Abstract:
The class of Elementary Discernible (ED) relations and properties is
defined by induction. It is the smallest class containing
the properties: x="a", x="b", the relations: x=y and
x*y=z
and closed under the operations of: Propositional calculus
and Quantifiers
restricted to subtexts. (x is subtext of y iff x is a text contained
in
the text y)
The relation R is Generally Discernible (GD) iff
there are two
Elementary Discernible ED relation S and T such that:
G1 R(x...) iff
there is a text y such that S(x...,y)
G2 nonR(x...) iff there is a text y such
that T(x...,y)
The advantage of this exhibition of decidability (and computability)
consists in simplified knowledge-structure: The metalogic becomes
released from mathematical debts.
We do not need arithmetization of the language because we can directly
describe the formulas and define all metalogical concepts. Further
on
without any application of mathematical truths we can prove that:
All discernible relation are representable in a first order theory
of
concatenation. Hence: A first order theory of concatenation is
essentially undecidable. Then also:
Logic is undecidable.
Thus the essential observations concerning logic may be proved without
any recurrence to the mathematical machinery.
For the first more detailed version of the paper look:
http://www.calculemus.org/forum/3/comput.doc