I want to record, in text writing here, that I would now view
the problems of (mathematical) logic differently, although I do not
have a good theory to be working on (and hoping to complete and
publish it in my lifetime!).
In particular, one simple idea that came to me was that (in
normal human language words and wording) of simply adding a
supplementary axiom to a first presented set of axioms, with
the additional axiom saying "The list of axioms specified before
this axiom forms a consistent set of axioms, and furthermore, with
this axiom also included in the complete set of axioms the whole
set of axioms remains consistent."
It is easy to note that the "additional axiom" is not expressed
in conventional logical symbols and notation, but rather simply in
words of human language communication. (English, rather than French
or another language is not essential here.)
Goedel incompleteness derived from the insufficiency of the system
of Russell and Whitehead to affirm its own consistency.
But it is natural for whoever accepts for use a set of axioms to
also presume and/or accept the concept of the consistency of those axioms
(for example the consistency of the axioms of Euclidean Geometry).
////////////////////////////////////////////////////////////////
Another notable concept, in relation to the issue of "completeness"
for a system of logic, is that of "axioms of infinity" in relation
to the study of issues of "set theory". If all of the possible axioms
of infinity have not already (however that might be) be included in
the axiomatic foundations of the system then it seems fairly clear
that an axiom specifying the existence, for example, of a larger ordinal
number might be added to the axioms of the system with enlargement.
(This relates to the "Burali-Forti paradox".)
And there can be incompleteness through situations where we seem
to have options that do not affect "number theory" (for example) but
allow us either to believe or to disbelieve in the validity of the "Axiom
of Choice".