The non-normal abyss in Kleene's computability theory
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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 ∃^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 ∃^2 while the former are only computable in ∃^3. While there is a great divide separating ∃^2 and ∃^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.
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