Primitive recursion, equality, and a universal set

Pfender, Michael; Kröplin, M.; Pape, D.

doi:10.1017/s0960129500000499

Cited by 1 publication

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References 6 publications

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“…Each primitive recursive map can be generated from the basic maps 0, s, id, !, ∆, Θ, and ℓ by composition, cylindrification and iteration: substitution is realized via composition with the induced (f, g) = (f, g)(c) = (f (c), g(c)) which in turn is obtained via diagonal, cylindrification, transposition, and composition. Since iteration g § then gives the ("full") schema of primitive recursion (see Freyd 1972, Pfender et al 1994, ev in fact evaluates all Gödel numbers of (internal) p. r. map terms, recursively given in the above way.…”

Section: Appendix: Evaluationmentioning

confidence: 99%

On Hilbert's Tenth Problem

Pfender

2014

Preprint

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Using an iterated Horner schema for evaluation of diophantine polynomials, we define a partial µ-recursive "decision" algorithm decis as a "race" for a first nullstelle versus a first (internal) proof of non-nullity for such a polynomial -within a given theory T extending Peano Arithmétique PA. If T is diophantine sound, i. e. if (internal) provability implies truth -for diophantine formulae -, then the T-map decis gives correct results when applied to the codes of polynomial inequalities D(x 1 , . . . , x m ) = 0. The additional hypothesis that T be diophantine complete (in the syntactical sense) would guarantee in addition termination of decis on these formula, i. e. decis would constitute a decision algorithm for diophantine formulae in the sense of Hilbert's 10th problem. From Matiyasevich's impossibility for such a decision it follows, that a consistent theory T extending PA cannot be both diophantine sound and diophantine complete. We infer from this the existence of a diophantine formulae which is undecidable by T. Diophantine correctness is inherited by the diophantine completion T of T, and within this extension decis terminates on all externally given diophantine polynomials, correctly. Matiyasevich's theoremfor the strengthening T of T -then shows that T, and hence T, cannot be diophantine sound. But since the internal consistency formula Con T for T implies -within PA -diophantine soundness of T, we get PA ⊢ ¬Con T , in particular PA must derive its own internal inconsistency formula.

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Section: Appendix: Evaluationmentioning

confidence: 99%

On Hilbert's Tenth Problem

Pfender

2014

Preprint

View full text Add to dashboard Cite

show abstract

Primitive recursion, equality, and a universal set

Cited by 1 publication

References 6 publications

On Hilbert's Tenth Problem

On Hilbert's Tenth Problem

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