Long Sequences of Descending Theories and other Miscellenia on Slow Consistency∗

For a provably recursive function f : N→ N of PA one can consider the notion of f -consistency for PA, Conf (PA) := ∀xCon(PA f(x)), where PA k denotes the fragment of PA with induction restricted to Σk formulae. It was shown in [8] that for a certain slow growing function f the strength of PA + Conf (PA) lies strictly between PA and PA + Con(PA). Letting τBH be the Bachmann-Howard ordinal, this paper exhibits for every α < τBH, a hierarchy of ever slower functions (fξ)ξ<α of length α such that for ζ < ξ < α one has PA C PA + Confξ(PA) C PA + Confζ (PA) C PA + Con(PA), where T1 C T2 conveys that T2 interprets T1 but T1 does not interpret T2. This confirms a conjecture stated in [8, 3.2.2]. It is also observed that the axioms of Gödel-Löb logic, GL, hold for any provability interpretation embodying slow provability. As a result, one obtains the equivalent of Solovay’s completeness theorem for GL for all of these slow provability notions.