The strength of compactness in Computability Theory and Nonstandard Analysis

Abstract

Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space $2^{\mathbb{N}}$ for uncountable covers. The most basic question is: how hard is it to compute a finite sub-cover from such a cover of $2^{\mathbb{N}}$? Another natural question is: how hard is it to compute a sequence that covers $2^{\mathbb{N}}$ minus a measure zero set from such a cover? The special and weak fan functionals respectively compute such finite sub-covers and sequences. In this paper, we establish the connection between these new fan functionals on one hand, and various well-known comprehension axioms on the other hand, including arithmetical comprehension, transfinite recursion, and the Suslin functional. In the spirit of Reverse Mathematics, we also analyse the logical strength of compactness in Nonstandard Analysis. Perhaps surprisingly, the results in the latter mirror (often perfectly) the computational properties of the special and weak fan functionals. In particular, we show that compactness (nonstandard or otherwise) readily brings us to the outer edges of Reverse Mathematics (namely $\Pi_2^1$-CA$_0$), and even into Schweber's higher-order framework (namely $\Sigma_{1}^{2}$-separation).