N ov 1 99 6 Orbit cardinals : On the definable cardinalities of quotient spaces of the form X / G , where G acts on a Polish space X

§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 definable cardinaliites is intended to be an abstract investigation of classification problems, in that we may say that the classification of the equivalence relation E on X is harder than the classification of F on Y if the definable cardinality of X/E exceeds that of Y/F . Of course the notion of reasonably definable is vague and subject to personal taste and prejudice. I will choose to explicate this notion by taking perhaps the most generous definition in wide currency. For me, the reasonably definable sets are those that appear in L(R), the universe of all objects that arise from transfinite operations applied to R. This may very well be too liberal for some, and an alternative approach would be to restrict ourselves to say the Borel sets, thereby giving us the notion of Borel cardinality; alternatively we may diet on the sets and functions arising in the σ-algebra generated by the open sets and closed under continuous images. For most of the problems considered below the structure one obtains for the cardinals in L(R) closely resembles that suggested by the Borel sets and functions. Indeed, under the assumption of AD, the universe of L(R) fills out the sketch outlined for us by the Borel sets, providing a canonical model of ZF where every set of reals has the regularity properties such as being Lebesgue measurable and the cardinal structure plays out the suggestions made by the Borel equivalence relations. It should be stressed that L(R) is a model of ZF, but not of choice. Thus not every set can be wellordered, and consequently not every cardinal corresponds to an ordinal. For instance, the cardinality of 20 is not an ordinal in L(R) – just as there is no Borel wellordering of R in ZFC. Morever, the existence of a surjection π : A → B does not guarantee that |A|, the cardinality of A, is less than the cardinality of B, in the sense of there being an injection from B to A. For instance,