Re: ZFC

On 3 Jul 2005 01:12:57 -0700, examach...@xxxxxxxxx wrote in
comp.theory:

EO>> Although computation has not been given an independent axiomatic
ground from set theory ... <<EO

Eray
====
I suspect that may explain somewhat the difficulty in addressing issues
that you are attempting to raise within standard interpretations of
classical theory.

It seems that the concept of mathematical truth is perceived
differently by computational theory, and by standard interpretations of
classical logic.

For instance, in the abstract of his paper, ' Truth in Complex Adaptive
Systems models should be based on proof by constructive verification',
which is due to be presented at Complexity, Science and Society
Conference Liverpool, 11-14 September 2005, David Shipworth endorses
the remark that classical or Platonic truth is 'verification free'.

http://complexity.vub.ac.be/phil/abstracts/19-Shipworth.pdf

Now, although one may agree that truth in complex adaptive systems
should be effectively verifiable, such endorsement seems to
unnecessarily - and possibly unfairly - limit our interpretation of
classical methods of argumentation and proof in mathematical reasoning.

Although Platonic truth is, indeed, 'verification free', there is no
reason to believe that the classical concept of deductive truth was,
inherently, Platonic. The latter truth, embodied formally in Tarski's
definitions of the truth and satisfiability of the symbolic expressions
of a formal language under an interpretation, only appears
'verification free' when interpreted Platonically.

That there are constructive interpretations of Tarski's definitions in
which Tarskian-truth can always be made 'verifiable', either
instantiationally or algorithmically, forms the subject of my draft
paper:

PA is instantiationally complete, but algorithmically incomplete: An
alternative interpretation of Gödelian incompleteness under Church's
Thesis that links formal logic and computability.

http://alixcomsi.com/PA_is_instantiationally_complete.htm

I would be interested in knowing whether the arguments of this paper,
and its consequences, are consistent with the direction of your line of
thought; whether, from your point of view, they go some way in spanning
the gap between the perception of (at least) arithmetical truth in
standard interpretations of axiomatic systems, and that of the
corresponding truth in the theory of computation; and whether they help
you place the various remarks made in the present discussion in a less
controversial perspective.

Regards,

Bhup

.

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