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 . As a result, PA+Con 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 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 by introducing a notion of slow consistency. (shrink)

In 2011, Hibbard suggested an intelligence measure for agents who compete in an adversarial sequence prediction game. We argue that Hibbard’s idea should actually be considered as two separate ideas: first, that the intelligence of such agents can be measured based on the growth rates of the runtimes of the competitors that they defeat; and second, one specific (somewhat arbitrary) method for measuring said growth rates. Whereas Hibbard’s intelligence measure is based on the latter growth-rate-measuring method, we survey other methods (...) for measuring function growth rates, and exhibit the resulting Hibbard-like intelligence measures and taxonomies. Of particular interest, we obtain intelligence taxonomies based on Big-O and Big-Theta notation systems, which taxonomies are novel in that they challenge conventional notions of what an intelligence measure should look like. We discuss how intelligence measurement of sequence predictors can indirectly serve as intelligence measurement for agents with Artificial General Intelligence (AGIs). (shrink)