We consider Borel sets of the form A ! (with usual topology) where cardinality of is less than some uncountable regular cardinal. We obtain a \normal form" of A, by nding a Borel set () such that A and () continuously reduce to each other. We do so by deening Borel operations which are homomorphic to the rst Veblen ordinal functions of base required to compute the Wadge degree of the set A: the...

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t > 0 we show that in this category there exists a unique free Borel system (Y, S) which is strictly t-universal in the sense that all invariant me...

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable or affine half-spaces.

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinator...

We give a completely constructive solution to Tarski’s circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k ≥ 1 and A,B ⊆ Rk are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answ...

Let S and T be hyperfinite sets in an N]-saturated nonstandard universe. The following are equivalent: m 1*1 ~ l (l) rff ~ • • (ii) There is a bijection from 5 onto T whose graph is Borel (over the internal subsets of S x T). This follows from somewhat more general results about analytic partial functions on hyperfinite sets, the proofs of which use Choquet's theorem on the capacitibility of an...

We give several refinements of known theorems on Borel uniformizations of sets with ”large sections”. In particular, we show that a set B ⊂ [0, 1] × [0, 1] which belongs to Σ0α, α ≥ 2, and which has all ”vertical” sections of positive Lebesgue measure, has a Π0α uniformization which is the graph of a Σ0α-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollar...

§0 Preface This paper is part of a project to obtain a structure theory for the simplest and most general objects of mathematics. I wish to consider the definable cardinalities that arise from the continuous actions of Polish groups. The philosophy is to calculate cardinalities using only sets and functions that are in some sense reasonably definable. As with [16] and [5], the study of definabl...