Antichain of ordinals in intuitionistic set theory
Shuwei Wang
TL;DR
The paper shows that intuitionistic set theory can admit incomparable ordinals and provides a canonical, $\Sigma$-definable method to encode any set as an antichain of ordinals starting from two such ordinals. By iterating and consolidating this encoding, it demonstrates that, assuming $\exists \alpha \perp \beta \in \mathrm{Ord}$, the statements $\mathrm{Ord} \subseteq L$ and $V = L$ are equivalent in IKP, with the construction remaining valid inside $L$ thanks to $L$’s definability and a key absoluteness result. This reveals a dramatic contrast with classical set theory and offers a route to translate questions about $L$-definability into statements about arbitrary sets. The work also discusses implications for the consistency of anti-classical axioms and the challenges of building intuitionistic $V \neq L$ models, highlighting an open problem in the field.
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.