FOM: Large cardinals as integers?

[Ara R. Aslyan]

Dear Professor Friedman:

The results you presented, at least for me, sound intriguing, since
examples discovered by you are of purely combinatorial nature and I
guess could be formulated in finite mathematics.                     
The large cardinal existence axioms, are formulated in ZFC, and I would
say is of ontological nature. Do you think there is a  fundamental
connection between this two, or this phenomena have to do with syntactical
peculiarities  of ZFC. 
	More precisely in order to avoid from this, is it possible to
formulate this combinatorial statements say in PA, or even in PRA and show
them provable in transfinite progressions of this theories. If the answer
is yes it would be interesting to estimate the ordinal length of
progression needed for this statements to be provable.  


Best wishes,

Dr. Ara Aslyan