Tag Archives: universal program

Potentialism and implicit actualism in the foundations of mathematics, Jowett Society lecture, Oxford, February 2019

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This will be a talk for the Jowett Society on 8 February, 2019. The talk will take place in the Oxford Faculty of Philosophy, 3:30 – 5:30pm, in the Lecture Room of the Radcliffe Humanities building.

Abstract. Potentialism is the view, originating in the classical dispute between actual and potential infinity, that one’s mathematical universe is never fully completed, but rather unfolds gradually as new parts of it increasingly come into existence or become accessible or known to us. Recent work emphasizes the modal aspect of potentialism, while decoupling it from arithmetic and from infinity: the essence of potentialism is about approximating a larger universe by means of universe fragments, an idea that applies to set-theoretic as well as arithmetic foundations. The modal language and perspective allows one precisely to distinguish various natural potentialist conceptions in the foundations of mathematics, whose exact modal validities are now known. Ultimately, this analysis suggests a refocusing of potentialism on the issue of convergent inevitability in comparison with radical branching. I shall defend 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 surprising parallels to the case of arithmetic potentialism.

Jowett Society talk entry | my posts on potentialism | Slides

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

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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

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

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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

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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.