Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T02:57:56.381Z Has data issue: false hasContentIssue false
  • English
  • Français

AN EXTENSION OF A THEOREM OF ZERMELO

Published online by Cambridge University Press:  06 March 2019

JOUKO VÄÄNÄNEN
JOUKO VÄÄNÄNEN*
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS PO BOX 64 (GUSTAF HÄLLSTRÖMIN KATU 2) FI-00014 UNIVERSITY OF HELSINKI, FINLAND and INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM SCIENCE PARK107 1098XG AMSTERDAM, NETHERLANDSE-mail: jouko.vaananen@helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that if $(M,{ \in _1},{ \in _2})$ satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is ${ \in _1}$ and also when the membership relation is ${ \in _2}$, and in both cases the formulas are allowed to contain both ${ \in _1}$ and ${ \in _2}$, then $\left( {M, \in _1 } \right) \cong \left( {M, \in _2 } \right)$, and the isomorphism is definable in $(M,{ \in _1},{ \in _2})$. This extends Zermelo’s 1930 theorem in [6].

Keywords

03E7003C35categoricityZermelosecond-order logicset theory
Type
Communications
Information
Bulletin of Symbolic Logic , Volume 25 , Issue 2 , June 2019 , pp. 208 - 212
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

REFERENCES

Dedekind, R., Was sind und was sollen die Zahlen? Vieweg and Sohn, Braunschweig, 1888.Google Scholar
Martin, D., Completeness or incompleteness of basic mathematical concepts (draft), 2018, http://math.ucla.edu/∼dam/booketc/efi.pdf.Google Scholar
Väänänen, J., Second-order logic or set theory? this Bulletin, vol. 18 (2012), no. 1, pp. 91–121.Google Scholar
Väänänen, J., Tracing internal categoricity, Proceedings of the conference “Mapping Traces”, Bogotá (2014), to appear.Google Scholar
Welch, P., Global reflection principles, Logic, Methodology and Philosophy of Science—Proceedings of the 15th International Congress, 2015 (Leitgeb, H., Niiniluoto, I., Seppälä, P., and Sober, E., editors), vol. 97, College Publications, London, 2017, pp. 82–100.Google Scholar
Zermelo, E., Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47.CrossRefGoogle Scholar