On the consistency of ZF set theory and its large cardinal extensions

I consider the question of the consistency of ZF set theory and of its large cardinal extensions, from both a historical and a theoretical point of view, touching on some epistemological aspects of the problem. First I recall the reasons why in the case of set theory neither model- theoretic nor proof-theoretic methods seem suitable for the question of consistency. Then I show how set-theorists have dealt with the problem by means of large cardinals and inner models, with a remarkable confidence in a sort of direct intuition of consistency. I argue that the relationship between intuition and formalization is the crucial point regarding the metamathematical treatment of the problem. Finally, I show how finitary versions of Goedel sentence constructions due to W. H. Woodin could give evidence that there is no substantial dissimilarity, from an epistemological point of view, between the problems of consistency for arithmetic and for large cardinal hypotheses. The ideas of Skolem, Zermelo, Goedel, Kreisel and Cohen on the topic are touched upon.