This will be a talk for the Notre Dame Logic Seminar on 6 February 2024, 2:00 pm.
Abstract. The principle of covering reflection holds of a cardinal $\kappa$ if for every structure $B$ in a countable first-order language there is a structure $A$ of size less than $\kappa$, such that $B$ is covered by elementary images of $A$ in $B$. Is there any such cardinal? Is the principle consistent? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.
This will be a talk at the XVII International Luminy Workshop in Set Theory at the Centre International de Rencontres Mathématiques (CIRM) near Marseille, France, held 9-13 October 2023.
Abstract. I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the abundant atom axiom, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal is supercompact if and only if for every second-order sentence $\psi$ true in some structure $\langle M,\ldots\rangle$ (of any size) in a language of size less than $\kappa$ is also true in a first-order elementary substructure $m\prec M$ of size less than $\kappa$ with $m\cap\kappa\in\kappa$. This is joint work with Bokai Yao.
This will be talk for the workshop Philosophy of Set Theory held at the University of Konstanz, 3 – 4 December 2021 — in person!
Update: Unfortunately, the workshop has been cancelled (perhaps postponed to next year) in light of the Covid resurgence.
Abstract. I shall analyze the roles and interaction of reflection and urelements in second-order set theory. Second-order reflection already exhibits large cardinal strength even without urelements, but recent work shows that in the presence of abundant urelements, second-order reflection is considerably stronger than might have been expected—at the level of supercompact cardinals. This is joint work with Bokai Yao (Notre Dame).
[bibtex key=”HamkinsSolberg:Categorical-large-cardinals”]
Abstract. Inspired by Zermelo’s quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. The heights of these models, we define, are the categorical large cardinals. We subsequently consider various philosophical aspects of categoricity for structuralism and realism, including the tension between categoricity and set-theoretic reflection, and we present (and criticize) a categorical characterization of the set-theoretic universe $\langle V,\in\rangle$ in second-order logic.
Categorical accounts of various mathematical structures lie at the very core of structuralist mathematical practice, enabling mathematicians to refer to specific mathematical structures, not by having carefully to prepare and point at specially constructed instances—preserved like the one-meter iron bar locked in a case in Paris—but instead merely by mentioning features that uniquely characterize the structure up to isomorphism.
The natural numbers $\langle \mathbb{N},0,S\rangle$, for example, are uniquely characterized by the Dedekind axioms, which assert that $0$ is not a successor, that the successor function $S$ is one-to-one, and that every set containing $0$ and closed under successor contains every number. We know what we mean by the natural numbers—they have a definite realness—because we can describe features that completely determine the natural number structure. The real numbers $\langle\mathbb{R},+,\cdot,0,1\rangle$ similarly are characterized up to isomorphism as the unique complete ordered field. The complex numbers $\langle\mathbb{C},+,\cdot\rangle$ form the unique algebraically closed field of characteristic $0$ and size continuum, or alternatively, the unique algebraic closure of the real numbers. In fact all our fundamental mathematical structures enjoy such categorical characterizations, where a theory is categorical if it identifies a unique mathematical structure up to isomorphism—any two models of the theory are isomorphic. In light of the Löwenheim-Skolem theorem, which prevents categoricity for infinite structures in first-order logic, these categorical theories are generally made in second-order logic.
In set theory, Zermelo characterized the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$ in his famous quasi-categoricity result:
Theorem. (Zermelo, 1930) The models of $\text{ZFC}_2$ are precisely those isomorphic to a rank-initial segment $\langle V_\kappa,\in\rangle$ of the cumulative set-theoretic universe $V$ cut off at an inaccessible cardinal $\kappa$.
It follows that for any two models of $\text{ZFC}_2$, one of them is isomorphic to an initial segment of the other. These set-theoretic models $V_\kappa$ have now come to be known as Zermelo-Grothendieck universes, in light of Grothendieck’s use of them in category theory (a rediscovery several decades after Zermelo); they feature in the universe axiom, which asserts that every set is an element of some such $V_\kappa$, or equivalently, that there are unboundedly many inaccessible cardinals.
In this article, we seek to investigate the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the $V_\kappa$ universes are categorically characterized by their sentences or theories.
Question. Which models of $\text{ZFC}_2$ satisfy fully categorical theories?
If $\kappa$ is the smallest inaccessible cardinal, for example, then up to isomorphism $V_\kappa$ is the unique model of $\text{ZFC}_2$ satisfying the first-order sentence “there are no inaccessible cardinals.” The least inaccessible cardinal is therefore an instance of what we call a first-order sententially categorical cardinal. Similar ideas apply to the next inaccessible cardinal, and the next, and so on for quite a long way. Many of the inaccessible universes thus satisfy categorical theories extending $\text{ZFC}_2$ by a sentence or theory, either in first or second order, and we should like to investigate these categorical extensions of $\text{ZFC}_2$.
In addition, we shall discuss the philosophical relevance of categoricity and point particularly to the philosophical problem posed by the tension between the widespread support for categoricity in our fundamental mathematical structures with set-theoretic ideas on reflection principles, which are at heart anti-categorical.
Our main theme concerns these notions of categoricity:
Main Definition.
Follow through to the arxiv for the pdf to read more:
[bibtex key=”HamkinsSolberg:Categorical-large-cardinals”]
Related talk: Categorical cardinals, CUNY Set Theory Seminar, June 2020
[bibtex key=”BartonCaicedoFuchsHamkinsReitzSchindler2020:Inner-model-reflection-principles”]
Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH.
Every set theorist is familiar with the classical Lévy-Montague reflection principle, which explains how truth in the full set-theoretic universe $V$ reflects down to truth in various rank-initial segments $V_\theta$ of the cumulative hierarchy. Thus, the Lévy-Montague reflection principle is a form of height-reflection, in that truth in $V$ is reflected vertically downwards to truth in some $V_\theta$.
In this brief article, in contrast, we should like to introduce and consider a form of width-reflection, namely, reflection to nontrivial inner models. Specifically, we shall consider the following reflection principles.
Definition.
We explain how to force the principles, how to separate them, how they are consequences of various large cardinal assumptions, consequences of the maximality principle and of the inner model hypothesis. Kindly proceed to the article (pdf available at the arxiv) for more. [bibtex key=”BartonCaicedoFuchsHamkinsReitz:Inner-model-reflection-principles”]
This article grew out of an exchange held by the authors on math.stackexchange
in response to an inquiry posted by the first author concerning the nature of width-reflection in comparison to height-reflection: What is the consistency strength of width reflection?
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.
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”]
This will be a talk for the CUNY Set Theory seminar on September 1, 2017, 10 am. GC 6417.
Abstract. The inner model reflection principle asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. Both of these principles, expressing a form of width-reflection in constrast to the usual height-reflection, are equiconsistent with ZFC and an outright consequence of the existence of sufficient large cardinals, as well as a consequence (in lightface form) of the maximality principle and also of the inner-model hypothesis. This is joint work with Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, myself and Jonas Reitz.
I taught a course in Fall 2011 at NYU entitled Topics in Logic: set theory and the philosophy of set theory, aimed at graduate students in philosophy and others who want to gain greater understanding of some of the set-theoretic topics central to work in the philosophy of set theory. The course began with a review of the mathematical ideas, including a presentation of large cardinals, strong axioms of infinity and their associated elementary embeddings of the universe, and forcing, emphasizing the connection with the Boolean ultrapower and Boolean-valued models, but discussing the alternative formalizations. The second part of the course covers some of the philosophical literature, including what it means to accept or believe mathematical axioms, whether mathematics needs new axioms, the criteria one might use when adopting new axioms, and the question of pluralism and categoricity in set theory.
Here is a partial list of our readings:
1. Mathematical background.
2. Penelope Maddy, “Believing the axioms”, in two parts. JSL vols. 52 and 53. Part 1, Part 2
3. Chris Freiling, “Axioms of Symmetry: throwing darts at the real number line,”
JSL, vol. 51. http://www.jstor.org/stable/2273955
4. W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205.
5. Donald Martin, “Multiple universes of sets and indeterminate truth values,” Topoi 20, 5–16, 2001.
6. Hartry Field, “Which undecidable mathematical sentences have determinate truth values,” as reprinted in his book Truth and the Absence of Fact, Oxford University Press, 2001.
7. A brief selection from Marc Balaguer, Platonism and Anti-Platonism in Mathematics, Oxford University Press, 1998, describing the plenitudinous Platonism position.
8. Daniel Isaacson, “The reality of mathematics and the case of set theory,” 2007.
9. J. D. Hamkins, “The set-theoretic multiverse,” to appear in the Review of Symbolic Logic.
10. Solomon Feferman, Does mathematics need new axioms? Text of an invited AMS-MAA joint meeting, San Diego, January, 1997.
11. Solomon Feferman, Is the continuum hypothesis a definite mathematical problem? Draft article for the Exploring the Frontiers of Independence lecture series at Harvard University, October, 2011.
12. Peter Koellner, Feferman On the Indefiniteness of CH, a commentary on Feferman’s EFI article.
13. Interpretability of theories, the interpretability degrees and Orey sentences in set theory and arithmetic. Some of the basic material is found in Per Lindström’s book Aspects of Incompleteness, available at http://projecteuclid.org/euclid.lnl/1235416274, particularly chapter 6, and some later chapters.
14. Haim Gaifman, “On ontology and realism in mathematics,” to appear in the Review of Symbolic Logic (special issue connected with the NYU philosophy of mathematics conference 2009).
15. Saharon Shelah, “Logical dreams,” Bulletin of the AMS, 40(20):203–228, 2003. (Pre-publication version available at:http://arxiv.org/abs/math.LO/0211398)
16. For mathematical/philosophical amusement, Philip Welch and Leon Horsten, “The aftermath.”
It’s been a great semester!