2011/06/14 by Wim Veldman, Veldman, Wim
Computer Science · Mathematics · #03F55 03F60 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Numerical Methods and Algorithms
paper · pdf · doi:10.48550/arxiv.1106.2738
openalex publication_date 2011/06/14 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
The paper is a contribution to intuitionistic reverse mathematics. We\nintroduce a formal system called Basic Intuitionistic Mathematics BIM, and then\nsearch for statements that are, over BIM, equivalent to Brouwer's Fan Theorem\nor to its positive denial, Kleene's Alternative to the Fan Theorem. The Fan\nTheorem is true under the intended intuitionistic interpretation and Kleene's\nAlternative is true in the model of BIM consisting of the Turing-computable\nfunctions. The task of finding equivalents of Kleene's Alternative is,\nintuitionistically, a nontrivial extension of finding equivalents of the Fan\nTheorem, although there is a certain symmetry in the arguments that we shall\ntry to make transparent.\n We introduce closed-and-separable subsets of Baire space and of the set of\nthe real numbers. Such sets may be compact and also positively noncompact. The\nFan Theorem is the statement that Cantor space, or, equivalently, the unit\ninterval, is compact, and Kleene's Alternative is the statement that Cantor\nspace, or, equivalently, the unit interval is positively noncompact. The class\nof the compact closed-and-separable sets and also the class of the\nclosed-and-separable sets that are positively noncompact are characterized in\nmany different ways and a host of equivalents of both the Fan Theorem and\nKleene's Alternative is found.\n