arXiv subjects

Øystein Linnebo

Publications and source records attributed to Øystein Linnebo.

3 recordsLinked to original sources

arXiv2026

Strict potentialism in modal mirrors

Potentialism is the view that objects are successively generated in an incompletable process. A strict version of the view adds that truths are successively determined. Strict potentialism can be analyzed using two modalities: one for the generation of objects, another for truths becoming determined. The result is a classical bimodal logic. We obtain simpler and more user-friendly theories by invoking so-called mirroring theorems to ``switch off'' one or both modalities, in return for a less classical logic. When the modality of object generation is switched off, we obtain a restricted plural logic. When the modality of truth determination is switched off, the logic becomes intuitionistic. Finally, the value of this general approach to strict potentialism is illustrated by applications to a Weyl-inspired predicative set theory, Cantor's domain principle, and strict potentialism about Cantorian sets.

math.LO↗

arXiv2026

Feasible constructivism

Dummett's argument for intuitionism is well known. There is a concern that the argument proves too much, specifically, that it supports the extreme and apparently incoherent position of strict finitism. The central question is how to explicate the notion that it is possible in practice to construct an arithmetical term or verify a statement. The strict finitist answer is plagued by the sorites paradox. We propose and develop feasibilism as a more plausible view, where computational feasibility, as captured by the class of polynomial-time problems, yields a robust and expedient explication of "possible in practice". In this approach, the complexity is bounded by a polynomial function of the input size, rather than bounded by a constant (as in strict finitism), thus resolving the sorites issues. We show that a system of strictly bounded arithmetic, introduced by Sam Buss, precisely formalizes the feasibilist view so as to satisfy Dummett's requirements.

math.LO↗

arXiv2018

The modal logic of set-theoretic potentialism and the potentialist maximality principles

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe, including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_β$); Grothendieck-Zermelo potentialism (true in all larger $V_κ$ for inaccessible cardinals $κ$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

math.LO↗