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
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
The last sentence can be read as: “S makes all theorems of PA true”. It is equivalent to the
The system of ω-logic described above can be iterated in the transfinite and one can axiomatize theories
Define for an ordinal α a sequence
←34 | 35→
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
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.
←35 | 36→
Similar distinctions can be made with respect to the concept of truth. The semantical concept of truth precisely defined by Tarski is in fact a mathematical notion. It provides a definition of truth in mathematics;20 it is the concept of truth for a model in a formal language (its essential feature is to define truth in terms of reference or satisfaction on the basis of a particular kind of syntactico-semantical analysis of the language). But one can also speak about epistemic truth – cf., e.g. Isaacson (1987, 1992) where it is argued that Peano arithmetic is complete with respect to an epistemic notion of arithmetical truth.
The distinction between proof and truth in mathematics presupposes of course some philosophical assumptions. In fact for pure formalists and for intuitionists there exists no truth/proof problem. For them a mathematical statement is true just in case it is provable, and proofs are syntactic or mental constructions of our own making. In the case of a Platonist (realist) philosophy of mathematics, the situation is different. One can say that Platonist approach to mathematics enabled Gödel to state the problem and to be able to distinguish between proof and truth, between syntax and semantics.21
14 ,,Wenn wir die Tatsachen eines bestimmtenmehr oder minder umfassenden Wissensgebiete zusammenstellen, so bemerken wir bald, daß diese Tatsachen einer Ordnung fähig sind. Diese Ordnung erfolgt jedesmal mit Hilfe eines gewissen Fachwerkes von Begriffen in der Weise, daß dem einzelnen Gegenstande des Wissensgebietes ein Begriff dieses Fachwerkes und jeder Tatsache innerhalb des Wissensgebietes eine logische Beziehung zwischen den Begriffen entspricht. Das Fachwerk der Begriffe ist nicht Anderes als die Theorie des Wissensgebietes”.
