On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018

This will be a talk for the Philosophy of Mathematics Seminar in Oxford, October 29, 2018, 4:30-6:30 in the Ryle Room of the Philosopher Centre.

Abstract. In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, it is natural to wonder whether one might find a similar success for set-theoretic mereology, based upon the set-theoretic inclusion relation $\subseteq$ rather than the element-of relation $\in$.  How well does set-theoretic mereological serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics in terms of the subset relation to the same extent that set theorists have argued (with whatever degree of success) that we may find faithful representations in terms of the membership relation? Basically, can we get by with merely $\subseteq$ in place of $\in$? Ultimately, I shall identify grounds supporting generally negative answers to these questions, concluding that set-theoretic mereology by itself cannot serve adequately as a foundational theory.

This is joint work with Makoto Kikuchi, and the talk is based on our joint articles:

The talk will also mention some related recent work with Ruizhi Yang (Shanghai).

Parallels in universality between the universal algorithm and the universal finite set, Oxford Math Logic Seminar, October 2018

This will be a talk for the Logic Seminar in Oxford at the Mathematics Institute in the Andrew Wiles Building on October 9, 2018, at 4:00 pm, with tea at 3:30.

Abstract. The universal algorithm is a Turing machine program $e$ that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a set-theoretic analogue, a locally verifiable definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. Recent work has uncovered a $\Sigma_1$-definable version that works with respect to end-extensions. I shall give an account of all three results, which have a parallel form, and describe applications to the model theory of arithmetic and set theory.

The rearrangement number: how many rearrangements of a series suffice to validate absolute convergence? Warwick Mathematics Colloquium, October 2018

This will be a talk for the Mathematics Colloquium at the University of Warwick, to be held October 19, 2018, 4:00 pm in Lecture Room B3.02 at the Mathematics Institute. I am given to understand that the talk will be followed by a wine and cheese reception.Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

  • The rearrangement number.
    • A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, “The rearrangement number,” ArXiv e-prints, 2016. (manuscript under review)  
      @ARTICLE{BlassBrendleBrianHamkinsHardyLarson:TheRearrangementNumber,
      author = {Andreas Blass and Jörg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
      title = {The rearrangement number},
      journal = {ArXiv e-prints},
      year = {2016},
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      note = {manuscript under review},
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      eprint = {1612.07830},
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  • The subseries number.
    • J. Brendle, W. Brian, and J. D. Hamkins, “The subseries number,” ArXiv e-prints, 2018. (manuscript under review)  
      @ARTICLE{BrendleBrianHamkins:The-subseries-number,
      author = {Jörg Brendle and Will Brian and Joel David Hamkins},
      title = {The subseries number},
      journal = {ArXiv e-prints},
      year = {2018},
      volume = {},
      number = {},
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      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-subseries-number},
      eprint = {1801.06206},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
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Plenary talk, 16th International Congress of Logic, Methodology and Philosophy of Science and Technology, CLMPST 2019, Prague

I shall be giving a keynote plenary talk for the 16th International Congress of Logic, Methodology and Philosophy of Science and Technology (CLMPST 2019), to be held 5-10 August 2019  at the Institute of Philosophy of the Czech Academy of Sciences in the beautiful city of Prague .  The CLMPST congress is held every four years, and the theme of the 2019 meeting is, “Bridging across academic cultures.”

 

I shall announce the title and abstract for the talk on this post at a later date when it becomes available.

Meanwhile, please join me in Prague!  See the Call for Papers, requesting contributed papers and contributed symposia on twenty different thematic sections, from mathematical and philosophical logic to the philosophy of science, philosophy of computing and many other areas. I am given to understand that this will be a large meeting, with about 800 participants expected.

Set-theoretic potentialism and the universal finite set, Scandinavian Logic Symposium, June 2018

This will be an invited talk at the Scandinavian Logic Symposium SLS 2018, held at the University of Gothenburg in Sweden, June 11-13, 2018.

Abstract. Providing a set-theoretic analogue of the universal algorithm, 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 sufficiently large $V_\theta$; the set is empty in any transitive model; 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 $N$ of ZFC in which $\varphi$ defines the new set $z$. I shall draw out consequences of the universal finite set for set-theoretic potentialism and discuss several issues it raises in the philosophy of set theory.

The talk will include joint work with W. Hugh Woodin, Øystein Linnebo and others.

Slides: Set-theoretic potentialism and universal finite set SLS 2018

The universal finite set, Rutgers Logic Seminar, April 2018

This will be a talk for the Rutgers Logic Seminar, April 2, 2018. Hill Center, Busch campus.

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 $z$.  The definition can be thought of as an idealized diamond sequence, and there are consequences for the philosophical theory of set-theoretic top-extensional potentialism.

This is joint work with W. Hugh Woodin.

Determinacy for open class games is preserved by forcing, CUNY Set Theory Seminar, April 2018

This will be a talk for the CUNY Set Theory Seminar, April 27, 2018, GC Room 6417, 10-11:45am (please note corrected date).

Abstract. Open class determinacy is the principle of second order set theory asserting of every two-player game of perfect information, with plays coming from a (possibly proper) class $X$ and the winning condition determined by an open subclass of $X^\omega$, that one of the players has a winning strategy. This principle finds itself about midway up the hierarchy of second-order set theories between Gödel-Bernays set theory and Kelley-Morse, a bit stronger than the principle of elementary transfinite recursion ETR, which is equivalent to clopen determinacy, but weaker than GBC+$\Pi^1_1$-comprehension. In this talk, I’ll given an account of my recent joint work with W. Hugh Woodin, proving that open class determinacy is preserved by forcing. A central part of the proof is to show that in any forcing extension of a model of open class determinacy, every well-founded class relation in the extension is ranked by a ground model well-order relation. This work therefore fits into the emerging focus in set theory on the interaction of fundamental principles of second-order set theory with fundamental set theoretic tools, such as forcing. It remains open whether clopen determinacy or equivalently ETR is preserved by set forcing, even in the case of the forcing merely to add a Cohen real.

Nonamalgamation in the Cohen generic multiverse, CUNY Logic Workshop, March 2018

This will be a talk for the CUNY Logic Workshop on March 23, 2018, GC 6417 2-3:30pm.

Abstract. Consider a countable model of set theory $M$ in the context of all its successive forcing extensions and grounds. This generic multiverse has long been known to exhibit instances of nonamalgamation: one can have two extensions $M[c]$ and $M[d]$, both adding a merely a generic Cohen real, which have no further extension in common. In this talk, I shall describe new joint work that illuminates the extent of non-amalgamation: every finite partial order (and more) embeds into the generic multiverse over any given model in a way that preserves amalgamability and non-amalgamability. The proof uses the set-theoretic blockchain argument (pictured above), which has affinities with constructions in computability theory in the Turing degrees. Other arguments, which also resemble counterparts in computability theory, show that the generic multiverse exhibits the exact pair phenonemon for increasing chains. This is joint work with Miha Habič, myself, Lukas Daniel Klausner and Jonathan Verner. The paper will be available this Spring.

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.

On the strengths of the class forcing theorem and clopen class game determinacy, Prague set theory seminar, January 2018

This will be a talk for the Prague set theory seminar, January 24, 11:00 am to about 2pm (!).

Abstract. The class forcing theorem is the assertion that every class forcing notion admits corresponding forcing relations. This assertion is not provable in Zermelo-Fraenkel ZFC set theory or Gödel-Bernays GBC set theory, if these theories are consistent, but it is provable in stronger second-order set theories, such as Kelley-Morse KM set theory. In this talk, I shall discuss the exact strength of this theorem, which turns out to be equivalent to the principle of elementary transfinite recursion ETRord for class recursions on the ordinals. The principle of clopen determinacy for class games, in contrast, is strictly stronger, equivalent over GBC to the full principle of ETR for class recursions over arbitrary class well-founded relations. These results and others mark the beginnings of the emerging subject I call the reverse mathematics of second-order set theory.

The exact strength of the class forcing theorem | Open determinacy for class games

Self reference in computability theory and the universal algorithm, Ouroboros: Formal Criteria of Self-Reference in Mathematics and Philosophy, Bonn, February 2018

This will be a talk for the conference: Ouroboros: Formal Criteria of Self-Reference in Mathematics and Philosophy, held in Bonn, February 16-18, 2018.

Abstract. I shall give an elementary account of the universal algorithm, due to Woodin, 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. Furthermore, the algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. Thus, the algorithm sheds some light on the debate between free will and determinism, if one should imagine extending the universe into a nonstandard time scale. 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.

Ouroboros Bonn 2018 Conference Poster | Slides

The universal algorithm and the universal finite set, Prague 2018

This will be a talk at the Prague Gathering of Logicians & Beauty of Logic 2018, January 25-27, 2018.

Abstract. The universal algorithm is a Turing machine program $e$ that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a $\Sigma_2$ definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. I shall give an account of both results and describe applications to the model theory of arithmetic and set theory.

Slides | Lecture notes

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

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.

The hierarchy of second-order set theories between GBC and KM and beyond

This was a talk at the upcoming International Workshop in Set Theory at the Centre International de Rencontres Mathématiques at the Luminy campus in Marseille, France, October 9-13, 2017.

Hierarchy between GBC and KM

Abstract. Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, strictly between GBC and GBC+$\Pi^1_1$-comprehension; open determinacy for class games, in contrast, is strictly stronger; meanwhile, the class forcing theorem, asserting that every class forcing notion admits corresponding forcing relations, is strictly weaker, and is exactly equivalent to the fragment $\text{ETR}_{\text{Ord}}$ and to numerous other natural principles. What is emerging is a higher set-theoretic analogue of the familiar reverse mathematics of second-order number theory.

Slides