Perfect-Set Properties in L(R)[U]

It is well-known that various forms of the axiom of choice lead to sets of reals with singular properties. One of the most familiar examples is Bernstein's totally imperfect set of reals obtained using a well-ordering of R, i.e., a set of reals X which is neither disjoint nor includes a nonempty perfect set of reals (see [Be]). That some form of AC is needed to get such a set was proved much later by Solovay [So] who produced a model in which every definable set of reals X is Lebesgue measurable, has the property of Baire, and contains a nonempty perfect set whenever uncountable. Here, ``definable'' means that X=[x # R : .(x, r, :)] for some formula . and parameters r # R and : # Ord. Thus, a natural model where the pathology of totally imperfect sets of reals might not occur is the model L(R), the constructible closure of the reals. Solovay constructed his model using a Le vy collapse of an inaccessible cardinal } to the first uncountable ordinal |1 , so it is appropriate to call Article No. AI981752