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

By Ulrich Kohlenbach

ISBN-10: 3540775323

ISBN-13: 9783540775324

ISBN-10: 3540775331

ISBN-13: 9783540775331

Ulrich Kohlenbach offers an utilized type of facts idea that has led lately to new ends up in quantity conception, approximation idea, nonlinear research, geodesic geometry and ergodic idea (among others). This utilized procedure is predicated on logical modifications (so-called evidence interpretations) and matters the extraction of powerful facts (such as bounds) from prima facie useless proofs in addition to new qualitative effects reminiscent of independence of options from yes parameters, generalizations of proofs by way of removing of premises.

The e-book first develops the required logical equipment emphasizing novel sorts of Gödel's well-known sensible ('Dialectica') interpretation. It then establishes common logical metatheorems that attach those concepts with concrete arithmetic. eventually, prolonged case reviews (one in approximation concept and one in fastened aspect idea) convey intimately how this equipment should be utilized to concrete proofs in numerous parts of mathematics.

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Additional resources for Applied Proof Theory: Proof Interpretations and their Use in Mathematics

Example text

Many theories, such as PA, allow the contraction of tuples of variables into single variables. As we discussed above, infinity statements (for quantifier-free properties) in number theory have the form of Π20 -formulas. e. ∃x∀y > x ¬A0 (y, a), where a, y are the only free variables in A0 (y, a). e. e. N(a) ≥ |{y : A0 (y, a)}|. It is clear that any h also is an upper bound N on the number of solutions but in general the existence of a computable function N does not imply the existence of a computable height function h as the following example (due to [267]) shows: consider again Kleene’s T -predicate and define A0 (y, a) :≡ T (a, a, y).

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. e. |= A ⇔ |= AH (this fact is also expressed by saying that AH is a validity normal form) but are not logically equivalent since in general PL AH → A.

Tn,1 , . . 3 Herbrand’s theorem and the no-counterexample interpretation 25 formation of AH ) such that k1 AH,D :≡ kn ... j1 =1 A0 t1, j1 , f1 (t1, j1 ), . . ,tn, jn , fn (t1, j1 , . . ,tn, jn ) jn =1 is a tautology. The terms ti, j can be extracted constructively from a given PL−= -proof of A and conversely one can construct a PL−= -proof for A out of a given tautology AH,D . The theorem holds for PL if ‘tautology’ is replaced by ‘quasi-tautology’. g. [332]. The most difficult part of the proof of Herbrand’s theorem is the construction of the Herbrand terms ti, j .

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

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