Second-order Logic and the Power Set

Ethan BRAUER*

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)

1 Scopus Citations

Abstract

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments.

Original languageEnglish
Pages (from-to)123-142
Number of pages20
JournalJournal of Philosophical Logic
Volume47
Issue number1
Early online date19 Dec 2016
DOIs
Publication statusPublished - 1 Feb 2018
Externally publishedYes

Keywords

  • Axiom of separation
  • Entanglement of logic and mathematics
  • Jane
  • Power set
  • Second-order logic
  • Set theory

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