Antichain of ordinals in intuitionistic set theory

TL;DR

This paper explores the structure of ordinals in intuitionistic set theory, showing that incomparable ordinals can generate complex set-theoretic constructs, leading to surprising equivalences in certain theories.

Contribution

It demonstrates that in intuitionistic set theory, incomparable ordinals can be used to define any subset of a set, revealing new structural properties.

Findings

In classical set theory, ordinals form a linear chain.

In intuitionistic set theory, ordinals can be incomparable.

In theories with incomparable ordinals, certain set-theoretic statements are equivalent.

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 paper, 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 "$\mathrm{IKP} + {}$there exist two incomparable ordinals", the statements $\mathrm{Ord} \subseteq L$ and $V = L$ are equivalent.