Oxford Set Theory Seminar

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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.

Trinity Term 2021

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

For the Zoom access code, contact Samuel Adam-Day at me@samadamday.com.

19 May 2021 4:30 pm UK

Samuel Adam-Day

University of Oxford

The continuous gradability of the cut-point orders of $\newcommand\R{\mathbb{R}}\R$-trees

Abstract. An $\R$-tree is a metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying $\R$-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part of the talk, I answer this question in the negative: there is a branchwise-real tree order which is not continuously gradable. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every embedded well-stratified (i.e. set-theoretic) tree is $\R$-gradable. This tighter link with set theory is put to work in the third part answering a number of refinements of the main question, yielding several independence results.

26 May 2021 4:30 pm BST

Sandra Müller

TU Wien

The strength of determinacy when all sets are universally Baire

Abstract. The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin’s derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan’s conjecture.

1 June 2021 4:30 pm BST

Christopher Turner

Bristol

Forcing Axioms and Name Principles

Abstract. Forcing axioms are a well-known way of expressing the concept ”there are filters in V which are close to being generic”. Name principles are another expression of this concept. A name principle says: ”Let $\sigma$ be any sufficiently nice name which is forced to have some property. Then there is a filter $g\in V$ such that $\sigma^g$ has that property.” Name principles have often been used on an ad-hoc basis in proofs, but have not been studied much as axioms in their own right. In this talk, I will present some of the connections between different name principles, and between name principles and forcing axioms. This is based on joint work with Philipp Schlicht.

16 June 2021 4:30 pm BST

Joan Bagaria

Barcelona

Some recent results on Structural Reflection

Abstract. The general Structural Reflection (SR) principle asserts that for every definable, in the first-order language of set theory, possibly with parameters, class $\mathcal{C}$ of relational structures of the same type there exists an ordinal $\alpha$ that reflects $\mathcal{C}$, i.e.,  for every $A$ in $\mathcal{C}$ there exists $B$ in $\mathcal{C}\cap V_\alpha$ and an elementary embedding from $B$ into $A$. In this form, SR is equivalent to Vopenka’s Principle (VP). In my talk I will present some different natural variants of SR which are equivalent to the existence some well-known large cardinals weaker than VP. I will also consider some forms of SR, reminiscent of Chang’s Conjecture, which imply the existence of large cardinal principles stronger than VP, at the level of rank-into-rank embeddings and beyond. The latter is a joint work with Philipp Lücke. 

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.

For the Zoom access code, contact Samuel Adam-Day at me@samadamday.com.

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

20 January 2021 4pm UK

Dima Sinapova

University of Illinois at Chicago

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.

3 February 2021 4pm UK

Spencer Unger

University of Toronto

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.

24 March 2021 4pm UK

Andrew Marks

UCLA

The decomposability conjecture

Abstract. We characterize which Borel functions are decomposable into a countable union of functions which are piecewise continuous on $\Pi^0_n$ domains, assuming projective determinacy. One ingredient of our proof is a new characterization of what Borel sets are $\Sigma^0_n$ complete. Another important ingredient is a theorem of Harrington that there is no projective sequence of length $\omega_1$ of distinct Borel sets of  bounded rank, assuming projective determinacy. This is joint work with Adam Day.

10 March 2021 4pm UK

Peter Koellner

Harvard University

Minimal Models and $\beta$ Categoricity

Abstract. Let us say that a theory $T$ in the language of set theory is $\beta$-consistent at $\alpha$ if there is a transitive model of $T$ of height $\alpha$, and let us say that it is $\beta$-categorical at $\alpha$ iff there is at most one transitive model of $T$ of height $\alpha$. Let us also assume, for ease of formulation, that there are arbitrarily large $\alpha$ such that $\newcommand\ZFC{\text{ZFC}}\ZFC$ is $\beta$-consistent at $\alpha$.

The sentence $\newcommand\VEL{V=L}\VEL$ has the feature that $\ZFC+\VEL$ is $\beta$-categorical at $\alpha$, for every $\alpha$. If we assume in addition that $\ZFC+\VEL$ is $\beta$-consistent at $\alpha$, then the uniquely determined model is $L_\alpha$, and the minimal such model, $L_{\alpha_0}$, is model of determined by the $\beta$-categorical theory $\ZFC+\VEL+M$, where $M$ is the statement “There does not exist a transitive model of $\ZFC$.”

It is natural to ask whether $\VEL$ is the only sentence that can be $\beta$-categorical at $\alpha$; that is, whether, there can be a sentence $\phi$ such that $\ZFC+\phi$ is $\beta$-categorical at $\alpha$, $\beta$-consistent at $\alpha$, and where the unique model is not $L_\alpha$.  In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal $\alpha$, let $n(\alpha)$ be the next admissible ordinal above $\alpha$, and, for the purposes of this discussion, let us say that an ordinal $\alpha$ is minimal iff a bounded subset of $\alpha$ appears in $L_{n(\alpha)}\setminus L_\alpha$. [Note that $\alpha_0$ is minimal (indeed a new subset of $\omega$ appears as soon as possible, namely, in a $\Sigma_1$-definable manner over $L_{\alpha_0+1}$) and an ordinal $\alpha$ is non-minimal iff $L_{n(\alpha)}$ satisfies that $\alpha$ is a cardinal.] Friedman showed that for all $\alpha$ which are non-minimal, $\VEL$ is the only sentence that is $\beta$-categorical at $\alpha$. The question of whether this is also true for $\alpha$ which are minimal has remained open.

In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a “lightface” question (such as the one under consideration) it is easier to first address the “boldface” analogue of the question by shifting from the context of $L$ to the context of $L[x]$, where $x$ is a real. In this new setting everything is relativized to the real $x$: For an ordinal $\alpha$, we let $n_x(\alpha)$ be the first $x$-admissible ordinal above $\alpha$, and we say that $\alpha$ is $x$-minimal iff a bounded subset of $\alpha$ appears in $L_{n_x(\alpha)}[x]\setminus L_{\alpha}[x]$.

Theorem. Assume $M_1^\#$ exists and is fully iterable. There
  is a sentence $\phi$ in the language of set theory with two
  additional constants, c and d, such that for a Turing cone
  of $x$, interpreting c by $x$, for all $\alpha$

  1. if $L_\alpha[x]\models\ZFC$ then there is an interpretation of d by something in $L_\alpha[x]$ such that there is a $\beta$-model of $\ZFC+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$, and
  2. if, in addition, $\alpha$ is $x$-minimal, then there is a unique $\beta$-model of $\ZFC+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$.

The sentence $\phi$ asserts the existence of an object which is external to $L_\alpha[x]$ and which, in the case where $\alpha$ is minimal, is canonical. The object is a branch $b$ through a certain tree in $L_\alpha[x]$, and the construction uses techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.

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.

21 October 2020 4 pm UK

Andreas Blass

University of Michigan

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.

4 November 4 pm UK

Mirna Džamonja

Institut for History and Philosophy of Sciences and Techniques, CNRS & Université Panthéon Sorbonne, Paris and Institute of Mathematics, Czech Academy of Sciences, Prague

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.

18 November 4 pm UK

Gabriel Goldberg

Harvard University

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 hierarchy. Or 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.

 

2 December 4 pm UK

Kameryn J Williams

University of Hawai’i at Mānoa

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.

For the Zoom access code, contact Samuel Adam-Day me@samadamday.com.

 

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.

Some-Set-Theory-of-Kaufmann-Models-SLIDESDownload

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

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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. 

ProspectusTT2020-Hamkins-WrigleyDownload

Readings as detailed below:

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-1-Reading-Disjunctive-ArgumentDownload

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

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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

Apple podcast | Google podcast | Spotify podcast

The real numbers are not interpretable in the complex field

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Consider the real numbers $\newcommand\R{\mathbb{R}}\R$ and the complex numbers $\newcommand\C{\mathbb{C}}\C$ and the question of whether these structures are interpretable in one another as fields.

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

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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.

Slides – Bi-interpretation-in-set-theory-Oberwolfach-2020Download

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

Bi-interpretation in weak set theories

  • [bibtex key=”FreireHamkins:Bi-interpretation-in-weak-set-theories”]

Bi-interpretation in set theory, Bristol, February 2020

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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-set-theory-Bristol-2020Download

Bi-interpretation in weak set theories

[bibtex key=”HamkinsFreire:Bi-interpretation-in-weak-set-theories”]

Bi-interpretation in weak set theories

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[bibtex key=”FreireHamkins:Bi-interpretation-in-weak-set-theories”]

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.

Read more in the full article: [bibtex key=”FreireHamkins:Bi-interpretation-in-weak-set-theories”]

Philosophy meets maths, Oxford, January 2020

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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!

Philosophy-meets-maths-Oxford-PhilosophyScience-Taster-Day-Jan-2020Download

Modal model theory, STUK 4, Oxford, December 2019

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This will be my talk for the Set Theory in the United Kingdom 4, a conference to be held in Oxford on 14 December 2019. I am organizing the conference with Sam Adam-Day. 

Modal model theory

Abstract. I shall introduce the subject of modal model theory, a research effort bringing modal concepts and vocabulary into model theory. For any first-order theory T, we may naturally consider the models of T as a Kripke model under the submodel relation, and thereby naturally expand the language of T to include the modal operators. In the class of all graphs, for example, a statement is possible in a graph, if it is true in some larger graph, having that graph as an induced subgraph, and a statement is necessary when it is true in all such larger graphs. The modal expansion of the language is quite powerful: in graphs it can express k-colorability and even finiteness and countability. The main idea applies to any collection of models with an extension concept. The principal questions are: what are the modal validities exhibited by the class of models or by individual models? For example, a countable graph validates S5 for graph theoretic assertions with parameters, for example, just in case it is the countable random graph; and without parameters, just in case it is universal for all finite graphs. Similar results apply with digraphs, groups, fields and orders. This is joint work with Wojciech Wołoszyn.

Hand-written lecture notes

The $\Sigma_1$-definable universal finite sequence

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[bibtex key=”HamkinsWilliams2021:The-universal-finite-sequence”]

Abstract. We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.


I know that you know that I know that you know…. Oxford, October 2019

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This will be a fun start-of-term Philosophy Undergraduate Welcome Lecture for philosophy students at Oxford in the Mathematics & Philosophy, Physics & Philosophy, Computer Science & Philosophy, and Philosophy & Linguistics degrees. New students are especially encouraged, but everyone is welcome! The talk is open to all. The talk will be Wednesday 16th October, 5-6 pm in the Mathematical Institute, with wine and nibbles afterwards.

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!

Philosophy Faculty announcement of talk

I-know-you-know-I-know-you-know-Oxford-October-2019.pdfDownload Slides

Lectures on the philosophy of mathematics, Oxford, Michaelmas term 2019

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This will be a series of self-contained lectures on the philosophy of mathematics, given at Oxford University in Michaelmas term 2019. We will be meeting in the Radcliffe Humanities Lecture Room at the Faculty of Philosophy every Friday 12-1 during term.

All interested parties are welcome. The lectures are intended principally for students preparing for philosophy exam paper 122 at the University of Oxford.

Euclid detail from The School of Athens painting by Raphael

The lectures will be organized loosely around mathematical themes, in such a way I hope that brings various philosophical issues naturally to light. The lectures will be based on my new book, forthcoming with MIT Press.

There are tentative plans to make the lectures available by video. I shall post further details concerning this later.

Lecture 1. Numbers. Numbers are perhaps the essential mathematical idea, but what are numbers? We have many kinds of numbers—natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more—and these number systems provide a fruitful background for classical arguments on incommensurability and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and structuralism.

Lecture 2. Rigour. Let us consider the problem of mathematical rigour in the development of the calculus. Informal continuity concepts and the use of infinitesimals ultimately gave way to formal epsilon-delta limit concepts, which provided a capacity for refined notions, such as uniform continuity, equicontinuity and uniform convergence. Nonstandard analysis resurrected the infinitesimal concept on a more secure foundation, providing a parallel development of the subject, which can be understood from various sweeping perspectives. Meanwhile, increasing abstraction emerged in the function concept, which we shall illustrate with the Devil’s staircase, space-filling curves and the Conway base 13 function. Whether the indispensibility of mathematics for science grounds mathematical truth is put in question on the view known as fictionalism.

Lecture 3. Infinity. We shall follow the allegory of Hilbert’s hotel and the paradox of Galileo to the equinumerosity relation and the notion of countability. Cantor’s diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity; some arguments give rise to the distinction between constructive and non-constructive proof. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Time permitting, we shall count into the transfinite ordinals.

Lecture 4. Geometry. Classical Euclidean geometry, accompanied by its ideal of straightedge and compass construction and the Euclidean concept of proof, is an ageless paragon of deductive mathematical reasoning. Yet, the impossibility of certain constructions, such as doubling the cube, trisecting the angle or squaring the circle, hints at geometric realms beyond Euclid, and leads one to the concept of constructible and non-constructible numbers. The rise of non-Euclidean geometry, especially in light of scientific observations and theories suggesting that physical reality may not be Euclidean, challenges previous accounts of what geometry is about and changes our understanding of the nature of geometric and indeed mathematical ontology. New formalizations, such as those of Hilbert and Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure hints at the tantalizing possibility of automation in our geometrical reasoning.

Lecture 5. Proof. What is proof? What is the relation between proof and truth? Is every mathematical truth, true for a reason? After clarifying the distinction between syntax and semantics, we shall discuss new views on the dialogical nature of proof. With formal proof systems, we shall highlight the importance of soundness, completeness and verifiability in any such system, outlining the central ideas used in proving the completeness theorem. The compactness theorem distills the finiteness of proofs into an independent purely semantic consequence. Computer-verified proof promises increasing significance; it’s role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakenings of the logical rules.

Lecture 6. Computability. What is computability? Gödel defined the primitive recursive functions, a robust class of computable functions, yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Turing’s machine concept, growing out of a careful philosophical analysis of computability, laid a foundation for the contemporary computer era; the widely accepted Church-Turing thesis asserts that Turing has the right notion. The distinction between computable decidability and computable enumerability, highlighted by the undecidability of the halting problem, shows that not all mathematical problems can be solved by machine, and a vast hierarchy looms in the Turing degrees, an infinitary information theory. Complexity theory refocuses the subject on the realm of feasible computation, with the still-unsolved P vs. NP problem standing in the background of nearly every serious issue in theoretical computer science.

Lecture 7. Incompleteness. The Hilbert program, seeking to secure the consistency of higher mathematics by finitary reasoning about the formal system underlying it, was dashed by Gödel’s incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, via self-reference, and via definability. After this, we’ll discuss the second incompleteness theorem, the Rosser variation, and Tarski on the non-definability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength underlying all mathematical theories.

Lecture 8. Set theory. We shall discuss the emergence of set theory as a foundation of mathematics. Cantor founded the subject with key set-theoretic insights, but Frege’s formal theory was naive, refuted by the Russell paradox. Zermelo’s set theory, in contrast, grew ultimately into the successful contemporary theory, founded upon the cumulative conception. Set theory was simultaneously a new mathematical subject, with its own motivating questions and tools, but also a new foundational theory, with a capacity to represent essentially arbitrary abstract mathematical structure. Sophisticated technical developments, including especially the forcing method and discoveries in the large cardinal hierarchy, led to a necessary engagement with deep philosophical concerns, such as the criteria by which one adopts new mathematical axioms and set-theoretic pluralism.

Introducing modal model theory

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Let me introduce to you the topic of modal model theory, injecting some ideas from modal logic into the traditional subject of model theory in mathematical logic.

For example, we may consider the class of all models of some first-order theory, such as the class of all graphs, or the class of all groups, or all fields or what have you. In general, we have $\newcommand\Mod{\text{Mod}}\Mod(T)$, where $T$ is a first-order theory in some language $L$.

We may consider $\Mod(T)$ as a potentialist system, a Kripke model of possible worlds, where each model accesses the larger models, of which it is a submodel. So $\newcommand\possible{\Diamond}\possible\varphi$ is true at a model $M$, if there is a larger model $N$ in which $\varphi$ holds, and $\newcommand\necessary{\Box}\necessary\varphi$ is true at $M$, if $\varphi$ holds in all larger models.

In this way, we enlarge the language $L$ to include these modal operators. Let $\possible(L)$ be the language obtained by closing $L$ under the modal operators and Boolean connectives; and let $L^\possible$ also close under quantification. The difference is whether a modal operator falls under the scope of a quantifier.

Recently, in a collaborative project with Wojciech Aleksander Wołoszyn, we made some progress, which I’d like to explain. (We also have many further results, concerning the potentialist validities of various natural instances of $\Mod(T)$, but those will wait for another post.)

Theorem. If models $M$ and $N$ are elementarily equivalent, that is, if they have the same theory in the language of $L$, then they also have the same theory in the modal language $\possible(L)$.

Proof. We show that whenever $M\equiv N$ in the language of $L$, then $M\models\varphi\iff N\models\varphi$ for sentences $\varphi$ in the modal language $\possible(L)$, by induction on $\varphi$.

Of course, by assumption the statement is true for sentences $\varphi$ in the base language $L$. And the property is clearly preserved by Boolean combinations. What remains is the modal case. Suppose that $M\equiv N$ and $M\models\possible\varphi$. So there is some extension model $M\subset W\models\varphi$.

Since $M\equiv N$, it follows by the Keisler-Shelah theorem that $M$ and $N$ have isomorphic ultrapowers $\prod_\mu M\cong\prod_\mu N$, for some ultrafilter $\mu$. It is easy to see that isomorphic structures satisfy exactly the same modal assertions in the class of all models of a theory. Since $M\subset W$, it follows that the ultrapower of $M$ is extended to (a copy of) the ultrapower of $W$, and so $\prod_\mu M\models\possible\varphi$, and therefore also $\prod_\mu N\models\possible\varphi$. From this, since $N$ embeds into its ultrapower $\prod_\mu N$, it follows also that $N\models\possible\varphi$, as desired. $\Box$

Corollary. If one model elementarily embeds into another $M\prec N$, in the language $L$ of these structures, then this embedding is also elementary in the language $\possible(L)$.

Proof. To say $M\prec N$ in language $L$ is the same as saying that $M\equiv N$ in the language $L_M$, where we have added constants for every element of $M$, and interpreted these constants in $N$ via the embedding. Thus, by the theorem, it follows that $M\equiv N$ in the language $\possible(L_M)$, as desired. $\Box$

For example, every model $M$ is elementarily embedding into its ultrapowers $\prod_\mu M$, in the language $\possible(L)$.

We’d like to point out next that these results do not extend to elementary equivalence in the full modal language $L^\possible$.

For a counterexample, let’s work in the class of all simple graphs, in the language with a binary predicate for the edge relation. (We’ll have no parallel edges, and no self-edges.) So the accessibility relation here is the induced subgraph relation.

Lemma. The 2-colorability of a graph is expressible in $\possible(L)$. Similarly for $k$-colorability for any finite $k$.

Proof. A graph is 2-colorable if we can partition its vertices into two sets, such that a vertex is in one set if and only if all its neighbors are in the other set. This can be effectively coded by adding two new vertices, call them red and blue, such that every node (other than red and blue) is connected to exactly one of these two points, and a vertex is connected to red if and only if all its neighbors are connected to blue, and vice versa. If the graph is $2$-colorable, then there is an extension realizing this statement, and if there is an extension realizing the statement, then (even if more than two points were added) the original graph must be $2$-colorable. $\Box$

A slightly more refined observation is that for any vertex $x$ in a graph, we can express the assertion, “the component of $x$ is $2$-colorable” by a formula in the language $\possible(L)$. We simply make the same kind of assertion, but drop the requirement that every node gets a color, and insist only that $x$ gets a color and the coloring extends from a node to any neighbor of the node, thereby ensuring the full connected component will be colored.

Theorem. There are two graphs that are elementary equivalent in the language $L$ of graph theory, and hence also in the language $\possible(L)$, but they are not elementarily equivalent in the full modal language $L^\possible$.

Proof. Let $M$ be a graph consisting of disjoint copies of a 3-cycle, a 5-cycle, a 7-cycle, and so on, with one copy of every odd-length cycle. Let $M^*$ be an ultrapower of $M$ by a nonprincipal ultrafilter.

Thus, $M^*$ will continue to have one 3-cycle, one 5-cycle, one 7-cycle and on on, for all the finite odd-length cycles, but then $M^*$ will have what it thinks are non-standard odd-length cycles, except that it cannot formulate the concept of “odd”. What it actually has are a bunch of $\mathbb{Z}$-chains.

In particular, $M^*$ thinks that there is an $x$ whose component is $2$-colorable, since a $\mathbb{Z}$-chain is $2$-colorable.

But $M$ does not think that there is an $x$ whose component is $2$-colorable, because an odd-length finite cycle is not $2$-colorable. $\Box$.

Since we used an ultrapower, the same example also shows that the corollary above does not generalize to the full modal language. That is, we have $M$ embedding elementarily into its ultrapower $M^*$, but it is not elementary in the language $L^\possible$.

Let us finally notice that the Łoś theorem for ultraproducts fails even in the weaker modal language $\possible(L)$.

Theorem. There are models $M_i$ for $i\in\mathbb{N}$ and a sentence $\varphi$ in the language of these models, such that every nonprincipal ultraproduct $\prod_\mu M_i$ satisfies $\possible\varphi$, but no $M_i$ satisfies $\possible\varphi$. .

Proof. In the class of all graphs, using the language of graph theory, let the $M_i$ be all the odd-length cycles. The ultraproduct $\prod_\mu M_i$ consists entirely of $\mathbb{Z}$-chains. In particular, the ultraproduct graph is $2$-colorable, but none of the $M_i$ are $2$-colorable. $\Box$

Alan Turing’s theory of computation, Oxford and Cambridge Club, June 2019

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I shall speak for the Oxford and Cambridge Club, in a joint event hosted by Maths and Science Group and the Military History Group, an evening (6 June 2019) with dinner and talks on the theme of the Enigma and Code breaking. 

Oxford and Cambridge Club

Abstract: I shall describe Alan Turing’s transformative philosophical analysis of the nature of computation, including his argument that some mathematical questions must inevitably remain beyond our computational capacity to answer.

The talk will highlight ideas from Alan Turing’s phenomenal 1936 paper on computable numbers:

Turing’s theory of computation – Oxford Cambridge Club 2019.pdfDownload

Computational self-reference and the universal algorithm, Queen Mary University of London, June 2019

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This will be a talk for the Theory Seminar for the theory research group in Theoretical Computer Science at Queen Mary University of London. The talk will be held 4 June 2019 1:00 pm, ITL first floor.

Abstract. Curious, often paradoxical instances of self-reference inhabit deep parts of computability theory, from the intriguing Quine programs and Ouroboros programs to more profound features of the Gödel phenomenon. In this talk, I shall give an elementary account of the universal algorithm, showing how the capacity for self-reference in arithmetic gives rise to a Turing machine program $e$, which provably enumerates a finite set of numbers, but which can in principle enumerate any finite set of numbers, when it is run in a suitable model of arithmetic. In this sense, every function becomes computable, computed all by the same universal program, if only it is run in the right world. Furthermore, the universal algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. An analogous result holds in set theory, where Woodin and I have provided a universal locally definable finite set, which can in principle be any finite set, in the right universe, and which can furthermore be successively extended to become any desired finite superset of that set in a suitable top-extension of that universe.

Computational self-reference and the universal algorithm QMUL 2019.pdfDownload