Investigating separation, reduction and uniformization

The project will examine definable subsets of the real numbers with regard to three structurally interesting properties: separation, reduction and uniformization. For the sake of brevity, only the uniformization property will be presented here. Subsets of the real numbers often appear in mathematical considerations. These sets are usually not arbitrary objects but are definable, in the sense that their elements must satisfy certain precise formulas. The formulas can be simpler or more complicated, which in turn results in a hierarchy of definable subsets of the real numbers, ordered according to the complexity of their definitions. One begins with subsets that are very easy to define (e.g. with intervals, lines, planes, etc.) and works one`s way step by step to objects that are increasingly difficult to describe. One now says that a definable, two-dimensional set A of complexity n can be uniformized if, for every given real number x, if there exists at least one number y such that the pair (x,y) lies in A, then such a y can actually be found, using a formula of complexity n. In other words, if A can be uniformized, then one can conclude from the mere existence of a y such that (x,y) is in A that such a y can also be effectively found, using a formula of the same complexity as the set A. Since it is often crucial in mathematics not only to know that there must be numbers with certain properties, but also to specify such numbers concretely, uniformization is very important in many areas. The uniformization property (of complexity n) relevant to this project states that all sets A of complexity n can be uniformized, and the question of whether there are such sets n such that the uniformization property of complexity n is valid has proven to be very fruitful since it was asked almost a hundred years ago. As a result, mathematicians have found a large number of deep and surprising results. We now know that the question of the validity of the uniformization property is an undecidable statement in the Gödelian sense. More precisely, this means that there are different mathematical universes that have different opinions about the validity of the uniformization property. And furthermore, we know that assuming gigantic infinities (so-called large cardinals), this undecidability disappears. In this case, a clear, complete picture emerges of how the uniformization property behaves. Despite this work, many fundamental questions are still open. This project aims to investigate some of them. Two exemplary examples should be mentioned. First, is it possible to construct universes that provide complete pictures of the uniformization property other than the picture produced by large cardinals? Second, are large cardinals necessary to provide certain pictures?