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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 X... ..."
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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 X...
Generalizing Theorems in Real Closed Fields
, 1995
"... Jan Krajicek posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A(1 + ... + 1) (n occurrences of 1) is provable in length k for all n, then (x)A(x) is provable? It is argued that the answer to this question depends on the particular f ..."
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Jan Krajicek posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A(1 + ... + 1) (n occurrences of 1) is provable in length k for all n, then (x)A(x) is provable? It is argued that the answer to this question depends on the particular formulation of the "theory of real closed fields." Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive answer to Kraj'icek's question for (1) the axiom system RCF of ArtinSchreier with Gentzen's LK as underlying logical calculus, (2) RCF with the variant LKB of LK allowing introduction of several quantifiers of the same type in one step, (3) LKB and the firstorder schemata corresponding to Dedekind cuts and the supremum principle. A negative answer is given for (4) any system containing the schema of extensionality.
Term Induction
, 2001
"... In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called ter ..."
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In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called term induction, it derives a restricted term built from successor s and the constant 0. We call such terms numerals. To characterise the difference between T (tind) and pure logic, we employ proof theoretic methods. Firstly we establish a variant of Herbrand’s Theorem for T (tind). Let ∃¯xF (¯x) be a Σ1 formula; provable by Π. Then there exists a disjunction � N i1 · · · � N il M1(s i1 (0),..., s il(0)) ∨ · · · ∨ Mm(s i1 (0),..., s il(0)), denoted by H that is valid for some N ∈ IN, furthermore the Mi are instances of F (ā). In T (tind) it is not possible to bound the length of Herbrand disjunctions in terms of proof length and logical complexity of the endformula as usual. The main result is that we can bound the length of the {s, 0}matrix of the above disjunctions in this way.