# The surprising strength of reflection in second-order set theory with abundant urelements, Konstanz, December 2021

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).

# Categorical large cardinals and the tension between categoricity and set-theoretic reflection

• J. D. Hamkins and R. Solberg, “Categorical large cardinals and the tension between categoricity and set-theoretic reflection,” Mathematics ArXiv, 2020.
[Bibtex]
@ARTICLE{HamkinsSolberg:Categorical-large-cardinals,
author = {Joel David Hamkins and Robin Solberg},
title = {Categorical large cardinals and the tension between categoricity and set-theoretic reflection},
journal = {Mathematics ArXiv},
year = {2020},
volume = {},
number = {},
pages = {},
month = {},
note = {Under review},
abstract = {},
keywords = {under-review},
url = {http://jdh.hamkins.org/categorical-large-cardinals/},
source = {},
doi = {},
eprint = {2009.07164},
archivePrefix ={arXiv},
primaryClass = {math.LO}
}

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.

• A cardinal $\kappa$ is first-order sententially categorical, if there is a first-order sentence $\sigma$ in the language of set theory, such that $V_\kappa$ is categorically characterized by $\text{ZFC}_2+\sigma$.
• A cardinal $\kappa$ is first-order theory categorical, if there is a first-order theory $T$ in the language of set theory, such that $V_\kappa$ is categorically characterized by $\text{ZFC}_2+T$.
• A cardinal $\kappa$ is second-order sententially categorical, if there is a second-order sentence $\sigma$ in the language of set theory, such that $V_\kappa$ is categorically characterized by $\text{ZFC}_2+\sigma$.
• A cardinal $\kappa$ is second-order theory categorical, if there is a second-order theory $T$ in the language of set theory, such that $V_\kappa$ is categorically characterized by $\text{ZFC}_2+T$.

• J. D. Hamkins and R. Solberg, “Categorical large cardinals and the tension between categoricity and set-theoretic reflection,” Mathematics ArXiv, 2020.
[Bibtex]
@ARTICLE{HamkinsSolberg:Categorical-large-cardinals,
author = {Joel David Hamkins and Robin Solberg},
title = {Categorical large cardinals and the tension between categoricity and set-theoretic reflection},
journal = {Mathematics ArXiv},
year = {2020},
volume = {},
number = {},
pages = {},
month = {},
note = {Under review},
abstract = {},
keywords = {under-review},
url = {http://jdh.hamkins.org/categorical-large-cardinals/},
source = {},
doi = {},
eprint = {2009.07164},
archivePrefix ={arXiv},
primaryClass = {math.LO}
}

# Inner-model reflection principles

• N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, J. Reitz, and R. Schindler, “Inner-model reflection principles,” Studia Logica, vol. 108, p. 573–595, 2020.
[Bibtex]
@ARTICLE{BartonCaicedoFuchsHamkinsReitzSchindler2020:Inner-model-reflection-principles,
author = {Neil Barton and Andr\'es Eduardo Caicedo and Gunter Fuchs and Joel David Hamkins and Jonas Reitz and Ralf Schindler},
title = {Inner-model reflection principles},
journal = {Studia Logica},
year = {2020},
volume = {108},
number = {},
pages = {573--595},
month = {},
note = {},
abstract = {},
keywords = {},
source = {},
doi = {10.1007/s11225-019-09860-7},
eprint = {1708.06669},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/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.

1. The inner-model reflection principle asserts that if a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then there is a proper inner model $W$, a transitive class model of ZF containing all ordinals, with $a\in W\subsetneq V$ in which $\varphi(a)$ is true.
2. The ground-model reflection principle asserts that if $\varphi(a)$ is true in $V$, then there is a nontrivial ground model $W\subsetneq V$ with $a\in W$ and $W\models\varphi(a)$.
3. Variations of the principles arise by insisting on inner models of a particular type, such as ground models for a particular type of forcing, or by restricting the class of parameters or formulas that enter into the scheme.
4. The lightface forms of the principles, in particular, make their assertion only for sentences, so that if $\sigma$ is a sentence true in $V$, then $\sigma$ is true in some proper inner model or ground $W$, respectively.

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.

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?

# The inner-model and ground-model reflection principles, CUNY Set Theory seminar, September 2017

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.

# Philosophy of set theory, Fall 2011, NYU PH GA 1180

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 1Part 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!