  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.
