Can there be natural instances of nonlinearity in the hierarchy of consistency strength? UWM Logic Seminar, January 2021

This is a talk for the University of Wisconsin, Madison Logic Seminar, 25 January 2020 1 pm (7 pm UK).

The talk will be held online via Zoom ID: 998 6013 7362.

Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to overcome that criticism—as well as I am able to—by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.

The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.

Set-theoretic and arithmetic potentialism: the state of current developments, CACML 2020

This will be a plenary talk for the Chinese Annual Conference on Mathematical Logic (CACML 2020), held online 13-15 November 2020. My talk will be held 14 November 17:00 Beijing time (9 am GMT).

Potentialist perspectives

Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspective—a statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.

A new proof of the Barwise extension theorem, and the universal finite sequence, Barcelona Set Theory Seminar, 28 October 2020

This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.

Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1 definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism.  This work is joint with Kameryn J. Williams.

Article: The $\Sigma_1$-definable universal finite sequence

  • J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” ArXiv e-prints, 2019. (Under review)  
    @ARTICLE{HamkinsWilliams:The-universal-finite-sequence,
    author = {Joel David Hamkins and Kameryn J. Williams},
    title = {The $\Sigma_1$-definable universal finite sequence},
    journal = {ArXiv e-prints},
    year = {2019},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    eprint = {1909.09100},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    source = {},
    doi = {},
    }

The $\Sigma_1$-definable universal finite sequence

  • J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” ArXiv e-prints, 2019. (Under review)  
    @ARTICLE{HamkinsWilliams:The-universal-finite-sequence,
    author = {Joel David Hamkins and Kameryn J. Williams},
    title = {The $\Sigma_1$-definable universal finite sequence},
    journal = {ArXiv e-prints},
    year = {2019},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    eprint = {1909.09100},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    source = {},
    doi = {},
    }

Abstract. We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

  • The universal algorithm,
    • J. D. Hamkins and H. W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (Manuscript under review)  
      @ARTICLE{HamkinsWoodin:The-universal-finite-set,
      author = {Joel David Hamkins and W. Hugh Woodin},
      title = {The universal finite set},
      journal = {ArXiv e-prints},
      year = {2017},
      volume = {},
      number = {},
      pages = {1--16},
      month = {},
      note = {Manuscript under review},
      abstract = {},
      keywords = {under-review},
      source = {},
      doi = {},
      eprint = {1711.07952},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      url = {http://jdh.hamkins.org/the-universal-finite-set},
      }
  • The modal logic of arithmetic potentialism,
    • J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” ArXiv e-prints, pp. 1-35, 2018. (Under review)  
      @ARTICLE{Hamkins:The-modal-logic-of-arithmetic-potentialism,
      author = {Joel David Hamkins},
      title = {The modal logic of arithmetic potentialism and the universal algorithm},
      journal = {ArXiv e-prints},
      year = {2018},
      volume = {},
      number = {},
      pages = {1--35},
      month = {},
      eprint = {1801.04599},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      note = {Under review},
      url = {http://wp.me/p5M0LV-1Dh},
      abstract = {},
      keywords = {under-review},
      source = {},
      doi = {},
      }
  • A new proof of the Barwise extension theorem
  • Kameryn’s blog post about the paper


Computational self-reference and the universal algorithm, Queen Mary University of London, June 2019

This will be a talk for the Theory Seminar for the theory research group in Theoretical Computer Science at Queen Mary University of London. The talk will be held 4 June 2019 1:00 pm, ITL first floor.

Abstract. Curious, often paradoxical instances of self-reference inhabit deep parts of computability theory, from the intriguing Quine programs and Ouroboros programs to more profound features of the Gödel phenomenon. In this talk, I shall give an elementary account of the universal algorithm, 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. In this sense, every function becomes computable, computed all by the same universal program, if only it is run in the right world. Furthermore, the universal algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. 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.

The modal logic of potentialism, ILLC Amsterdam, May 2019

This will be a talk at the Institute of Logic, Language and Computation (ILLC) at the University of Amsterdam for events May 11-12, 2019. See Joel David Hamkins in Amsterdam 2019.

Job Adriaenszoon Berckheyde [Public domain]

Abstract: Potentialism can be seen as a fundamentally model-theoretic notion, in play for any class of mathematical structures with an extension concept, a notion of substructure by which one model extends to another. Every such model-theoretic context can be seen as a potentialist framework, a Kripke model whose modal validities one can investigate. In this talk, I’ll explain the tools we have for analyzing the potentialist validities of such a system, with examples drawn from the models of arithmetic and set theory, using the universal algorithm and the universal definition.