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The subject of the paper is the ω-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.¹ 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 internal 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, ω-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.

DOI: 10.1007/978-90-481-3529-5_6

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