Typeless theories of truth and the built-in provability of consistency: parallels and intertwining

The concept of Tarski's truth, long dominating in the world literature and based on the hierarchy of languages, gradually gives way to typeless concepts of truth. The most important difference between these two concepts is the impossibility of the definition in the formal language of the predicate of truth in the first concept, and the presence of universal predicates of truth in the second. Both concepts differ significantly in the epistemological characteristics of the key concept of truth. At present, it is urgent to determine the scope of the significance of the typeless concept of truth. Although the idea of the typeless concept of truth appeared three to four decades ago, it became widely known after the publication of S. Kripke's theory of truth. Further development focused on the actual problems of the theory of truth. And only recently there appeared works devoted to broader aspects of the application of typeless concepts of truth, in particular, the expressive possibilities of formal languages regarding the analysis of these languages themselves. The combination of the typeless concept of truth and the search for the possibility of proving certain properties of language within language itself is of considerable interest, since both approaches are options for solving the Liar's paradox or for avoiding this paradox by various kinds of technical methods. The objective of this study is to demonstrate the possibility of proving the consistency of formal languages within languages themselves. This kind of "built-in consistency", in which the second theorem of Godel is not satisfied, becomes possible when using typeless concepts of truth. A specific scheme is proposed for combining Kripke's typeless concept of truth and B. Whittle's programmed "built-in consistency" approach to obtain systems that give considerable freedom in expressing our intuitions as to what a theory should be, the expressive means of which allow to prove important facts about this very theory. The use of analogies between typeless theories of truth and theories with "built-in" consistency allows to make a conclusion about the sphere of applicability of the Second Godel Theorem without questioning its validity under the Hilbert-Bernays derivability conditions. Rather, it is about the fact that translating formal results into everyday language leads to significant distortions in the content of mathematical results. The analogies proposed above are also useful in the sense that such a distortion is easier to trace in the example of Tarski's theory of the indeterminateness of truth, which is often translated as an assertion that no intelligent language contains its own predicate of truth.

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