# Workshop on the Set-theoretic Multiverse, Konstanz, September 2022

Masterclass of “The set-theoretic multiverse” ten years after

Focused on mathematical and philosophical aspects of the set-theoretic multiverse and the pluralist debate in the philosophy of set theory, this workshop will have a master class on potentialism, a series of several speakers, and a panel discussion. To be held 21-22 September 2022 at the University of Konstanz, Germany. (Contact organizers for Zoom access.)

I shall make several contributions to the meeting.

### Master class tutorial on potentialism

I shall give a master class tutorial on potentialism, an introduction to the general theory of potentialism that has been emerging in recent work, often developed as a part of research on set-theoretic pluralism, but just as often branching out to a broader application. Although the debate between potentialism and actualism in the philosophy of mathematics goes back to Aristotle, recent work divorces the potentialist idea from its connection with infinity and undertakes a more general analysis of possible mathematical universes of any kind. Any collection of mathematical structures forms a potentialist system when equipped with an accessibility relation (refining the submodel relation), and one can define the modal operators of possibility $\Diamond\varphi$, true at a world when $\varphi$ is true in some larger world, and necessity $\Box\varphi$, true in a world when $\varphi$ is true in all larger worlds. The project is to understand the structures more deeply by understanding their modal nature in the context of a potentialist system. The rise of modal model theory investigates very general instances of potentialist system, for sets, graphs, fields, and so on. Potentialism for the models of arithmetic often connects with deeply philosophical ideas on ultrafinitism. And the spectrum of potentialist systems for the models of set theory reveals fundamentally different conceptions of set-theoretic pluralism and possibility.

### The multiverse view on the axiom of constructibility

I shall give a talk on the multiverse perspective on the axiom of constructibility. Set theorists often look down upon the axiom of constructibility V=L as limiting, in light of the fact that all the stronger large cardinals are inconsistent with this axiom, and furthermore the axiom expresses a minimizing property, since $L$ is the smallest model of ZFC with its ordinals. Such views, I argue, stem from a conception of the ordinals as absolutely completed. A potentialist conception of the set-theoretic universe reveals a sense in which every set-theoretic universe might be extended (in part upward) to a model of V=L. In light of such a perspective, the limiting nature of the axiom of constructibility tends to fall away.

### Panel discussion: The multiverse view—challenges for the next ten years

This will be a panel discussion on the set-theoretic multiverse, with panelists including myself, Carolin Antos-Kuby, Giorgio Venturi, and perhaps others.

# Pointwise definable end-extensions of the universe, Sophia 2022, Salzburg

This will be an online talk for the Salzburg Conference for Young Analytical Philosophy, the SOPhiA 2022 Salzburgiense Concilium Omnibus Philosophis Analyticis, with a special workshop session Reflecting on ten years of the set-theoretic multiverse. The workshop will meet Thursday 8 September 2022 4:00pm – 7:30pm.

The name of the workshop (“Reflecting on ten years…”), I was amazed to learn, refers to the period since my 2012 paper, The set-theoretic multiverse, in the Review of Symbolic Logic, in which I had first introduced my arguments and views concerning set-theoretic pluralism. I am deeply honored by this workshop highlighting my work in this way and focussing on the developments growing out of it.

In this talk, I shall engage in that discussion by presenting some very new work connecting several topics that have been prominent in discussions of the set-theoretic multiverse, namely, set-theoretic potentialism and pointwise definability.

Abstract. Using the universal algorithm and its generalizations, I shall present new work on the possibility of end-extending any given countable model of arithmetic or set theory to a pointwise definable model, one in which every object is definable without parameters. Every countable model of Peano arithmetic, for example, admits an end-extension to a pointwise definable model. And similarly, every countable model of ZF set theory admits an end-extension to a pointwise definable model of ZFC+V=L, as well as to pointwise definable models of other sufficient theories, accommodating large cardinals. I shall discuss the philosophical significance of these results in the philosophy of set theory with a view to potentialism and the set-theoretic multiverse.

# Nonlinearity and illfoundedness in the hierarchy of consistency strength and the question of naturality, Italy (AILA), September 2022

This will be a talk for the meeting of The Italian Association for Logic and its Applications (AILA) in Caserta, Italy 12-15 September 2022.

Abstract. Set theorists and philosophers of mathematics often point to a mystery in the foundations of mathematics, namely, that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Why should it be? The phenomenon is thought to carry profound significance for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories, the “one road upward.” And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features, however, are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to rebut that criticism by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.

# Set theory inside out: realizing every inner model theory in an end extension, European Set Theory Conference, September 2022

This will be a talk for the European Set Theory Conference 2022 in Turin, Italy 29 August – 2 September 2022.

Abstract. Every countable model of ZFC set theory with an inner model satisfying a sufficient theory must also have an end-extension satisfying that theory. For example, every countable model with a measurable cardinal has an end-extension to a model of $V=L[\mu]$; every model with extender-based large cardinals has an end-extension to a model of $V=L[\vec E]$; every model with infinitely many Woodin cardinals and a measurable above has an end-extension to a model of $\text{ZF}+\text{DC}+V=L(\mathbb{R})+\text{AD}$. These results generalize the famous Barwise extension theorem, of course, asserting that every countable model of ZF set theory admits an end-extension to a model of $\text{ZFC}+{V=L}$, a theorem which was simultaneously a technical culmination of Barwise’s pioneering methods in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the Barwise theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory, while also providing the generalization I mentioned. This proof furthermore leads directly to the universal finite sequence, a $\Sigma_1$-definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe, a result holding important consequences for the nature of set-theoretic potentialism.  This work is joint with Kameryn J. Williams.

• J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” Journal of Symbolic Logic, 2021.
[Bibtex]
@ARTICLE{HamkinsWilliams2021:The-universal-finite-sequence,
author = {Joel David Hamkins and Kameryn J. Williams},
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# Potentialism and implicit actualism in the foundations of mathematics, Notre Dame, March 2021

This will be a talk for the Department Colloquium of the Philosophy Department of the University of Notre Dame, 26 March 12 pm EST (4pm GMT).

Abstract: Potentialism is the view, originating in the classical dispute between actual and potential infinity, that one’s mathematical universe is never fully completed, but rather unfolds gradually as new parts of it increasingly come into existence or become accessible or known to us. Recent work emphasizes the modal aspect of potentialism, while decoupling it from arithmetic and from infinity: the essence of potentialism is about approximating a larger universe by means of universe fragments, an idea that applies to set-theoretic as well as arithmetic foundations. The modal language and perspective allows one precisely to distinguish various natural potentialist conceptions in the foundations of mathematics, whose exact modal validities are now known. Ultimately, this analysis suggests a refocusing of potentialism on the issue of convergent inevitability in comparison with radical branching. I shall defend the theses, first, that convergent potentialism is implicitly actualist, and second, that we should understand ultrafinitism in modal terms as a form of potentialism, one with surprising parallels to the case of arithmetic potentialism.

# Set-theoretic and arithmetic potentialism: the state of current developments, CACML 2020

This will be a plenary talk for the Chinese Annual Conference on Mathematical Logic (CACML 2020), held online 13-15 November 2020. My talk will be held 14 November 17:00 Beijing time (9 am GMT).

Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspective—a statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.

# A new proof of the Barwise extension theorem, and the universal finite sequence, Barcelona Set Theory Seminar, 28 October 2020

This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.

Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1 definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism.  This work is joint with Kameryn J. Williams.

# Choiceless large cardinals and set-theoretic potentialism

• R. Cutolo and J. D. Hamkins, “Choiceless large cardinals and set-theoretic potentialism,” Mathematical Logic Quarterly, p. 10 pages, 2022.
[Bibtex]
@ARTICLE{CutoloHamkins:Choiceless-large-cardinals-and-set-theoretic-potentialism,
author = {Raffaella Cutolo and Joel David Hamkins},
title = {Choiceless large cardinals and set-theoretic potentialism},
journal = {Mathematical Logic Quarterly},
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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.

# 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.

# Modal model theory, STUK 4, Oxford, December 2019

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

• J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” Journal of Symbolic Logic, 2021.
[Bibtex]
@ARTICLE{HamkinsWilliams2021:The-universal-finite-sequence,
author = {Joel David Hamkins and Kameryn J. Williams},
title = {The $\Sigma_1$-definable universal finite sequence},
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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.

• The universal algorithm,
• J. D. Hamkins and H. W. Woodin, “The universal finite set,” Mathematics ArXiv, p. 1–16, 2017.
[Bibtex]
@ARTICLE{HamkinsWoodin:The-universal-finite-set,
author = {Joel David Hamkins and W. Hugh Woodin},
title = {The universal finite set},
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pages = {1--16},
month = {},
note = {Manuscript under review},
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url = {http://jdh.hamkins.org/the-universal-finite-set},
}
• The modal logic of arithmetic potentialism,
• J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” Mathematics ArXiv, p. 1–35, 2018.
[Bibtex]
@ARTICLE{Hamkins:The-modal-logic-of-arithmetic-potentialism,
author = {Joel David Hamkins},
title = {The modal logic of arithmetic potentialism and the universal algorithm},
journal = {Mathematics ArXiv},
year = {2018},
volume = {},
number = {},
pages = {1--35},
month = {},
eprint = {1801.04599},
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• A new proof of the Barwise extension theorem
• Kameryn’s blog post about the paper

# Introducing modal model theory

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 $L$.


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. $\Box$

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

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.