Modal model theory

This is joint work with Wojciech Aleksander Wołoszyn, who is about to begin as a DPhil student with me in mathematics here in Oxford. We began and undertook this work over the past year, while he was a visitor in Oxford under the Recognized Student program.

[bibtex key=”HamkinsWoloszyn:Modal-model-theory”]

Abstract. We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement $\varphi$ is possible in a structure (written $\Diamond\varphi$) if $\varphi$ is true in some extension of that structure, and $\varphi$ is necessary (written $\Box\varphi$) if it is true in all extensions of the structure. A principal case for us will be the class $\text{Mod}(T)$ of all models of a given theory $T$—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, $k$-colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_\omega$, $\beth_\omega$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. A graph obeys the maximality principle $\Diamond\Box\varphi(a)\to\varphi(a)$ with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.

Follow through the arXiv for a pdf of the article.

[bibtex key=”HamkinsWoloszyn:Modal-model-theory”]

Modal model theory as mathematical potentialism, Oslo online Potentialism Workshop, September 2020

This will be a talk for the Oslo potentialism workshop, Varieties of Potentialism, to be held online via Zoom on 23 September 2020, from noon to 18:40 CEST (11am to 17:40 UK time). My talk is scheduled for 13:10 CEST (12:10 UK time). Further details about access and registration are availavle on the conference web page.

Abstract. I shall introduce and describe the subject of modal model theory, in which one studies a mathematical structure within a class of similar structures under an extension concept, giving rise to mathematically natural notions of possibility and necessity, a form of mathematical potentialism. We study the class of all graphs, or all groups, all fields, all orders, or what have you; a natural case is the class $\text{Mod}(T)$ of all models of a fixed first-order theory $T$. In this talk, I shall describe some of the resulting elementary theory, such as the fact that the $\mathcal{L}$ theory of a structure determines a robust fragment of its modal theory, but not all of it. The class of graphs illustrates the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_\omega$, $\beth_\omega$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. When augmented with the actuality operator @, modal graph theory becomes fully bi-interpretable with truth in the set-theoretic universe. This is joint work with Wojciech Wołoszyn.

Potentialism and implicit actualism in the foundations of mathematics, Jowett Society lecture, Oxford, February 2019

This will be a talk for the Jowett Society on 8 February, 2019. The talk will take place in the Oxford Faculty of Philosophy, 3:30 – 5:30pm, in the Lecture Room of the Radcliffe Humanities building.

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.

Jowett Society talk entry | my posts on potentialism | Slides

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:

https://photos.app.goo.gl/DcaXV3sLVU4enaxv8

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.

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.

[bibtex key=”HamkinsLinnebo:Modal-logic-of-set-theoretic-potentialism”]

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.

[bibtex key=”HamkinsLinnebo:Modal-logic-of-set-theoretic-potentialism”]

Structural connections between a forcing class and its modal logic

[bibtex key=HamkinsLeibmanLoewe2015:StructuralConnectionsForcingClassAndItsModalLogic]

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting $\square$ as “in all forcing extensions” and $\Diamond$ as “in some forcing extension”. In this modal language one may easily express sweeping general forcing principles, such as $\Diamond\square\varphi\to\square\Diamond\varphi$, the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or $\Diamond\square\varphi\to\varphi$, the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (see A simple maximality principle).

Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2 (see The modal logic of forcing). In this article, we prove that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of $\omega_1$-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise, including the connection between various control statements, such as buttons, switches and ratchets, and their corresponding forcing validities. These structural connections therefore support a forcing-only analysis of other diverse forcing classes.

Preprints available at:  ar$\chi$iv | NI12055-SAS | UvA ILLC PP-2012-19 | HBM 446

The modal logic of forcing

[bibtex key=HamkinsLoewe2008:TheModalLogicOfForcing]

What are the most general principles in set theory relating forceability and truth? As with Solovay’s celebrated analysis of provability, both this question and its answer are naturally formulated with modal logic. We aim to do for forceability what Solovay did for provability. A set theoretical assertion $\psi$ is forceable or possible, if $\psi$ holds in some forcing extension, and necessary, if $\psi$ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory known as S4.2.

Follow-up article:  Structural connections between a forcing class and its modal logic