Omega-Incompleteness, Truth, Intentionality

The subject of the paper is the omega-incompleteness of a formal theory which seeks to formalize finitist arithmetic. PRA (i.e. primitive recursive arithmetic) is normally considered to be the theory that formalizes finitist arithmetic. But the arguments which follow also hold if one assumes PA (i.e. Peano arithmetic) as the theory formalizing finitist arithmetic (in a broader sense, of course). I take two points of view: one internai to the theory, and one relative to some suitable non-conservative extension of it. I shall seek to show that: (i) with respect to the first point of view, omega-incompleteness entails an irreducible distinction between truth in finitist arithmetic and provability through methods based on finitist (finitary and concrete) evidence; (ii) with respect to the second point of view, this irreducible distinction can be overcome, but only if one accepts a form of evidence (non-finitary with respect to content, finitary in form but abstract). Abstract evidence is thus the finite expression of an intensional relationship between the subject and an infinite reality. Point (ii) wiIl be subsequently confirmed by analysis of certain kinds of prototypical proof. My thesis is developed on the basis of detailed formal analysis of the omega-incompleteness of first-order numerical theories (PRA in particular), and of certain kinds of prototypical proof: (1) the Euclidean proposition concerning the relationship between lowest common multiple and greatest common divisor; (2) the Euclidean algorithm of the remainders; (3) Friedman's finite form of Kruskal' s theorem.