The rearrangement and subseries numbers: how much convergence suffices for absolute convergence? Mathematics Colloquium, University of Münster, January 2019

This will be a talk for the Mathematics Colloquium at the University of Münster, January 10, 2019.

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 subseries number is defined similarly, as the smallest number of subseries whose convergence suffices to test a series for absolute convergence. The exact values of the rearrangement and subseries numbers turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement and subseries numbers 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, Joerg Brendle, 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},
      volume = {},
      number = {},
      pages = {},
      month = {},
      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-rearrangement-number},
      eprint = {1612.07830},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
      source = {},
      }

  • 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 = {},
      pages = {},
      month = {},
      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-subseries-number},
      eprint = {1801.06206},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
      source = {},
      }

An infinitary-logic-free proof of the Barwise end-extension theorem, with new applications, University of Münster, January 2019

This will be a talk for the Logic Oberseminar at the University of Münster, January 11, 2019.

Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of ZF has an end-extension to a model of ZFC + V=L. This theorem is both (i) a technical culmination of Barwise’s pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory. The new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal $\Sigma_1$-definable finite set, a set-theoretic version of Woodin’s universal algorithm.

A new proof of the Barwise extension theorem, without infinitary logic, CUNY Logic Workshop, December 2018

I’ll be back in New York from Oxford, and this will be a talk for the CUNY Logic Workshop, December 14, 2018.

Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of ZF has an end-extension to a model of ZFC + V=L. This theorem is both (i) a technical culmination of Barwise’s pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory. The new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal $\Sigma_1$-definable finite set, a set-theoretic version of Woodin’s universal algorithm.

My lecture notes are available.

Faculty respondent to paper of Ethan Jerzak on Paradoxical Desires, Oxford Graduate Philosophy Conference, November 2018

The Oxford Graduate Philosophy Conference will be held at the Faculty of Philosophy November 10-11, 2018, with graduate students from all over the world speaking on their papers, with responses and commentary by Oxford faculty.

I shall be the faculty respondent to the delightful paper, “Paradoxical Desires,” by Ethan Jerzak of the University of California at Berkeley, offered under the following abstract.

Ethan Jerzak (UC Berkeley): Paradoxical Desires
I present a paradoxical combination of desires. I show why it’s paradoxical, and consider ways of responding to it. The paradox saddles us with an unappealing disjunction: either we reject the possibility of the case by placing surprising restrictions on what we can desire, or we revise some bit of classical logic. I argue that denying the possibility of the case is unmotivated on any reasonable way of thinking about propositional attitudes. So the best response is a non-classical one, according to which certain desires are neither determinately satisfied nor determinately not satisfied. Thus, theorizing about paradoxical propositional attitudes helps constrain the space of possibilities for adequate solutions to semantic paradoxes more generally.

The conference starts with coffee at 9:00 am.  This session runs 11 am to 1:30 pm on Saturday 10 November in the Lecture Room.

Conference Program | Conference web page

Here are the notes I used for my response.

 

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).

Slides

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.

Slides

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},
      volume = {},
      number = {},
      pages = {},
      month = {},
      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-rearrangement-number},
      eprint = {1612.07830},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
      source = {},
      }

  • 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 = {},
      pages = {},
      month = {},
      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-subseries-number},
      eprint = {1801.06206},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
      source = {},
      }

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