  We show that it is provable in PA that there is an arithmetically definable
sequence $\{\phi_{n}:n \in \omega\}$ of $\Pi^{0}_{2}$-sentences, such that
  - PRA+$\{\phi_{n}:n \in \omega\}$ is $\Pi^{0}_{2}$-sound and
$\Pi^{0}_{1}$-complete
  - the length of $\phi_{n}$ is bounded above by a polynomial function of $n$
with positive leading coefficient
  - PRA+$\phi_{n+1}$ always proves 1-consistency of PRA+$\phi_{n}$.
  One has that the growth in logical strength is in some sense "as fast as
possible", manifested in the fact that the total general recursive functions
whose totality is asserted by the true $\Pi^{0}_{2}$-sentences in the sequence
are cofinal growth-rate-wise in the set of all total general recursive
functions. We then develop an argument which makes use of a sequence of
sentences constructed by an application of the diagonal lemma, which are
generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction
as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth
in Mathematics". The argument establishes the result that it is provable in PA
that $P \neq NP$. We indicate how to pull the argument all the way down into
EFA.
