A Sharp Phase Transition Threshold for Elementary Descent Recursive Functions

Harvey Friedman introduced natural independence results for the Peano axioms via certain schemes of combinatorial well-foundedness. We consider here parameterized versions of this scheme and classify exactly the threshold for the transition from provability to unprovability in PA. For this purpose we fix a natural bijection between the ordinals below ε0 and the positive integers and obtain an induced natural well ordering on the positive integers. We classify the asymptotic of the associated global count functions. Using these asymptotics we classify precisely the phase transition for the parameterized hierarchy of elementary descent recursive functions and hence for the combinatorial well-foundedness scheme. Let CWF(g) be the assertion (∀K)(∃M)(∀m0, . . . , mM )[∀i ≤ M(mi ≤ K+g(i)) → ∃i < M(mi mi+1)]. Let fα(i) := iH−1 α (i) where H−1 α denotes the functional inverse of the α-th function from the Hardy hierarchy. Then PA CWF(fα) ⇐⇒ α < ε0.

• Type: Journal
• Year: 2007
• Where: LOGCOM
• Authors: Arnoud den Boer, Andreas Weiermann
• Keywords: Combinatorial Well-foundedness, Harvey Friedman, LOGCOM 2007, Positive Integers