English

Polymorphic Ordinal Notations

Logic 2025-06-23 v2

Abstract

We give an alternative presentation of the ordinal notation at the strength of Π11CA0\Pi^1_1-CA_0 which allows the "uncountable" notation Ω\Omega to be interpreted "polymorphically" - that is, we allow the notation to be interpreted as different cardinals depending on their context. This gives us a way to represent functions on ordinals within our ordinal notation system. We then use this idea to present an ordinal notation system for a system a bit weaker than parameter-free Π21CA0\Pi^1_2-CA_0.

Keywords

Cite

@article{arxiv.2504.02131,
  title  = {Polymorphic Ordinal Notations},
  author = {Henry Towsner},
  journal= {arXiv preprint arXiv:2504.02131},
  year   = {2025}
}