The truth of undecidable sentences in the perspective of Godel's first incompleteness theorem

The first theorem of Godel on the incompleteness of arithmetic asserts that for a wide class of formal systems of arithmetic there are true but unsolvable sentences. Thus, the concepts of truth in informal arithmetic and formally provable statements diverge, which gives rise to the whole complex of epistemological problems. The purpose of the article is to analyze the problems of the truth of the unprovable Godel sentence, in particular, to consider the role of methods for constructing the Godel sentence, the ideas and the mechanism of coding and self-reference, balancing on the verge of a paradox. The standard establishment of the truth of the Godel sentence in a syntactic and model-theoretic way was challenged by M. Dummett, who advanced interesting arguments in favor of the intuitive origin of the truth of the Godel sentence, which has an important epistemological significance. The author proposes a comparison of the standard approach with Dummett's understanding of the nature of the Godel proposition. The problem solved in the article: the use of the dual nature of the Godel sentence in the context of establishing its truth as (a) the metamathematical sentence and (b) the arithmetic assertion; proof of the truth of the Godel proposition as a universal quantification by establishing the truth of its examples; demonstration of the need for a standard interpretation of the formal system of arithmetic for the truth of the Godel sentence; establishing a connection between these two ideas. Methods used: reconstructing Dummett's arguments using first-order logic; use of the metamathematical result on the relation of the Godel proposition of the provability predicate in the Diagonal Lemma, or the Fixed Point Theorem. Conclusions: Dummett's argument about the nature of the truth of the Godel sentence is alternative with respect to the standard explanation of this truth. The argument of G. Sereni on the reducibility of Dummet's argument to the standard way of establishing the truth of the Godel sentence is investigated. The reason for reducibility is the existence of different epistemo-logical standards for the usual arithmetic and meta-arithmetic assertions, since the sentence can be interpreted in a dual way - as arithmetic and as metamathematical. The comparison of Dummett's and Sereni's arguments rests on the assumption of the incomprehensibility of the arithmetic interpretation of the predicative of provability and the sufficiency of the metamathematical interpretation of the Godel sentence. It is shown that the undertaken explication of Dummett's argument in favor of the truth of the Godel sentence can not be considered as a final, since it leaves a considerable number of questions open. The relation between metatheoretical and arithmetic methods includes the questions of self-referential assertions, various ways of proving the First Theorem of Godel, the role of the concept of consistency in connection with the truth of the Godel sentence, the ways of constructing the Godel sentence, the role of coding, etc. This is a genuine tangle of problems that are currently the subject of intense research into the phenomenon of incompleteness.

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