Reconsidering Ordered Pairs
Dana Scott, Dominic McCarty · Bulletin of Symbolic Logic · 2008
Abstract 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 arecursivedefinition 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 twoinfinite setsunder the new definition turns out to be equal to the maximum of the ranks of the sets.