Re: My investigations into Godels Incompleteness Theorem


On Fri, Sep 29 2006 11:51 am, Peter_Smith wrote in sci logic:

PS>> ... if M, the standard interpretation, verifies the axioms, then
PA is omega-consistent, by the usual argument. <<PS

Peter
=====
Well, I suppose we can get away with this argument so long as we do not
specify whether the "verification" is intuitive or formal.

This is the implicit ambiguity in Tarski's definitions of
satisfiability and truth - of the formulas of a formal language under
an interpretation - that admits intuitive (read non-constructive)
interpretations as mathematically valid.

However, if we interpret "verifiable" constructively, to mean
"effectively verifiable", the consequences are significantly different.

Note that the axioms of PA are, indeed, effectively verifiable, and
that the rules of inference preserve effective verifiability (I think
that Turing's 1936 paper on computable numbers establishes this point).

In other words, we should be defining "true in M" formally as
"Turing-decidable as TRUE in M". This is my M*.

I introduce it as a separate, "constructive, standard" interpretation
only to emphasise, first, that the classical "Platonic, standard"
interpretion of PA - in which we define "true" intuitively - has
mathematical significance at the individual, conceptual, level, and,
second, that the two interpretations should be viewed as complementing,
and not contradicting, each other.

I briefly discuss this in Appendix A of my investigations on:

"Why we shouldn't fault Lucas and Penrose for continuing to believe in
the Goedelian argument against computationalism"

http://alixcomsi.com/Why_we_shouldnt.pdf
http://alixcomsi.com/Why_we_shouldnt.htm

My thesis is that if our intention is to use mathematical languages to
effectively communicate the concept of "truth" under an interpretation
unambiguously, and in an intuitionistically unobjectionable manner,
then we may need to define it as in M*.

The underlying issue is the one raised by Brouwer. However - since his
analysis pre-dated a precise definition of "truth" under an
interpretation, and of "effective decidability / computability" - his
objection was limited to noting that we may not unconditionally
conclude from the PA-provability of [(Ex)R(x)] that there must be some
natural number n such that R(n) holds, unless we can actually
"construct" n.

I highlight the significance - and validity - of Brouwer's argument
(and the point of divergence between his constructive interpretation of
mathematical languages, and the Platonic interpretation that was - and
is still - implicit in the standard interpretation of PA) in my
investigations on:

"Why Brouwer was right in suggesting that Hilbert's Law of the Excluded
Middle needed qualification"

http://alixcomsi.com/Why_Brouwer_was_right.pdf
http://alixcomsi.com/Why_Brouwer_was_right.htm

Regards,

Bhup

.

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