[FOM] Re: 218:Unexpected Pi01 statements

[Timothy Y. Chow]

Harvey Friedman <friedman at math.ohio-state.edu> wrote:
>This simplifies posting #217 in a particularly satisfying way.
>We present a new explicitly P01 statement provable from Mahlo cardinals 
>of finite order but not from ZFC. We first state an infinite form.

Could you give some indication of what is going on "behind the scenes"  
here?  In your Annals paper, you gave a very enlightening sketch of a
strategy for using large cardinals to obtain finitary results: Define a
notion of completion; show that if a completion of a finite structure
contains a certain finite configuration then the finite structure itself
must have the configuration; show that under certain hypotheses on the
finite structure, it has completions of every well-ordered type; show
that a suitable large cardinal hypothesis implies the existence of the 
finite configuration.

I presume the same general strategy is at play here?  What new ideas are
you using in order to get your new Pi01 statements?  Can you give some
intuition as to why order invariance and strict domination and so forth
are showing up as relevant concepts in this context?

Tim