By Carlos J. Moreno
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Additional resources for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods
G. . The most difficult part of the proof of Herbrand’s theorem is the construction of the Herbrand terms ti, j . The reverse direction for PL−= follows similar to the special case treated above: the fi -terms in AH,D are replaced by new variables (starting from terms of maximal size) yielding an index-function-free Herbrand disjunction AD . From this A is derived by a direct proof. For PL the reverse direction is more complicated to establish since also instances of equality axioms x = y → fi (x) = fi (y) are now allowed in the proof of AH,D .
Instead of a single variable we may have (here and in the following) also a tuple x = x1 , . . , xn of variables. e. A ≡ ∀xA0 (x), where A0 is quantifier-free. Such sentences A, sometimes called complete, don’t ask for any witnessing data. So the problem of extracting data is empty here. e. A ≡ ∃x A0 (x). We treat this as a special case of 3) A ≡ ∀x∃y A0 (x, y). Let’s consider the case where x, y ∈ N and A0 ∈ L (PA) (here PA denotes first order Peano arithmetic which we assume to contain all primitive recursive functions; see chapter 3 for a precise definition).
For another proof (in fact a variant of proof 3) see the exercise 1. Still further proofs can be found in . Discussion: 1) All three proofs provide more information than the mere fact that ‘there are infinitely many primes’ is true. By making their quantitative content explicit one can compare them with respect to their numerical quality. 2) The unwindings of the proofs 1)–3) were straightforward and didn’t require any tools from logic as guiding principles. g. [122, 267, 204, 205]). The final verification of the data extracted will always be again an ordinary mathematical proof (obtained by a proof-theoretic transformation of the original proof) which does not rely on any logical metatheorems (in contrast to the verification of the general procedure of transformation).