Catagory, Measure, Inductive Inference: A Triality Theorem and its Applications
Carl H. Smith

Abstract:
The famous Sierpinski­Erd¨os Duality Theorem states, informally, that any 
theorem about effective measure 0 and/or first category sets is also true 
when all occurrences of ``effective measure 0'' are replaced by ``first 
category'' and vice versa. This powerful and nice result shows that 
``measure'' and ``category'' are equally useful notions neither of which 
can be preferred to the other one when making formal the intuitive notion 
``almost all sets.'' Effective versions of measure and category are used 
in recursive function theory and related areas, and resource­bounded 
versions of the same notions are used in Theory of Computation. Again they 
are dual in the same sense.
We show that in the world of recursive functions there is a third equipotent 
notion dual to both measure and category. This new notion is related to 
learnability (also known as inductive inference or identifiability). We use 
the term ``triality'' to describe this three­party duality.