On Wed, 13 Nov 2002, Tyler Durden wrote:
> Damn what a pack of geeks! (Looks like I might end up liking this list!)
>
> When we say "complete", are we talking about completeness in the Godelian
> sense? According to Godel, and formal system (except for the possibility of
> the oddballs mentioned below--I hadn't heard of this possibility) is
> "incomplete" in that there will exist true statements that can not be proven
> given the axioms of that system.
Sorry for the long delay...very busy at work.
As far as I am aware formal language definitions of 'complete' and Godel's
are the same. I've never seen anything from Godel that indicated
otherwise.
Your comment is almost correct. Complete means that all true statements
can be written (which of course implies that all other statements -must-
be false). Incomplete means there are true statements which can't be
written in the -limited- syntax of the incomplete system (which again
implies there are false statements which can't be made).
The sticking point isn't the writing of the statements (irrespective of
complete or not) but rather the -proof- of their truth. Godel doesn't
address the list of statements or how the list was generated but rather
the algorithmic process of proof of value. We know axiomatically that we
can't write all true statemens in a incomplete system, that's what makes
it incomplete. The test is can we tell them apart algorithmically? That
question is what puts Godel in the "Halting Problem" category.
The sticking point, and what upset everyone, was that 'complete' and
'provable' aren't the same thing. if you can't write the complete list
you of course can't prove all of them. No surpise here. But even if you
can write all of the true statements you still can't prove they are true
within the system. In other words self-reference destroys the concept of
proof as we normally think of it. If you want to prove something you have
to step outside of its context (ie meta- views). That destroys the
classical/intuitive/naive concept of proof.
Godel upsets people because he demonstrates that 'truth' in whatever
context one wishes to discuss it is relative. There is no -universal-
definition of 'truth'.
Reality is observer dependent. Abandon transcendence.
--
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