Lifting countable to uncountable mathematics
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Ordinary, i.e.non-set theoretic mathematics is generally formalised in second-order arithmetic, a direct descendant of Hilbert-Bernays' Grundlagen der Mathematik with a language restricted to the countable: only natural numbers and sets thereof are directly given, while uncountable objects are available indirectly via countable representations. For instance, Turing's computational framework, nowadays called computability theory, and the associated program Reverse Mathematics (essentially) take place in second-order arithmetic. The aim of this paper is to lift some of these 'countable' results to uncountable mathematics. In particular, we show that with little modification recursive counterexamples from computability theory and reversals from Reverse Mathematics, second-order/countable as they may be, yield interesting results in higher-order/uncountable mathematics. We shall treat the following topics/theorems: the monotone convergence theorem/Specker sequences, compact and closed sets in metric spaces, the Rado selection lemma, the ordering and algebraic closures of fields, and ideals of rings.
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