Title: Relative randomness and cardinality

Abstract: 

Given two oracles A,B we say that A is weaker than B
as an oracle for testing randomness if every sequence that is fails a
randomness test relative to A, it fails a randomness test relative to
B. By a theorem of Joe Miller this is equivalent to saying that B can
compress (i.e. give short descriptions of) sequences at least as well as A
can.

A fundamental question on relative randomness is, given an oracle B,
how many weaker oracles than B exist?

This is a Borel class, so it is either countable or as big as the set
of reals. We will survay the work that has been done toward solving this
question and present a new definitive result for the case where B has a 
\Delta^0_2 definition. We show that in this case there are uncountably 
many oracles which are weaker than B, unless
B is trivial (i.e. no better for testing randomness than a computable 
oracle), in which case it is known to be countable. It follows that 
\Delta^0_2 is the largest arithmetical class with this property.
Moreover, if B is not trivial then the class contains a perfect
\Pi^0_1 set of reals.