By now there is a long list of questions in analysis and algebra which are known to be undecided in the standard set theory (Zermelo-Fraenkel). In particular, no standard methods accepted and used by mathematicians can provide a proof deciding such questions. Yet, a definitive answer is often desirable. I will discuss some axioms that settle most of these open questions, provide useful extensions of standard set theory, and are intersting on their own. These axioms rely on the existence of sets that are significantly "larger" than any sets mainstream mathematics works with.