Some Consequences of a Recursive Number-Theoretic Relation that is Not the Standard Interpretation of Any of its Formal Representations

We give precise definitions of primitive and formal mathematical objects, and show: there is an elementary, recursive, number-theoretic relation that is not a formal mathematical object in Gödel's formal system P, since it is not the standard interpretation of any of its representations in P; the range of a recursive number-theoretic function does not always define a formal mathematical object (recursively enumerable set) consistently in any Axiomatic Set Theory that is a model for P; there is no P-formula, (P), whose standard interpretation is unambiguously equivalent to Gödel'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 has a unique, constructive, extension as a total function, and, secondly, that we can define effectively computable number-theoretic functions that are not classically Turing-computable; Turing's and Cantor's diagonal arguments do not necessarily define Cauchy sequences