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LOGCOM
2007

A Sharp Phase Transition Threshold for Elementary Descent Recursive Functions

9 years 11 months ago
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.
Arnoud den Boer, Andreas Weiermann
Added 16 Dec 2010
Updated 16 Dec 2010
Type Journal
Year 2007
Where LOGCOM
Authors Arnoud den Boer, Andreas Weiermann
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