Abolishing Platonism in Multiverse Theories
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ORIGINAL PAPER
Abolishing Platonism in Multiverse Theories Stathis Livadas1 Received: 22 June 2020 / Accepted: 4 November 2020 Springer Nature B.V. 2020
Abstract A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis (CH), by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known settheoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily address ontological multiversism as advocated, for instance, by Hamkins or Va¨a¨ta¨nen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter. Keywords Absoluteness Concept expansion Constituting subjectivity Continuum hypothesis Forcing extension Multiverse Multiverse dependent logic Ontological multiversism Ordinals Subjective interpretation
1 Introduction A key issue in the current debate among set-theorists about the concept of multiverse is the way one may assess the argumentation in favor or against multiversism in set-theory and the mathematical foundations. In a deeper sense this & Stathis Livadas [email protected] 1
Patras, Greece
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Axiomathes
debate concerns the content and the terms in which the conflicting views about multiversism or universism are brought out to the fore. Of course the debate on the mathematical universe vs. multiverse approach has almost nothing to do with the corresponding debate among cosmologists regarding the single vs. multiple universes approach insofar as by mathematical universe one normally understands the conventional set-theoretical universe V in terms of which all meaningful mathematics can be done.1 As to the multiverse approach the question of whether this term stands for a multiplicity of mathematical universes distinct from the conventional one V, the answer depends on the philosophical leanings one might have in terms of a presumed ontological (or other) content of mathematics, something that will be made clear concerning my own philosophical attitude in the next sections. A key motivation in the setting and elaboration of the pro-multiversism arguments is the fact that some set-theoret
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