ANTICHAIN OF ORDINALS IN INTUITIONISTIC SET THEORY

Abstract In classical set theory, the ordinals form a linear chain that we often think of as a very thin portion of the set-theoretic universe. In intuitionistic set theory, however, this is not the case and there can be incomparable ordinals. In this article, we shall show that starting from two incomparable ordinals, one can construct canonical bijections from any arbitrary set to an antichain of ordinals, and consequently any subset of the given set can be defined using ordinals as parameters. This implies the surprising result that in the theory “ upper I upper K upper P plus empty set $\mathrm {IKP} + {}$ I K P + { } there exist two incomparable ordinals,” the statements upper O r d subset of or equal to upper L $\mathrm {Ord} \subseteq L$ O r d ⊆ L and upper V equals upper L $V = L$ V = L are equivalent.