Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-10T02:25:41.593Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ackermann, W. (1935). Zum Eliminationsproblem der mathematischen Logik. Mathematische Annalen, 111, 61–63.10.1007/BF01472201CrossRefGoogle Scholar
Asser, G. (1981). Einführung in die mathematische Logik , Vol III. Leipzig, Germany: Teubner-Verlag.Google Scholar
Gassner, C. (1985). Das Auswahlaxiom im Prädikatenkalkül zweiter Stufe. Dissertation, Greifswald.Google Scholar
Gassner, C. (1994). The axiom of choice in second-order predicate logic. Mathematical Logic Quarterly, 40, 533–546.10.1002/malq.19940400410CrossRefGoogle Scholar
van Heijenoort, J., ed. (1967). From Frege to Gödel. A Source Book in Mathematical Logic, 1897–1931. Cambridge, MA: Harvard University Press.Google Scholar
Hilbert, D. & Ackermann, W. (1938). Grundzüge der mathematischen Logik. Berlin: Julius Springer Verlag.10.1007/978-3-662-41928-1CrossRefGoogle Scholar
Jech, T. (2003). Set Theory . Berlin: Springer-Verlag.Google Scholar
Kanamori, A. (1997). The mathematical import of Zermelo’s well-ordering theorem. The Bulletin of Symbolic Logic, 3(3), 281–311.10.2307/421146CrossRefGoogle Scholar
Kanamori, A. (2004). Zermelo and set theory. The Bulletin of Symbolic Logic, 10(4), 487–553.10.2178/bsl/1102083759CrossRefGoogle Scholar
Mancosu, P. & Siskind, B. (2019). Neologicist Foundations: Inconsistent abstraction principles and part-whole, In Mras, G. M., Weingartner, P., and Ritter, B., editors. Philosophy of Logic and Mathematics: Proceedings of the 41st International Wittgenstein Symposium. Berlin: De Gruyter, 2019, pp. 215–248.CrossRefGoogle Scholar
Moore, G. H. (1982). Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. New York: Springer-Verlag.10.1007/978-1-4613-9478-5CrossRefGoogle Scholar
Rubin, H. & Rubin, J. E. (1985). Equivalents of the Axiom of Choice, II. Studies in Logic and the Foundations of Mathematics, Vol. 116. Amsterdam: North-Holland.Google Scholar
Sargsyan, G. (2014). Hod mice and the mouse set conjecture. Memoirs of the American Mathematical Society, 236(1111).Google Scholar
Scholz, H. & Hasenjaeger, G. (1961). Grundzüge der mathematischen Logik . Berlin: Springer-Verlag.10.1007/978-3-642-94814-5CrossRefGoogle Scholar
Shapiro, S. (1991). Foundations without Foundationalism. Oxford: Oxford University Press.Google Scholar
Shapiro, S. & Weir, A. (1999). New V, ZF and abstraction. Philosophia Mathematica, 7(3), 293–321.10.1093/philmat/7.3.293CrossRefGoogle Scholar
Steel, J. & Van Wesep, R. (1982). Two consequences of determinacy consistent with choice. Transactions of the American Mathematical Society, 272(1), 67–85.10.1090/S0002-9947-1982-0656481-5CrossRefGoogle Scholar
Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet Werden Kann. Mathematische Annalen, 59(4), 514–516. English translation in van Heijenoort 1967, 139–141.10.1007/BF01445300CrossRefGoogle Scholar
Zermelo, E. (1908). Neuer Beweis für die Möglichkeit einer Wohlordung . Mathematische Annalen, 65, 107–128. English translation in van Heijenoort 1967, 183–198.Google Scholar