Exploring the abyss in Kleene's computability theory
Sam Sanders
TL;DR
The paper investigates Kleene's higher-order computability, focusing on normal versus non-normal functionals and the computational gulf between $\exists^{2}$ and $\exists^{3}$. By grounding the analysis in quasi-continuity, Baire classes, semi-continuity, and bounded variation, it identifies non-normal functionals that are computable at different oracle levels and proves that seemingly similar notions can inhabit opposite sides of the abyss—e.g., quasi-continuous vs cliquish, Baire 1 vs Baire 2. It also highlights the critical role of representations (e.g., RM/reals) in determining computability boundaries and demonstrates that certain problems (like Cantor realisers) force higher-type power ($\exists^{3}$) while closely related problems can be tamed by $\exists^{2}$. Overall, the work clarifies how the computational content of foundational real-analysis theorems interacts with Kleene's computation schemes, revealing deep separations with potential implications for higher-order analysis and reverse mathematics.
Abstract
Kleene's computability theory based on the 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, the weakest among which seems to be 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 computable in $\exists^3$ but not in weaker oracles. Of course, there is a great divide or abyss separating $\exists^2$ and $\exists^3$ and we identify slight variations of our new non-normal functionals that are again computable in $\exists^2$, i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis.