actualism in the foundations of mathematics,” given for the Jowett Society in Oxford last year. The slides are available at: http://jdh.hamkins.org/potentialism-and-implicit-actualism-in-the-foundations-of-mathematics-jowett-society-oxford-february-2019.  Interested readers may also wish to consult the more extensive slides for the three-lecture workshop I gave on potentialism at the Hejnice Winter School in 2018; the slides are available at http://jdh.hamkins.org/set-theoretic-potentialism-ws2018. My intent is to concentrate on the nature and significance of control statements, such as buttons, switches, ratchets and railyards, for determining the modal logic of a potentialist system.

Week 7 (9th June)
Joel David Hamkins and Øystein Linnebo. “The modal logic of set-theoretic potentialism and the potentialist maximality principles”. Review of Symbolic Logic (2019). https://doi.org/10.1017/S1755020318000242. arXiv:1708.01644. http://wp.me/p5M0LV-1zC. This week, we shall see how the control statements allow us to analyze precisely the modal logic of various conceptions of set-theoretic potentialism.

Week 8 (16th June)
Joel David Hamkins, “Arithmetic potentialism and the universal algorithm,” arxiv: 1801.04599, available at http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm. Please feel free to skip over the more technical parts of this paper. In the seminar discussion, we shall concentrate on the basic idea of arithmetic potentialism, including a full account of the universal algorithm and the significance of it for potentialism, as well as remarks of the final section of the paper.

# Philosophical Trials interview: Joel David Hamkins on Infinity, Gödel’s Theorems and Set Theory

I was interviewed by Theodor Nenu as the first installment of his Philosophical Trials interview series with philosophers, mathematicians and physicists.

Theodor provided the following outline of the conversation:

• 00:00 Podcast Introduction
• 00:50 MathOverflow and books in progress
• 04:08 Mathphobia
• 05:58 What is mathematics and what sets it apart?
• 08:06 Is mathematics invented or discovered (more at 54:28)
• 09:24 How is it the case that Mathematics can be applied so successfully to the physical world?
• 12:37 Infinity in Mathematics
• 16:58 Cantor’s Theorem: the real numbers cannot be enumerated
• 24:22 Russell’s Paradox and the Cumulative Hierarchy of Sets
• 29:20 Hilbert’s Program and Godel’s Results
• 35:05 The First Incompleteness Theorem, formal and informal proofs and the connection between mathematical truths and mathematical proofs
• 40:50 Computer Assisted Proofs and mathematical insight
• 44:11 Do automated proofs kill the artistic side of Mathematics?
• 48:50 Infinite Time Turing Machines can settle Goldbach’s Conjecture or the Riemann Hypothesis
• 54:28 Nonstandard models of arithmetic: different conceptions of the natural numbers
• 1:00:02 The Continuum Hypothesis and related undecidable questions, the Set-Theoretic Multiverse and the quest for new axioms
• 1:10:31 Minds and computers: Sir Roger Penrose’s argument concerning consciousness

# The real numbers are not interpretable in the complex field


What does it mean to interpret one mathematical structure in another? It means to provide a definable copy of the first structure in the second, by providing a definable domain of $k$-tuples (not necessarily just a domain of points) and definable interpretations of the atomic operations and relations, as well as a definable equivalence relation, a congruence with respect to the operations and relations, such that the first structure is isomorphic to the quotient of this definable structure by that equivalent relation. All these definitions should be expressible in the language of the host structure.

One may proceed recursively to translate any assertion in the language of the interpreted structure into the language of the host structure, thereby enabling a complete discussion of the first structure purely in the language of the second.

For an example, we can define a copy of the integer ring $\langle\mathbb{Z},+,\cdot\rangle$ inside the semi-ring of natural numbers $\langle\mathbb{N},+,\cdot\rangle$ by considering every integer as the equivalence class of a pair of natural numbers $(n,m)$ under the same-difference relation, by which $$(n,m)\equiv(u,v)\iff n-m=u-v\iff n+v=u+m.$$ Integer addition and multiplication can be defined on these pairs, well-defined with respect to same difference, and so we have interpreted the integers in the natural numbers.

Similarly, the rational field $\newcommand\Q{\mathbb{Q}}\Q$ can be interpreted in the integers as the quotient field, whose elements can be thought of as integer pairs $(p,q)$ written more conveniently as fractions $\frac pq$, where $q\neq 0$, considered under the same-ratio relation
$$\frac pq\equiv\frac rs\qquad\iff\qquad ps=rq.$$
The field structure is now easy to define on these pairs by the familiar fractional arithmetic, which is well-defined with respect to that equivalence. Thus, we have provided a definable copy of the rational numbers inside the integers, an interpretation of $\Q$ in $\newcommand\Z{\mathbb{Z}}\Z$.

The complex field $\C$ is of course interpretable in the real field $\R$ by considering the complex number $a+bi$ as represented by the real number pair $(a,b)$, and defining the operations on these pairs in a way that obeys the expected complex arithmetic. $$(a,b)+(c,d) =(a+c,b+d)$$ $$(a,b)\cdot(c,d)=(ac-bd,ad+bc)$$ Thus, we interpret the complex number field $\C$ inside the real field $\R$.

Question. What about an interpretation in the converse direction? Can we interpret $\R$ in $\C$?

Although of course the real numbers can be viewed as a subfield of the complex numbers $$\R\subset\C,$$this by itself doesn’t constitute an interpretation, unless the submodel is definable. And in fact, $\R$ is not a definable subset of $\C$. There is no purely field-theoretic property $\varphi(x)$, expressible in the language of fields, that holds in $\C$ of all and only the real numbers $x$. But more: not only is $\R$ not definable in $\C$ as a subfield, we cannot even define a copy of $\R$ in $\C$ in the language of fields. We cannot interpret $\R$ in $\C$ in the language of fields.

Theorem. As fields, the real numbers $\R$ are not interpretable in the complex numbers $\C$.

We can of course interpret the real numbers $\R$ in a structure slightly expanding $\C$ beyond its field structure. For example, if we consider not merely $\langle\C,+,\cdot\rangle$ but add the conjugation operation $\langle\C,+,\cdot,z\mapsto\bar z\rangle$, then we can identify the reals as the fixed-points of conjugation $z=\bar z$. Or if we add the real-part or imaginary-part operators, making the coordinate structure of the complex plane available, then we can of course define the real numbers in $\C$ as those complex numbers with no imaginary part. The point of the theorem is that in the pure language of fields, we cannot define the real subfield nor can we even define a copy of the real numbers in $\C$ as any kind of definable quotient structure.

The theorem is well-known to model theorists, a standard observation, and model theorists often like to prove it using some sophisticated methods, such as stability theory. The main issue from that point of view is that the order in the real numbers is definable from the real field structure, but the theory of algebraically closed fields is too stable to allow it to define an order like that.

But I would like to give a comparatively elementary proof of the theorem, which doesn’t require knowledge of stability theory. After a conversation this past weekend with Jonathan Pila, Boris Zilber and Alex Wilkie over lunch and coffee breaks at the Robin Gandy conference, here is such an elementary proof, based only on knowledge concerning the enormous number of automorphisms of $\C$, a consequence of the categoricity of the complex field, which itself follows from the fact that algebraically closed fields of a given characteristic are determined by their transcedence degree over their prime subfield. It follows that any two transcendental elements of $\C$ are automorphic images of one another, and indeed, for any element $z\in\C$ any two complex numbers transcendental over $\Q(z)$ are automorphic in $\C$ by an automorphism fixing $z$.

Proof of the theorem. Suppose that we could interpret the real field $\R$ inside the complex field $\C$. So we would define a domain of $k$-tuples $R\subseteq\C^k$ with an equivalence relation $\simeq$ on it, and operations of addition and multiplication on the equivalence classes, such that the real field was isomorphic to the resulting quotient structure $R/\simeq$. There is absolutely no requirement that this structure is a submodel of $\C$ in any sense, although that would of course be allowed if possible. The $+$ and $\times$ of the definable copy of $\R$ in $\C$ might be totally strange new operations defined on those equivalence classes. The definitions altogether may involve finitely many parameters $\vec p=(p_1,\ldots,p_n)$, which we now fix.

As we mentioned, the complex number field $\C$ has an enormous number of automorphisms, and indeed, any two $k$-tuples $\vec x$ and $\vec y$ that exhibit the same algebraic equations over $\Q(\vec p)$ will be automorphic by an automorphism fixing $\vec p$. In particular, this means that there are only countably many isomorphism orbits of the $k$-tuples of $\C$. Since there are uncountably many real numbers, this means that there must be two $\simeq$-inequivalent $k$-tuples in the domain $R$ that are automorphic images in $\C$, by an automorphism $\pi:\C\to\C$ fixing the parameters $\vec p$. Since $\pi$ fixes the parameters of the definition, it will take $R$ to $R$ and it will respect the equivalence relation and the definition of the addition and multiplication on $R/\simeq$. Therefore, $\pi$ will induce an automorphism of the real field $\R$, which will be nontrivial precisely because $\pi$ took an element of one $\simeq$-equivalence class to another.

The proof is now completed by the observation that the real field $\langle\R,+,\cdot\rangle$ is rigid; it has no nontrivial automorphisms. This is because the order is definable (the positive numbers are precisely the nonzero squares) and the individual rational numbers must be fixed by any automorphism and then every real number is determined by its cut in the rationals. So there can be no nontrivial automorphism of $\R$, and we have a contradiction. So $\R$ is not interpretable in $\C$. $\Box$

# Bi-interpretation of weak set theories, Oberwolfach, April 2020

This will be a talk for the workshop in Set Theory at the Mathematisches Forschungsinstitute Oberwolfach, April 5-11, 2020.

Note: the conference has been cancelled due to concerns over the Coronavirus-19. (Meanwhile, I have given the talk for the Oxford Set Theory Seminar — see below.)

Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

Since the Oberwolfach meeting had been canceled, I gave the talk for the Oxford Set Theory Seminar on 20 May 2020.

• A. R. Freire and J. D. Hamkins, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020. (Under review)
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# Bi-interpretation in set theory, Bristol, February 2020

This will be a talk for the Logic and Set Theory seminar at the University of Bristol, on 25 February, 2020.

Abstract: In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

Bi-interpretation in weak set theories

# Bi-interpretation in weak set theories

• A. R. Freire and J. D. Hamkins, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020. (Under review)
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Abstract. In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory $\newcommand\ZFCm{\text{ZFC}^-}\ZFCm$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic — even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable — and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.

Set theory exhibits a robust mutual interpretability phenomenon: in a given model of set theory, we can define diverse other interpreted models of set theory. In any model of Zermelo-Fraenkel ZF set theory, for example, we can define an interpreted model of ZFC + GCH, via the constructible universe, as well as definable interpreted models of ZF + ¬AC, of ZFC + MA + ¬CH, of ZFC + $\mathfrak{b}<\mathfrak{d}$, and so on for hundreds of other theories. For these latter theories, set theorists often use forcing to construct outer models of the given model; but nevertheless the Boolean ultrapower method provides definable interpreted models of these theories inside the original model (explained in theorem 7). Similarly, in models of ZFC with large cardinals, one can define fine-structural canonical inner models with large cardinals and models of ZF satisfying various determinacy principles, and vice versa. In this way, set theory exhibits an abundance of natural mutually interpretable theories.

Do these instances of mutual interpretation fulfill the more vigourous conception of bi-interpretation? Two models or theories are mutually interpretable, when merely each is interpreted in the other, whereas bi-interpretation requires that the interpretations are invertible in a sense after iteration, so that if one should interpret one model or theory in the other and then re-interpret the first theory inside that, then the resulting model should be definably isomorphic to the original universe (precise definitions in sections 2 and 3). The interpretations mentioned above are not bi-interpretations, for if we start in a model of ZFC+¬CH and then go to L in order to interpret a model of ZFC+GCH, then we’ve already discarded too much set-theoretic information to expect that we could get a copy of our original model back by interpreting inside L. This problem is inherent, in light of the following theorem of Ali Enayat, showing that indeed there is no nontrivial bi-interpretation phenomenon to be found amongst the set-theoretic models and theories satisfying ZF. In interpretation, one must inevitably discard set-theoretic information.

Theorem. (Enayat 2016)

1. ZF is solid: no two models of ZF are bi-interpretable.
2. ZF is tight: no two distinct theories extending ZF are bi-interpretable.

The proofs of these theorems, provided in section 6, seem to use the full strength of ZF, and Enayat had consequently inquired whether the solidity/tightness phenomenon somehow required the strength of ZF set theory. In this paper, we shall find support for that conjecture by establishing nontrivial instances of bi-interpretation in various natural weak set theories, including Zermelo-Fraenkel theory $\ZFCm$, without the power set axiom, and Zermelo set theory Z, without the replacement axiom.

Main Theorems

1. $\ZFCm$ is not solid: there are well-founded models of $\ZFCm$ that are bi-interpretable, but not isomorphic.
2. Indeed, it is relatively consistent with ZFC that $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ are bi-interpretable.
3. $\ZFCm$ is not tight: there are distinct bi-interpretable extensions of $\ZFCm$.
4. Z is not solid: there are well-founded models of Z that are bi-interpretable, but not isomorphic.
5. Indeed, every model of ZF is bi-interpretable with a transitive inner model of Z in which the replacement axiom fails.
6. Z is not tight: there are distinct bi-interpretable extensions of Z.

These claims are made and proved in theorems 20, 17, 21 and 22. We shall in addition prove the following theorems on this theme:

7. Well-founded models of ZF set theory are never mutually interpretable.
8. The Väänänen internal categoricity theorem does not hold for $\ZFCm$, not even for well-founded models.

These are theorems 14 and 16. Statement (8) concerns the existence of a model $\langle M,\in,\bar\in\rangle$ satisfying $\ZFCm(\in,\bar\in)$, meaning $\ZFCm$ in the common language with both predicates, using either $\in$ or $\bar\in$ as the membership relation, such that $\langle M,\in\rangle$ and $\langle M,\bar\in\rangle$ are not isomorphic.

• A. R. Freire and J. D. Hamkins, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020. (Under review)
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# Philosophy meets maths, Oxford, January 2020

This will be a fun talk for the Philosophy Plus Science Taster Day, a fun day of events for prospective students in the joint philosophy degrees, whether Mathematics & Philosophy, Physics & Philosophy or Computer Science & Philosophy. The talk will be Friday 10th January in the Andrew Wiles building.

Abstract. In this talk, we shall pose and solve various fun puzzles in epistemic logic, which is to say, puzzles involving reasoning about knowledge, including one’s own knowledge or the knowledge of other people, including especially knowledge of knowledge or knowledge of the lack of knowledge. We’ll discuss several classic puzzles of common knowledge, such as the two-generals problem, Cheryl’s birthday problem, and the blue-eyed islanders, as well as several new puzzles. Please come and enjoy!