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The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt ..."
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The kprovability problem is, given a first order formula &phi; and an integer k, to determine if &phi; has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
ON THE NUMBER OF STEPS IN PROOFS
, 1989
"... In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. W ..."
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Cited by 15 (2 self)
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In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length ( = the number of symbols in the proof), depth ( = the maximal depth of a formula in the proof) and number o! steps ( = the number of formulas in the proof). For a particular formaI system and a given formula A we consider the shortest length of a proof of A, the minimal depth ofa proof of A and the minimal number of steps in a proof of A. The main results are the following: (1) a bound on the depth in terms of the number of steps: Theorem 2.2, (2) a bound on the depth in terms of the length: Theorem 2.3, (3) a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: (1) a bound on the number of steps in a cutfree proof, (2) some speedup results, (3) bounds on the number of steps in proofs of ParisHarrington sentences. Some paper
Kreisel’s Conjecture with minimality principle
, 2008
"... We prove that Kreisel’s Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms of identity (theory PAM). The result is independent on the choice of language of PAM. We also show that if infinitely many instances of A(x) are provable in a bounded number of steps ..."
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We prove that Kreisel’s Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms of identity (theory PAM). The result is independent on the choice of language of PAM. We also show that if infinitely many instances of A(x) are provable in a bounded number of steps in PAM then there exists k ∈ ω s.t. PAM ` ∀x> k A(x). The results imply that PAM does not prove scheme of induction or identity schemes in a bounded number of steps. 1 Introduction. Kreisel’s Conjecture (KC) is the following assertion: Let A(x) be a formula of PA with one free variable. Assume that there exists c ∈ ω s.t. for every n A(n) is provable in PA in c steps. Then ∀xA(x) is provable in PA.