Lógos and Máthma 2. Roman Murawski

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Lógos and Máthma 2 - Roman Murawski Polish Contemporary Philosophy and Philosophical Humanities

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gives a nonconservative extension of PA. In fact one can prove in this theory the consistency of PA.

      The theories PAT for T being PA+ “S is a full inductive satisfaction class” can be characterized by transfinite induction or the consistency of appropriate ω-logics. Denote by Γ − PA(S) the theory PA + “S is a full Γ-inductive satisfaction class” and by PA(S) the theory PA+ “S is a full inductive satisfaction class”.

      Consider the following sequence of formulas of the language L(PA) (one uses here arithmetization):

      Γn+1(φ) = “there exists a proof of the formula φ

      based on

      Observe that in this system of ω-logic only the application of the ω-rule increases the degree of complexity of a proof.

      Theorem 5 (Kotlarski 1986)

      It can also be proved (cf. Kotlarski 1986) that the theory PA(S) is equal to the theory

      The last sentence can be read as: “S makes all theorems of PA true”. It is equivalent to the -inductiveness of the satisfaction class S.

      The system of ω-logic described above can be iterated in the transfinite and one can axiomatize theories and PAPA(S) by consistency statements of appropriate systems of this logic (cf. Kotlarski and Ratajczyk 1990a).

      Define for an ordinal α a sequence in the following way: and put ωn = ωn(ω). Let now TI(ρ), where ρ is an ordinal, denote the scheme of transfinite induction up to ρ. Then the following theorem holds.

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      Theorem 6 (Kotlarski and Ratajczyk 1990b) Let m be a natural number. Then

      The above theorems show how strong is Peano arithmetic augmented with an appropriate notion of satisfaction (and truth). One can see that only by assuming that the added notion of satisfaction (truth) is full and at least inductive one obtains a proper extension of PA. It is interesting that such extensions are equivalent to PA extended by appropriate forms of transfinite induction or by the statements of the consistency of appropriate systems of ω-logic. In other words, the above theorems show in particular that what can be proved about natural numbers using Peano axioms and the notion of satisfaction (truth) that is assumed to be full and -inductive is exactly the same as what can be proved in PA plus transfinite induction for ordinals (for all k ∈N) or in PA plus appropriate consistency statements. Similarly for PA plus full inductive satisfaction (truth) on the one hand and PA plus transfinite induction for ordinals (for all k ∈N) or PA plus appropriate consistency statements on the other. They show also that by adding to PA the notion of satisfaction (truth) and assuming that it is full and makes all theorems of PA true, one obtains a theory with exactly the same theorems about natural numbers as by taking PA augmented with a concept of a full and inductive satisfaction (truth) or PA plus appropriate consistency statements. In the above considerations, we restricted ourselves to formal proofs and to the semantical notion of truth in mathematics. We tried to show how the awareness of differences between them has been developed – from the hopes that formal proofs provide sufficient means to exhaust the mathematical truth to the discovery of various limitations of them. Let us finish with some general remarks.

      Concepts of proof and truth are (even in mathematics) ambiguous. One should distinguish between working proofs of everyday mathematics and idealized formal proofs used by logicians. On the other hand, a proof in mathematics has various aspects and can be studied from various points of view. One can distinguish psychological, social, cultural and logical aspects of proofs. a proof can be studied as a mathematical or as an epistemological object. The former is precisely defined on the basis of mathematical logic, and the latter is a vague concept. The former is an idealization of proofs occurring in a research practice of mathematicians, is a reconstruction of them. Recently, one can observe in the philosophy of mathematics a tendency to concentrate on the actual research practice of mathematicians rather than on idealized foundational reconstructions of it and consequently to study the methods actually used by mathematicians.

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