Russell's typicality as another randomness notion

Abstract

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-L\"{o}f (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of ``small'' cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain $M$ satisfies $cf(|M|)>\aleph_0$, then there exist $|M|$ typical elements and only $<|M|$ non-typical ones. In particular this is true for the standard model ${\cal R}$ of second-order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to $\subseteq$) with the classes of ML-random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways).