Why Believe Infinite Sets Exist?

Axiomathes 28 (4):447-460 (2018)

Authors
Andrei Mărăşoiu
University of Virginia
Abstract
The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s :481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, and a theory of objects that theory calls “sets”. While Dedekind’s argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored?
Keywords infinite sets  Dedekind
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DOI 10.1007/s10516-018-9375-5
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References found in this work BETA

Measuring Coherence.Igor Douven & Wouter Meijs - 2007 - Synthese 156 (3):405 - 425.
Blind Reasoning.Paul Boghossian - 2003 - Aristotelian Society Supplementary Volume 77 (1):225–248.
Blind Reasoning.Paul Boghossian & Timothy Williamson - 2003 - Proceedings of the Aristotelian Society, Supplementary Volumes( 77:225-293.
Blind Reasoning.Paul Boghossian - 2003 - Supplement to the Proceedings of the Aristotelian Society 77 (1):225-248.

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