A reverse look at Brouwer’s Fan Theorem
Douglas Bridges · Birkhäuser Basel eBooks · 2008
Classical reverse mathematics is a research programme, started over thirty years ago by Harvey Friedman (Friedman 1975), in which the aim is to place the theorems of large parts (perhaps all?) of mathematics into a number of equivalence classes, the theorems in any given class being equivalent to some natural set-existence principle. Currently, as presented in the compendious text by Simpson (Simpson 1999), five such equivalence classes are used, whose representing set-existence principles can be placed in increasing order of logical complexity. For example, it is known that both the Bolzano-Weierstrass theorem on the extraction of convergent subsequences and the Ascoli-Arzelà theorem on compact subsets of function spaces are equivalent to the second principle in that hierarchy (Simpson 1984), Theorem 4.2). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.