7 research outputs found
Consistency Decision
The consistency formula for set theory can be stated in terms of the
free-variables theory of primitive recursive maps. Free-variable p. r.
predicates are decidable by set theory, main result here, built on recursive
evaluation of p. r. map codes and soundness of that evaluation in set
theoretical frame: internal p. r. map code equality is evaluated into set
theoretical equality. So the free-variable consistency predicate of set theory
is decided by set theory, {\omega}-consistency assumed. By G\"odel's second
incompleteness theorem on undecidability of set theory's consistency formula by
set theory under assumption of this {\omega}- consistency, classical set theory
turns out to be {\omega}-inconsistent.Comment: arXiv admin note: text overlap with arXiv:1312.727
On Hilbert's Tenth Problem
Using an iterated Horner schema for evaluation of diophantine polynomials, we
define a partial -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 Arithmetique 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 . 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 theorem -- for 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
that PA derives \neg Con_T, in particular PA must derive its own internal
inconsistency formula
Arithmetical Foundations - Recursion. Evaluation. Consistency
Primitive recursion, mu-recursion, universal object and universe theories,
complexity controlled iteration, code evaluation, soundness, decidability,
G\"odel incompleteness theorems, inconsistency provability for set theory,
constructive consistency