Read e-book online Applied Proof Theory: Proof Interpretations and their Use in PDF

By Ulrich Kohlenbach

ISBN-10: 3540775323

ISBN-13: 9783540775324

ISBN-10: 3540775331

ISBN-13: 9783540775331

Ulrich Kohlenbach offers an utilized type of evidence concept that has led lately to new ends up in quantity idea, approximation concept, nonlinear research, geodesic geometry and ergodic thought (among others). This utilized strategy relies on logical adjustments (so-called evidence interpretations) and matters the extraction of potent information (such as bounds) from prima facie useless proofs in addition to new qualitative effects resembling independence of recommendations from yes parameters, generalizations of proofs via removing of premises.

The publication first develops the mandatory logical equipment emphasizing novel kinds of Gödel's recognized useful ('Dialectica') interpretation. It then establishes basic logical metatheorems that attach those innovations with concrete arithmetic. ultimately, prolonged case reviews (one in approximation thought and one in mounted element conception) exhibit intimately how this equipment may be utilized to concrete proofs in several components of mathematics.

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X p−1 ) = xi p ≥ 1, i < p (Projections), S(x) = x + 1 (Successor) are primitive recursive. 2) If h0 (x0 , . . , x p−1 ), . . , hl−1 (x0 , . . , x p−1 ) and g(y0 , . . , yl−1 ) are primitive recursive functions, then also f (x0 , . . , x p−1 ) = g(h0 (x0 , . . , x p−1 ), . . , hl−1 (x0 , . . , x p−1 )) is primitive recursive. 3) If g(x0 , . . , x p−1 ) and h(z, y, x0 , . . , x p−1 ) are primitive recursive functions, then also f defined by f (0, x0 , . . , x p−1 ) = g(x0 , . . , x p−1 ), f (y + 1, x0 , .

5772... is the nr so-called Euler-Mascheroni constant. Hence for nr := e pr −C we have ∑ i=1 1 i > pr (and this is essentially optimal). ,αr ≤n p1 1 ≤ pr . · . . · pαr r Hence there must be an i (1 ≤ i ≤ nr ) which contains a prime factor p with pr < p ≤ i ≤ nr . So put together ∃p(p prime ∧ pr < p ≤ e pr −C ). Applying this argument to all prime numbers p1 < . . < prx ≤ x we obtain ∀x∃p(p prime ∧ x < p ≤ ex−C ). So we can take g(x) := ex−C (or an appropriate upper bound of this to make it computable).

G. Littlewood’s theorem on the sign changes of π (n) − li(n)) and algebra see Kreisel’s original papers on the subject [241, 242] and also [243]. As briefly discussed above, Luckhardt [267] presents (inspired by Kreisel [249]) an important application of Herbrand terms extracted from two proofs of Roth’s theorem in diophantine approximation resulting in the first polynomial bounds on the number of solutions 40 2 Unwinding proofs (see also Luckhardt [268]). Applications of the ε -substitution method to the solution of Hilbert’s 17th problem and subsequent work in this direction are discussed in Delzell [84] (see also Delzell’s papers [79, 80, 81, 82, 83] although some do not use proof theory directly).

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


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