Abstract
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.
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- 2.
“om” is the metatheoretical universal quantifier. It means “for all”.
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Smorinski (1977), p 847.
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Cfr. Tieszen (2005), p. 152: “Objects or concepts that can be completely represented in space-time as finitary, concrete, real, and immediately intuitable”.
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Cfr. Tieszen (2005), p 152: “Objects or concepts that are in some sense infinitary, ideal or abstract, and not immediately intuitable”.
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Of course, the formalists claim that the truth of the axioms of ZF, or PA about the correspondent abstract structure of sets is not required. Indeed, considering only the case of ZF, it follows from Kreisel’s conservation theorem that if α is a Π1-formula and ZF- ⊢ α then PRA- ConsZF ⊢ α. As a consequence, in order to obtain the consistency of PRA we just need to assume the truth of ConsZF, i.e. of a sentence regarding a concrete syntactical fact, and we must not committ ourselves to assume the truth of the ZF axioms about the abstract structure of sets. Actually, it is (trivially) true that the consistency of PRA may be obtained from the consistency of ZF, but the problem is precisely to justify the latter assertion. To require the existence of an abstract model is a way of solution. Another way consists in elaborating a constructive proof (that, of course, can be carried out with less difficulty for the included theory). In this case, however, to prove the truth of consistency means to show the truth of the syntactical fact of consistency by means of the abstract structural properties of the proof itself. This is typical of the constructivistic approch, about which we are going to speak below.
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Cfr. Gödel (1972), pp 271–273: “P. Bernays has pointed out on several occasions that, in view of the fact that the consistency of a formal system cannot be proved by any deduction procedures available in the system itself, it is necessary to go beyond the framework of finitary mathematics in Hilbert’s sense in order to prove the consistency of classical mathematics or even of classical number theory. Since finitary mathematics is defined as the mathematics of concrete intuition, this seem to imply that abstract concepts are needed for the proof of consistency of number theory … By abstract concepts, in this context, are meant concepts wich are essentially of the second or higher level, i.e. which do not have as their content properties or relations of concrete objects (such as combinations of symbols), but rather of thought structures or thought contents (e.g., proofs, meaningful propositions, and so on), where in the proofs of propositions about these mental objects insights are needed which are not derived from a reflection upon the combinatorial (space time) properties of the symbols representing them, but rather from a reflection upon the meanings involved.”
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On this see Longo (2002).
References
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Galvan, S. (2010). Ω-Incompleteness, Truth, Intentionality. In: Carsetti, A. (eds) Causality, Meaningful Complexity and Embodied Cognition. Theory and Decision Library A:, vol 46. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3529-5_6
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