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 $\eo$ and the positive integers
and obtain an induced natural well ordering $\prec$ 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
$$(\forall K)(\exists M)(\forall m_0,\ldots,m_M)[ \forall i\leq M(m_i\leq
K+g(i))\rightarrow \exists
i<M(m_i\preceq m_{i+1})].$$
Let $f_\al(i):=i^{H_\al^{-1}(i)}$ where $H_\al^{-1}$ denotes
the functional inverse of the $\al$-th function from the Hardy hierarchy.
Then $$\PA\vdash \CWF(f_\al)\iff \al<\eo.$$