Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations

Details Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations

2002-10-05

arxiv.org/abs/math/0210078v6

Mathematics

General Mathematics

v6; introduced ACI compliant notation for citations; replaced Corollary 1.2; corrected Corollary 14.3; other minor corrections

Scientific paper

We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting inconsistency. We then show: there is an elementary, recursive, number-theoretic relation that is not a formal mathematical object, since it is not the standard interpretation of any of its representations in Goedel's formal system P; the range of a recursive number-theoretic function does not always define a (recursively enumerable set) formal mathematical object consistently in any Axiomatic Set Theory that models P; there is no P-formula, Con(P), whose standard interpretation is unambiguously equivalent to Goedel's number-theoretic definition of "P is consistent"; every recursive number-theoretic function is not strongly representable in P; Tarski's definitions of "satisfiability" and "truth" can be made constructive, and intuitionistically unobjectionable, by reformulating Church's Thesis constructively; the classical definition of Turing machines can be extended to include self-terminating, converging, and oscillating routines; a constructive Church's Thesis implies, firstly, that every partial recursive number-theoretic function can be constructively extended as a unique total function, and, secondly, that we can define effectively computable functions that are not classically Turing-computable; Turing's and Cantor's diagonal arguments do not necessarily define Cauchy sequences.

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