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Reconsidering Ordered Pairs

  • Dana Scott (a1) and Dominic McCarty (a2)

The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets (x,y) = {{x}, {x,y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater length. There are many advantages to the new definition, for it allows for uniform definitions working equally well in a wide range of models for set theories. In ZFC and closely related theories, the rank of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets.

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[3] Jon Barwise , Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, 1975.

[4] John L. Bell , Set theory: Boolean-valued models and independence proofs, third ed., Oxford Logic Guides, vol. 47, Oxford University Press, 2005.

[5] George Boolos , On the semantics of the constructible levels, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, (1970), pp. 139148.

[7] Kurt Gödel , The consistency of the axiom of choice and of the generalised continuum hypothesis, Annals of Mathematical Studies, 1940.

[8] Nelson Goodman , Sequences, The Journal of Symbolic Logic, vol. 6 (1941), pp. 150153.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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