Does Normal Mathematics Need New Axioms ?

We present a range of mathematical theorems whose proofs require unexpectedly strong logical methods, which in some cases go well beyond the usual axioms for mathematics. ************************************************ There are a variety of mathematical results that can only be obtained by using more than the usual axioms for mathematics. For several decades there has been a gradual accumulation of such results that are more and more concrete, more and more connected with standard mathematical contexts, and more and more relevant to ongoing mathematical activity. Probably the most well known mathematical problem that cannot be proved or refuted with the usual axioms (ZFC) is the continuum hypothesis that every set of real numbers is either countable or of cardinality the continuum (Kurt Gödel and Paul Cohen). But mathematicians have instinctively learned to hide from this kind of problem by focusing on relatively “concrete” subsets of complete separable metric spaces. In particular, the Borel measurable sets and functions in and between complete separable metric spaces proves to be a natural boundary. By way of illustration, “every Borel set of real numbers is either countable or of cardinality the continuum via a Borel measurable function” is a well known theorem of descriptive set theory.