Applied Proof Theory: Proof Interpretations and their Use in by Ulrich Kohlenbach

By Ulrich Kohlenbach

Ulrich Kohlenbach offers an utilized type of facts conception that has led in recent times to new ends up in quantity thought, approximation concept, nonlinear research, geodesic geometry and ergodic thought (among others). This utilized process is predicated on logical alterations (so-called evidence interpretations) and matters the extraction of potent info (such as bounds) from prima facie useless proofs in addition to new qualitative effects similar to independence of suggestions from sure parameters, generalizations of proofs through removal of premises.

The booklet first develops the mandatory logical equipment emphasizing novel different types of Gödel's well-known practical ('Dialectica') interpretation. It then establishes basic logical metatheorems that attach those ideas with concrete arithmetic. ultimately, prolonged case reviews (one in approximation idea and one in fastened aspect idea) convey intimately how this equipment may be utilized to concrete proofs in several components of mathematics.

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By a recursion over the proof-tree which keeps the basic structure of the proof unchanged (which is of crucial importance for actually analyzing concrete and – in particular – not fully formalized proofs). In fact, Parsons’ and G¨odel’s results where obtained by using a different more complicated interpretation, the so-called G¨odel functional (‘Dialectica’) interpretation ([133]), which we will treat in chapters 8, 9,10. In contrast to the no-counterexample interpretation, which only refers to functionals of type level 2, functional interpretation uses – even for first order systems like PA – functionals of arbitrary finite types to achieve an interpretation which respects the modus ponens.

A ≡ (∀y0 )∃x1 ∀y1 . . ∃xn ∀yn A0 (y0 , x1 , y1 , . . , xn , yn ). Then the Herbrand normal form of A is defined as AH :≡ (∀y0 )∃x1 , . . , xn A0 (y0 , x1 , f1 (x1 ), . . , xn , fn (x1 , . . , xn )), where f1 , . . , fn are new function symbols, called Herbrand index functions. 15. In theories with function variables and function quantifiers we take the Herbrand normal form of A to be AH :≡ ∀(y0 ), f1 , . . , fn ∃x1 , . . , xn A0 (y0 , x1 , f1 (x1 ), . . , xn , fn (x1 , . . , xn )). In the following PL denotes first order predicate logic with equality.

In contrast to the no-counterexample interpretation, which only refers to functionals of type level 2, functional interpretation uses – even for first order systems like PA – functionals of arbitrary finite types to achieve an interpretation which respects the modus ponens. We conclude this chapter by indicating why functionals of type 2 are not sufficient whereas higher types allow one to resolve the problem. 3 Herbrand’s theorem and the no-counterexample interpretation 33 and A0 , B0 are quantifier-free and suppose we have functionals satisfying the nocounterexample interpretation of A and A → B.

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