# 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.

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.

# Is there just one mathematical universe? DRIFT, Amsterdam, May 2019

This will be a talk for the Wijsgerig Festival DRIFT 2019, held in Amsterdam May 11, 2019. The theme of the conference is: Ontology.

Abstract. What does it mean to make existence assertions in mathematics?
Is there a mathematical universe, perhaps an ideal mathematical reality, that the assertions are about? Is there possibly more than one such universe? Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic worlds.

# Ansten Mørch-Klev

Ansten Mørch-Klev earned his M.Sc. degree under my direction at Universiteit van Amsterdam in July, 2007.   For his thesis, Ansten undertook to investigate the infinite-time analogue of Kleene’s $\mathcal{O}$, the natural extension of Kleene’s concept to the case of infinite time Turing machines.  The result was a satisfying and robust theory, which revealed (as predicted by Philip Welch) the central importance of the eventually writable ordinals in the theory of infinite time computability.  This work eventually appeared as:  Ansten Mørch-Klev, “Infinite time analogues of Kleene’s $\mathcal{O}$,” Archive for Mathematical Logic, 48(7):2009, p. 691-703, DOI:10.1007/s00153-009-0146-2.

Ansten Mørch Klev

Ansten Mørch-Klev, “Extending Kleene’s O Using Infinite Time Turing Machines, or How With Time She Grew Taller and Fatter”, M.Sc. thesis for Institute of Logic, Language and Computation, Universiteit van Amsterdam, July, 2007.  ILLC publication

Abstract.  We define two successive extensions of Kleene’s $\mathcal{O}$ using infinite time Turing machines. The first extension, $\mathcal{O}^+$, is proved to code a tree of height $\lambda$, the supremum of the writable ordinals, while the second extension, $\mathcal{O}^{++}$, is proved to code a tree of height $\zeta$, the supremum of the eventually writable ordinals. Furthermore, we show that $\mathcal{O}^+$ is computably isomorphic to $h$, the lightface halting problem of infinite time Turing machine computability, and that $\mathcal{O}^{++}$ is computably isomorphic to $s$, the set of programs that eventually writes a real. The last of these results implies, by work of Welch, that $\mathcal{O}^{++}$ is computably isomorphic to the $\Sigma_2$ theory of $L_\zeta$, and, by work of Burgess, that $\mathcal{O}^{++}$ is complete with respect to the class of the arithmetically quasi-inductive sets. This leads us to conjecture the existence of a parallel of hyperarithmetic theory at the level of $\Sigma_2(L_\zeta)$, a theory in which $\mathcal{O}^{++}$ plays the role of $\mathcal{O}$, the arithmetically quasi-inductive sets play the role of $\Pi^1_1$, and the eventually writable reals play the role of $\Delta^1_1$.