Using the axiom of choice one can construct sets of reals which are pathological in some sense. Similar constructions can be produce such ”pathological” subsets of any non trivial Polish space (= a complete separable metric space). Typical examples of ”pathology” is non mea- surability, lacking the property of Baire (=not equivalent to an open set modulo meager set), contradicting an infinitary versions of Ramsey theorem , etc.
Another important example is the non determinacy of infinitary games. A subset of the Baire space NN is ”pathological” if in the associated game no player has a winning strategy.

A prevailing paradigm in Descriptive Set Theory is that sets that has a ”simple description” should not be pathological. Evidence for this maxim is the fact that Borel sets are not pathological in any of the senses described above.In this talk we shall present a notion of ”super regularity” for subsets of a Polish space, the family of universally Baire sets.

The universally Baire sets typically do not show the ”pathologies” we listed above, especially if one assumes the existence of large cardi- nals.Also the large cardinals imply that the family of Universally Baire sets is much richer that the class of Borel sets.

The intuitive principle ”try to minimize the family of pathological set of reals ” is fruitful guiding principle for extending the set of axioms for Set Theory . It is connected to the guiding principle of strong axioms of infinity. We shall make some remarks how these principles can shed some light on the Continuum problem.

The talk should be accessible to a wide mathematical audience .