Modal principles of potentialism, Oxford, January 2018

This was a talk I gave at University College Oxford to the philosophy faculty.

Abstract. One of my favorite situations occurs when philosophical ideas or issues inspire a bit of mathematical analysis, which in turn raises further philosophical questions and ideas, in a fruitful cycle. The topic of potentialism originates, after all, in the classical dispute between actual and potential infinity. Linnebo and Shapiro and others have emphasized the modal nature of potentialism, de-coupling it from infinity: the essence of potentialism is about approximating a larger universe or structure by means of partial structures or universe fragments. In several mathematical projects, my co-authors and I have found the exact modal validities of several natural potentialist concepts arising in the foundations of mathematics, including several kinds of set-theoretic and arithmetic potentialism. Ultimately, the variety of kinds of potentialism suggest a refocusing of potentialism on the issue of convergent inevitability in comparison with radical branching. I defended 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 suprising parallels to the case of arithmetic potentialism.

Here are my lecture notes that I used as a basis for the talk:

For a fuller, more technical account of potentialism, see the three-lecture tutorial series I gave for the Logic Winter School 2018 in Hejnice: Set-theoretic potentialism, and follow the link to the slides.

The modal logic of arithmetic potentialism and the universal algorithm

  • J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” ArXiv e-prints, pp. 1-35, 2018. (manuscript under review)  
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Abstract. Natural potentialist systems arise from the models of arithmetic when they are considered under their various natural extension concepts, such as end-extensions, arbitrary extension, $\Sigma_n$-elementary extensions, conservative extensions and more. For these potentialist systems, I prove, a propositional modal assertion is valid in a model of arithmetic, with respect to assertions in the language of arithmetic with parameters, exactly when it is an assertion of S4. Meanwhile, with respect to sentences, the validities of a model are always between S4 and S5, and these bounds are sharp in that both endpoints are realized. The models validating exactly S5 are the models of the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main proof makes fundamental use of the universal algorithm, of which this article provides a self-contained account.

 

In this article, I consider the models of arithmetic under various natural extension concepts, including end-extensions, arbitrary extensions, $\Sigma_n$-elementary extensions, conservative extensions and more. Each extension concept gives rise to an arithmetic potentialist system, a Kripke model of possible arithmetic worlds, and the main goal is to discover the modal validities of these systems.

For most of the extension concepts, a modal assertion is valid with respect to assertions in the language of arithmetic, allowing parameters, exactly when it is an assertion of the modal theory S4. For sentences, however, the modal validities form a theory between S4 and S5, with both endpoints being realized. A model of arithmetic validates S5 with respect to sentences just in case it is a model of the arithmetic maximality principle, and these models are equivalently characterized as those realizing a maximal $\Sigma_1$ theory.

The main argument relies fundamentally on the universal algorithm, the theorem due to Woodin that there is a Turing machine program that can enumerate any finite sequence in the right model of arithmetic, and furthermore this model can be end-extended so as to realize any further extension of that sequence available in the model. In the paper, I give a self-contained account of this theorem using my simplified proof.

The paper concludes with philosophical remarks on the nature of potentialism, including a discussion of how the linear inevitability form of potentialism is actually much closer to actualism than the more radical forms of potentialism, which exhibit branching possibility. I also propose to view the philosphy of ultrafinitism in modal terms as a form of potentialism, pushing the issue of branching possibility in ultrafinitism to the surface.

Set-theoretic potentialism, Winter School in Abstract Analysis 2018, Hejnice, Czech Republic

This will be a tutorial lecture series for the Winter School in Abstract Analysis 2018, held in Hejnice of the Czech Republic.

Abstract. I shall introduce and develop the theory of set-theoretic potentialism. A potentialist system is a collection of first-order structures, all in the same language $\mathcal{L}$, equipped with an accessibility relation refining the inclusion relation. Any such system, viewed as an inflationary-domain Kripke model, provides a natural interpretation for the modal extension of the underlying language $\mathcal{L}$ to include the modal operators. We seek to understand a given potentialist system by analyzing which modal assertions are valid in it.

Set theory exhibits an enormous variety of natural potentialist systems. For example, with forcing potentialism, one considers the models of set theory, each accessing its forcing extensions; with rank potentialism, one considers the collection of of rank-initial segments $V_\alpha$ of a given set-theoretic universe; with Grothendieck-Zermelo potentialism, one has the collection of $V_\kappa$ for (a proper class of) inaccessible cardinals $\kappa$; with top-extensional potentialism, one considers the collection of countable models of ZFC under the top-extension relation; and so on with many other natural examples.

In this tutorial, we shall settle the precise potentialist validities of each of these potentialist systems and others, and we shall develop the general tools that enable one to determine the modal theory of a given potentialist system. Many of these arguments proceed by building connections between certain sweeping general features of the models in the potentialist system and certain finite combinatorial objects such as trees or lattices. A key step involves finding certain kinds of independent control statements — buttons, switches, ratchets and rail-switches — in the collection of models.

Slides

The universal finite set

  • J. D. Hamkins and W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)  
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    title = {The universal finite set},
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Abstract. We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\varphi$ has complexity $\Sigma_2$, so that any affirmative instance of it $\varphi(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_\theta$; the set is empty in any transitive model and others; and if $\varphi$ defines the set $y$ in some countable model $M$ of ZFC and $y\subseteq z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $\varphi$ defines the new set $z$. Thus, the set shows that no model of set theory can realize a maximal $\Sigma_2$ theory with its natural number parameters, although this is possible without parameters. Using the universal finite set, we prove that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.

Woodin had established the universal algorithm phenomenon, showing that there is a Turing machine program with a certain universal top-extension property in models of arithmetic (see also work of Blanck and Enayat 2017 and upcoming paper of mine with Gitman and Kossak; also my post The universal algorithm: a new simple proof of Woodin’s theorem). Namely, the program provably enumerates a finite set of natural numbers, but it is relatively consistent with PA that it enumerates any particular desired finite set of numbers, and furthermore, if $M$ is any model of PA in which the program enumerates the set $s$ and $t$ is any (possibly nonstandard) finite set in $M$ with $s\subseteq t$, then there is a top-extension of $M$ to a model $N$ in which the program enumerates exactly the new set $t$. So it is a universal finite computably enumerable set, which can in principle be any desired finite set of natural numbers in the right arithmetic universe and become any desired larger finite set in a suitable larger arithmetic universe.

I had inquired whether there is a set-theoretic analogue of this phenomenon, using $\Sigma_2$ definitions in set theory in place of computable enumerability (see The universal definition — it can define any mathematical object you like, in the right set-theoretic universe). The idea was that just as a computably enumerable set is one whose elements are gradually revealed as the computation proceeds, a $\Sigma_2$-definable set in set theory is precisely one whose elements become verified at some level $V_\theta$ of the cumulative set-theoretic hierarchy as it grows. In this sense, $\Sigma_2$ definability in set theory is analogous to computable enumerability in arithmetic.

Main Question. Is there a universal $\Sigma_2$ definition in set theory, one which can define any desired particular set in some model of \ZFC\ and always any desired further set in a suitable top-extension?

I had noticed in my earlier post that one can do this using a $\Pi_3$ definition, or with a $\Sigma_2$ definition, if one restricts to models of a certain theory, such as $V\neq\text{HOD}$ or the eventual GCH, or if one allows $\{x\mid\varphi(x)\}$ sometimes to be a proper class.

Here, we provide a fully general affirmative answer with the following theorem.

Main Theorem. There is a formula $\varphi(x)$ of complexity $\Sigma_2$ in the language of set theory, provided in the proof, with the following properties:

  1. ZFC proves that $\{x\mid \varphi(x)\}$ is a finite set.
  2. In any transitive model of \ZFC\ and others, this set is empty.
  3. If $M$ is a countable model of ZFC in which $\varphi$ defines the set $y$ and $z\in M$ is any finite set in $M$ with $y\subseteq z$, then there is a top-extension of $M$ to a model $N$ in which $\varphi$ defines exactly $z$.

By taking the union of the set defined by $\varphi$, an arbitrary set can be achieved; so the finite-set result as stated in the main theorem implies the arbitrary set case as in the main question. One can also easily deduce a version of the theorem to give a universal countable set or a universal set of some other size (for example, just take the union of the countable elements of the universal set). One can equivalently formulate the main theorem in terms of finite sequences, rather than sets, so that the sequence is extended as desired in the top-extension. The sets $y$ and $z$ in statement (3) may be nonstandard finite, if $M$ if $\omega$-nonstandard.

We use this theorem to establish the fundamental validities of top-extensional set-theoretic potentialism. Specifically, in the potentialist system consisting of the countable models of ZFC, with each accessing its top extensions, the modal validities with respect to substitution instances in the language of set theory, with parameters, are exactly the assertions of S4. When only sentences are considered, the validities are between S4 and S5, with both endpoints realized.

In particular, we prove that if ZFC is consistent, then there is a model $M$ of ZFC with the top-extensional maximality principle: any sentence $\sigma$ in the language of set theory which is true in some top extension $M^+$ and all further top extensions of $M^+$, is already true in $M$.

This principle is true is any model of set theory with a maximal $\Sigma_2$ theory, but it is never true when $\sigma$ is allowed to have natural-number parameters, and in particular, it is never true in any $\omega$-standard model of set theory.

Click through to the arXiv for more, the full article in pdf.

  • J. D. Hamkins and W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)  
    @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},
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    note = {manuscript under review},
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    }

A universal finite set, CUNY Logic Workshop, November 2017

This will be a talk for the CUNY Logic Workshop, November 17, 2017, 2pm GC Room 6417. 

Abstract. I shall define a certain finite set in set theory $$\{x\mid\varphi(x)\}$$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\varphi$ has complexity $\Sigma_2$ and therefore any instance of it $\varphi(x)$ is locally verifiable inside any sufficient $V_\theta$; the set is empty in any transitive model and others; and if $\varphi$ defines the set $y$ in some countable model $M$ of ZFC and $y\subset z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $\varphi$ defines the new set $z$. In particular, although there are models of set theory with maximal $\Sigma_2$ theories, nevertheless no model of set theory realizes a maximal $\Sigma_2$ theory with its natural-number parameters. Using the universal finite set, it follows that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.

This is joint work with W. Hugh Woodin.

Arithmetic potentialism and the universal algorithm, CUNY Logic Workshop, September 2017

This will be a talk for the CUNY Logic Workshop at the CUNY Graduate Center, September 8, 2017, 2-3:30, room GC 6417.

Empire_State_Building_New_York_March_2015

Abstract. Consider the collection of all the models of arithmetic under the end-extension relation, which forms a potentialist system for arithmetic, a collection of possible arithmetic worlds or universe fragments, with a corresponding potentialist modal semantics. What are the modal validities? I shall prove that every model of arithmetic validates exactly S4 with respect to assertions in the language of arithmetic allowing parameters, but if one considers sentences only (no parameters), then some models can validate up to S5, thereby fulfilling the arithmetic maximality principle, which asserts for a model $M$ that whenever an arithmetic sentence is true in some end-extension of $M$ and all subsequent end-extensions, then it is already true in $M$. (We also consider other accessibility relations, such as arbitrary extensions or $\Sigma_n$-elementary extensions or end-extensions.)

The proof makes fundamental use of what I call the universal algorithm, a fascinating result due to W. Hugh Woodin, asserting that there is a computable algorithm that can in principle enumerate any desired finite sequence, if only it is undertaken in the right universe, and furthermore any given model of arithmetic can be end-extended so as to realize any desired additional behavior for that universal program. I shall give a simple proof of the universal algorithm theorem and explain how it can be used to determine the potentialist validities of a model of arithmetic. This is current joint work in progress with Victoria Gitman and Roman Kossak, and should be seen as an arithmetic analogue of my recent work on set-theoretic potentialism with Øystein Linnebo. The mathematical program is strongly motivated by philosophical ideas arising in the distinction between actual and potential infinity.

 

The modal principles of potentialism in mathematics, Logic and Metaphysics Workshop, CUNY, November 2017

This will be a talk on November 6, 2017 for the Logic and Metaphysics workshop at the CUNY Graduate Center, run by Graham Priest. Room GC 3209.

Morning_Fog_at_GGB

The modal principles of potentialism in mathematics

Abstract. Potentialism is the view in the philosophy of mathematics that one’s mathematical universe, whether in arithmetic or set theory, is never fully completed, but rather unfolds gradually as new parts of it increasingly come into existence or become accessible or known to us. As in the classical dispute between actual versus potential infinity, the potentialist holds that objects in the upper or outer reaches have potential as opposed to actual existence, in the sense that one can imagine forming or discovering always more objects from that realm, as many as desired, but the task is never completed.  Recent work has emphasized the modal aspect of potentialism, and in this talk, I shall describe a general model-theoretic account of the modal logic of potentialism, identifying specific modal principles that hold or fail depending on features of the potentialist system under consideration. This work makes use of modal control statements, such as buttons, switches, dials and ratchets and the connection of these kinds of statements with the modal theories S4, S4.2, S4.3 and S5. I shall take the various natural kinds of arithmetic and set-theoretic potentialism as illustrative cases.

This is joint work with Øystein Linnebo, University of Oslo (see our paper The modal logic of set-theoretic potentialism and the potentialist maximality principles), and further joint work in progress with Victoria Gitman and Roman Kossak, and very recent joint work in progress with W. Hugh Woodin.

Lecture Notes

The modal logic of set-theoretic potentialism and the potentialist maximality principles

Joint work with Øystein Linnebo, University of Oslo.

  • J. D. Hamkins and Ø. Linnebo, “The modal logic of set-theoretic potentialism and the potentialist maximality principles,” , 2017. (manuscript under review)  
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Abstract. We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe (Structural connections between a forcing class and its modal logic), including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta$); Grothendieck-Zermelo potentialism (true in all larger $V_\kappa$ for inaccessible cardinals $\kappa$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

Set-theoretic potentialism is the view in the philosophy of mathematics that the universe of set theory is never fully completed, but rather unfolds gradually as parts of it increasingly come into existence or become accessible to us. On this view, the outer reaches of the set-theoretic universe have merely potential rather than actual existence, in the sense that one can imagine “forming” or discovering always more sets from that realm, as many as desired, but the task is never completed. For example, height potentialism is the view that the universe is never fully completed with respect to height: new ordinals come into existence as the known part of the universe grows ever taller. Width potentialism holds that the universe may grow outwards, as with forcing, so that already existing sets can potentially gain new subsets in a larger universe. One commonly held view amongst set theorists is height potentialism combined with width actualism, whereby the universe grows only upward rather than outward, and so at any moment the part of the universe currently known to us is a rank initial segment $V_\alpha$ of the potential yet-to-be-revealed higher parts of the universe. Such a perspective might even be attractive to a Platonistically inclined large-cardinal set theorist, who wants to hold that there are many large cardinals, but who also is willing at any moment to upgrade to a taller universe with even larger large cardinals than had previously been mentioned. Meanwhile, the width-potentialist height-actualist view may be attractive for those who wish to hold a potentialist account of forcing over the set-theoretic universe $V$. On the height-and-width-potentialist view, one views the universe as growing with respect to both height and width. A set-theoretic monist, in contrast, with an ontology having only a single fully existing universe, will be an actualist with respect to both width and height. The second author has described various potentialist views in previous work.

Although we are motivated by the case of set-theoretic potentialism, the potentialist idea itself is far more general, and can be carried out in a general model-theoretic context. For example, the potentialist account of arithmetic is deeply connected with the classical debates surrounding potential as opposed to actual infinity, and indeed, perhaps it is in those classical debates where one finds the origin of potentialism. More generally, one can provide a potentialist account of truth in the context of essentially any kind of structure in any language or theory.

Our project here is to analyze and understand more precisely the modal commitments of various set-theoretic potentialist views.  After developing a general model-theoretic account of the semantics of potentialism and providing tools for establishing both lower and upper bounds on the modal validities for various kinds of potentialist contexts, we shall use those tools to settle exactly the propositional modal validities for several natural kinds of set-theoretic height and width potentialism.

Here is a summary account of the modal logics for various flavors of set-theoretic potentialism.

Flavours of potentialism

In each case, the indicated lower and upper bounds are realized in particular worlds, usually in the strongest possible way that is consistent with the stated inclusions, although in some cases, this is proved only under additional mild technical hypotheses. Indeed, some of the potentialist accounts are only undertaken with additional set-theoretic assumptions going beyond ZFC. For example, the Grothendieck-Zermelo account of potentialism is interesting mainly only under the assumption that there are a proper class of inaccessible cardinals, and countable-transitive-model potentialism is more robust under the assumption that every real is an element of a countable transitive model of set theory, which can be thought of as a mild large-cardinal assumption.

The upper bound of S5, when it is realized, constitutes a potentialist maximality principle, for in such a case, any statement that could possibly become actually true in such a way that it remains actually true as the universe unfolds, is already actually true. We identify necessary and sufficient conditions for each of the concepts of potentialism for a world to fulfill this potentialist maximality principle. For example, in rank-potentialism, a world $V_\kappa$ satisfies S5 with respect to the language of set theory with arbitrary parameters if and only if $\kappa$ is $\Sigma_3$-correct. And it satisfies S5 with respect to the potentialist language of set theory with parameters if and only if it is $\Sigma_n$-correct for every $n$.  Similar results hold for each of the potentialist concepts.

Finally, let me mention the strong affinities between set-theoretic potentialism and set-theoretic pluralism, particularly with the various set-theoretic multiverse conceptions currently in the literature. Potentialists may regard themselves mainly as providing an account of truth ultimately for a single universe, gradually revealed, the limit of their potentialist system. Nevertheless, the universe fragments of their potentialist account can often naturally be taken as universes in their own right, connected by the potentialist modalities, and in this way, every potentialist system can be viewed as a multiverse. Indeed, the potentialist systems we analyze in this article—including rank potentialism, forcing potentialism, generic-multiverse potentialism, countable-transitive-model potentialism, countable-model potentialism—each align with corresponding natural multiverse conceptions. Because of this, we take the results of this article as providing not only an analysis of the modal commitments of set-theoretic potentialism, but also an analysis of the modal commitments of various particular set-theoretic multiverse conceptions. Indeed, one might say that it is possible (ahem), in another world, for this article to have been entitled, “The modal logic of various set-theoretic multiverse conceptions.”

For more, please follow the link to the arxiv where you can find the full article.

  • J. D. Hamkins and Ø. Linnebo, “The modal logic of set-theoretic potentialism and the potentialist maximality principles,” , 2017. (manuscript under review)  
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Set-theoretic potentialism, CUNY Logic Workshop, September, 2016

This will be a talk for the CUNY Logic Workshop, September 16, 2016, at the CUNY Graduate Center, Room 6417, 2-3:30 pm.

Book 06487 20040730160046 droste effect nevit.jpgAbstract.  In analogy with the ancient views on potential as opposed to actual infinity, set-theoretic potentialism is the philosophical position holding that the universe of set theory is never fully completed, but rather has a potential character, with greater parts of it becoming known to us as it unfolds. In this talk, I should like to undertake a mathematical analysis of the modal commitments of various specific natural accounts of set-theoretic potentialism. After developing a general model-theoretic framework for potentialism and describing how the corresponding modal validities are revealed by certain types of control statements, which we call buttons, switches, dials and ratchets, I apply this analysis to the case of set-theoretic potentialism, including the modalities of true-in-all-larger-$V_\beta$, true-in-all-transitive-sets, true-in-all-Grothendieck-Zermelo-universes, true-in-all-countable-transitive-models and others. Broadly speaking, the height-potentialist systems generally validate exactly S4.3 and the height-and-width-potentialist systems generally validate exactly S4.2. Each potentialist system gives rise to a natural accompanying maximality principle, which occurs when S5 is valid at a world, so that every possibly necessary statement is already true.  For example, a Grothendieck-Zermelo universe $V_\kappa$, with $\kappa$ inaccessible, exhibits the maximality principle with respect to assertions in the language of set theory using parameters from $V_\kappa$ just in case $\kappa$ is a $\Sigma_3$-reflecting cardinal, and it exhibits the maximality principle with respect to assertions in the potentialist language of set theory with parameters just in case it is fully reflecting $V_\kappa\prec V$.

This is current joint work with Øystein Linnebo, in progress, which builds on some of my prior work with George Leibman and Benedikt Löwe in the modal logic of forcing.

CUNY Logic Workshop abstract | link to article will be posted later

The modal logic of set-theoretic potentialism, Kyoto, September 2016

Kyoto cuisineThis will be a talk for the workshop conference Mathematical Logic and Its Applications, which will be held at the Research Institute for Mathematical Sciences, Kyoto University, Japan, September 26-29, 2016, organized by Makoto Kikuchi. The workshop is being held in memory of Professor Yuzuru Kakuda, who was head of the research group in logic at Kobe University during my stay there many years ago.

Abstract.  Set-theoretic potentialism is the ontological view in the philosophy of mathematics that the universe of set theory is never fully completed, but rather has a potential character, with greater parts of it becoming known to us as it unfolds. In this talk, I should like to undertake a mathematical analysis of the modal commitments of various specific natural accounts of set-theoretic potentialism. After developing a general model-theoretic framework for potentialism and describing how the corresponding modal validities are revealed by certain types of control statements, which we call buttons, switches, dials and ratchets, I apply this analysis to the case of set-theoretic potentialism, including the modalities of true-in-all-larger-$V_\beta$, true-in-all-transitive-sets, true-in-all-Grothendieck-Zermelo-universes, true-in-all-countable-transitive-models and others. Broadly speaking, the height-potentialist systems generally validate exactly S4.3 and the height-and-width-potentialist systems validate exactly S4.2. Each potentialist system gives rise to a natural accompanying maximality principle, which occurs when S5 is valid at a world, so that every possibly necessary statement is already true.  For example, a Grothendieck-Zermelo universe $V_\kappa$, with $\kappa$ inaccessible, exhibits the maximality principle with respect to assertions in the language of set theory using parameters from $V_\kappa$ just in case $\kappa$ is a $\Sigma_3$-reflecting cardinal, and it exhibits the maximality principle with respect to assertions in the potentialist language of set theory with parameters just in case it is fully reflecting $V_\kappa\prec V$.

This is joint work with Øystein Linnebo, which builds on some of my prior work with George Leibman and Benedikt Löwe in the modal logic of forcing. Our research article is currently in progress.

Slides | Workshop program