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A NOTE ON CHOICE PRINCIPLES IN SECOND-ORDER LOGIC

Part of: Set theory

Published online by Cambridge University Press:  25 August 2020

BENJAMIN SISKIND ,
PAOLO MANCOSU  and
STEWART SHAPIRO
BENJAMIN SISKIND*
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720-2390, USA
PAOLO MANCOSU
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720-2390, USA E-mail: mancosu@socrates.berkeley.edu
STEWART SHAPIRO
Affiliation:
DEPARTMENT OF PHILOSOPHY THE OHIO STATE UNIVERSITY 350 UNIVERSITY HALL, 230 NORTH OVAL MALL COLUMBUS, OH 43210, USA E-mail: shapiro.4@osu.edu
*
E-mail: bsiskind@berkeley.edu
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Abstract

Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.

Keywords

second-order logichigher-order logicthe axiom of choice

MSC classification

Secondary: 03E25: Axiom of choice and related propositions 03E70: Nonclassical and second-order set theories
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 2 , June 2023 , pp. 339 - 350
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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