On the computational properties of basic mathematical notions
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We investigate the computational properties of basic mathematical notions pertaining to ℝ→ℝ-functions and subsets of ℝ, like finiteness, countability, (absolute) continuity, bounded variation, suprema, and regularity. We work in higher-order computability theory based on Kleene's S1-S9 schemes. We show that the aforementioned italicised properties give rise to two huge and robust classes of computationally equivalent operations, the latter based on well-known theorems from the mainstream mathematics literature. As part of this endeavour, we develop an equivalent λ-calculus formulation of S1-S9 that accommodates partial objects. We show that the latter are essential to our enterprise via the study of countably based and partial functionals of type 3. We also exhibit a connection to infinite time Turing machines.
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