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

By Ulrich Kohlenbach

Ulrich Kohlenbach provides an utilized type of evidence thought that has led lately to new ends up in quantity concept, approximation concept, nonlinear research, geodesic geometry and ergodic idea (among others). This utilized method is predicated on logical modifications (so-called facts interpretations) and matters the extraction of potent info (such as bounds) from prima facie useless proofs in addition to new qualitative effects akin to independence of recommendations from yes parameters, generalizations of proofs by means of removal of premises.

The e-book first develops the required logical equipment emphasizing novel types of Gödel's recognized practical ('Dialectica') interpretation. It then establishes basic logical metatheorems that attach those options with concrete arithmetic. ultimately, prolonged case reports (one in approximation thought and one in fastened element thought) exhibit intimately how this equipment will be utilized to concrete proofs in numerous parts of mathematics.

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Tk such that k ∀y A0 (ti , y). (1) PA i=1 Within PA each ti can be computed to a numeral ni : (2) PA ti = ni for 1 ≤ i ≤ k. By (1) and (2) we have k ∀y A0 (ni , y). 2 Informal treatment of ineffective proofs 19 By the consistency of PA we know that k ¬P(ni ). (4) N |= i=1 Hence by the numeralwise representability of primitive recursive predicates in PA we have k ¬P(ni ). (5) PA i=1 But (3) and (5) imply (6) PA ∀y¬ProvPA (y, 0 = 1 ), which contradicts G¨odel’s second incompleteness theorem. g. for 0, we know that ∀yA0 (t, y) is true in N simply because N |= ∀y¬ProvPA (y, 0 = 1 ).

Hence |N(x)| ≤ 2r x. Now if there √ were only finitely many primes p1 , . . , pr , then |N(x)| = x for every x and so 2r x ≥ x for all x which is a contradiction. From this proof one gets a bound as follows: √ Let p1 , . . , pr be the first r primes. Define x := (2r )2 + 1 = 22r + 1. Then 2r x < x. Hence ∃n ≤ x(n is divisible by some prime p > pr ) and so ∃p(p prime ∧ pr < p ≤ 22r + 1 = 4r + 1). So we get again a bound g(r) := 4r + 1 which is exponential in r rather than pr . For another proof (in fact a variant of proof 3) see the exercise 1.

Xn ∀yn ∃a1 , . . , am Gi0 (ai ) → A0 (x1 , y1 , . . , xn , yn ) . i=1 Warning: For the extension of Herbrand’s theorem to open theories T it is important that the index function used in defining AH are new and do not occur in the 26 2 Unwinding proofs non-logical axioms. In particular if we have a schema of purely universal axioms then in the statement of Herbrand’s theorem this schema is always understood with respect to the original language (without the index functions). Otherwise the reverse direction in Herbrand’s theorem in general would fail (see [202] for a discussion of this and related matters thereby pointing out errors in the literature).

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