Re: Goedel - interesting problem?

From: Acme Diagnostics (LFinezapthis_at_partpostmark.net)
Date: 06/03/04 

Date: 3 Jun 2004 01:30:20 -0500

"Tron Furu" <tronfuru@frisurf.no> wrote:
>
>As for the program, here is the condensed replay:
>
>Specifically, would /a refutation, kinda, sorta */ have to be couched in the
>same formal language that /Goedel/ used himself? Plainly speaking, if it is
>related to e.g. the Liar Paradox, but is developed as a formalized version
>of it, can a /refutation/ be given by comments based on plain language
>problems like e.g. the Liar Paradox, assuming later formalization? Or is the
>trick _in_ the formalization...?
>
>* "kinda, sorta" here means "any sort of modification of Goedel's results,
>ranging from total via partial refutation to partial modification,
>differentiation, or generalization up to but not excluding extension".
>
>To the degree that the working mechanism of Goedel's Proof involve what
>pre-Fregian Logic classified as fallacies, at least a, plain language
>analysis of the proof as a fallacy is not excluded, although I would extend
>it some. The extension of the analysis would be in showing that Goedel's
>general argument structure /in the paper as a whole/ is a "Cornutus"
>(dilemma, in this case the destructive version); further, that (the proof
>section of the Paper/ involves the figure I have learned to call the
>"Crocodile" (exemplified in the tale of the Mother, the crocodile and the
>baby /this is a standard elementary logic parable, probably googleable,
>which I didn't want to presume to consume anyone's bandwidth with/), and
>only then, in the third level, the Liar paradox. If one succeeds in
>dissolving the Liar Paradox, one can pursue the "profitable" solution to the
>"Crocodile" ( this obviously requires knowledge of the parable.../, and then
>try to demonstrate that the dilemma is false.

Hi Tron,

As part payment for your recent worthy criticism, I offer these
wonderful resources provided me by one of my favorite posters, Kent
Paul Dolan (aka xanthian).

First, is this page demonstrating (among other things) the difficulty
one would have in refuting Goedel due to the number of different ways
one can arrive at the incompleteness theorem:

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html

Second, is Kent's masterpiece of concise and precisely worded
explanatory text on the effect of the theorem (compiled from two
posts):

- - - - -
[Goedel] proved that any set of
axioms at least as rich as the axioms of arithmetic has
statements which are true in that set of axioms, but cannot
be proved by using that set of axioms.

That does not prevent that those true things can be proved
with a more powerful set of axioms. It only conveys that the
stronger set of axioms will in turn contain new truths which
cannot be proved using only those axioms.

Goedel's incompleteness theorem only shows that some true
math facts cannot be proved within math, not that none of
them can.

It isn't all that complicated to follow the proof, either,
since it uses only the axioms of arithmetic to achieve its
goal.
- - - - -

The above is also most useful as an unambiguous definition to
be included in argumentation about the Goedel theorem.

I feel unqualified to address your questions in terms of semantics
and most of the content in the link, but think these resources will
prove helpful to you.

Larry

P.S. Adding comp.ai.philosophy as what I think is the most
appropriate group for this subject which Kent variously reads.

Relevant Pages
Quantcast