The non-normal abyss in Kleene's computability theory

Abstract

Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier ${\exists }^n$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied, based on well-known theorems like the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from ${\exists }^2$ while the former are only computable in ${\exists }^3$. While there is a great divide separating ${\exists }^2$ and ${\exists }^3$, we identify certain closely related non-normal functionals that fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, and semi-continuity.

12 pages. arXiv admin note: text overlap with arXiv:2210.05251