[FOM] Might there be no inaccessible cardinals?

[Kaveh Ghasemloo]

Let I be the statement that "there is an strongly inaccessible cardinal"
Assume If ZFC is consistent. Then the smallest inaccessible cardinal
gives a model of ZFC + \lnot I, just take the hierarchy of sets up to
it (as an ordinal), on the other hand, there is no finitistic proof of
"if Con(ZFC) then Con(ZFC+I)", since ZFC + I proves consistency of
ZFC, and therefore itself, which is impossible by Godel's Second
Incompleteness Theorem. [cf. Kunnen 1980, page 145]

Can anyone explain why Kunnen said that there is no *finitistic*
proof? Can we substitute *finitistic* with *in ZFC*?
-- 
Best
Kaveh Ghasemloo