Randomness and Reality

As he indicated, Benioff takes a random outcome sequence, for the purposes of the first premiss, to be one that avoids every Borel set (of sequences or reals) of measure zero (on a fixed probability measure) that is definable in the language of ZF (with one fixed parameter, interpreted as the probability measure). (Equivalently, a Benioffrandom sequence must belong to every such Borel set of measure 1.) Below I will have more to say concerning this definition. For now, let us note the motivation: a measure 0 Borel set is very special, its members are highly improbable. Intuitively, we would like to say that a random sequence avoids any such special set, or, equivalently, that it possesses every measure 1 property, often called a "property of randomness". The trouble with this, however, is that no sequence could