Slow consistency

The fact that “natural” theories, i.e. theories which have something like an “idea” to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of mathematics. However, one easily establishes the existence of theories with incomparable logical strengths using self-reference (Rosser-style). As a result, PA+Con(PA) is not the least theory whose strength is greater than that of PA. But still we can ask: is there a sense in which PA+Con(PA) is the least “natural” theory whose strength is greater than that of PA? In this paper we exhibit natural theories in strength strictly between PA and PA+Con(PA) by introducing a notion of slow consistency.