Re: [-empyre-] replying to several posts

On 04.05.03 23:48, "Jim Andrews" <> wrote:

> The Incompleteness Theorem establishes that in a system/language of formal
> logic at least as
> powerful as one capable of supporting arithmetic, there will always be
> 'undecidable
> propositions'. That is, one cannot form a system of formal logic in which all
> well-formed
> propositions are either false or provably true. There are *always* truths that
> cannot be proved
> within the system (whatever the system).
Borges' Library of Babel puts an interesting spin on this...a library that
contains all books, including the book that contains all books. Of course
Goedel shows this is impossible (mathematically) and intuitively we all know
it as well.

> By the way, I think the argument on "that
> a computer can
> never be as smart as a human being because the extent of its knowledge is
> limited by a fixed set
> of axioms, whereas people can discover unexpected truths" is mistaken: Godel's
> proof says that
> even any formal system used by humans is incomplete. So the only question is
> whether the 'human
> system' is necessarily more comprehensive or flexible than a computer's system
> and it doesn't
> seem likely to me.
Depends if you think humans are limited to operating within formal systems.
True, formal human systems are as limiting as artificial formal systems -
although it should be pointed out that artificial systems, at least at this
point in our technological development, remain entirely human formal
systems, as we create them. This aside, I would argue that people do not
operate entirely within and as formal systems. In fact, it would seem that
much in human behaviour and intent is informal. In this respect humans can
discover what computers and other formal systems cannot as they can operate
outside the strictures a formal system requires to function. That said, one
responsibility of the artist working digitally could be seen as finding how
to make computers, and other formal systems, work as informal systems.

> I wonder whether, given any particular undecidable proposition, there exists
> an axiom that one
> could introduce into the system that would render the undecidable proposition
> a theorem? Of
> course, even if there does exist such an axiom for each undecidable
> proposition, even then, with
> infinitely many axioms, one could still apply the argument to show that that
> system also was
> incomplete.
Perhaps meta-coding comes close to this. That is, you let the axiom's
auto-generate further axiom's, ad infinitum. Logically this is not
consistent with Goedel. In a sense this is computability and thus the Turing
machine is a disproof of Goedel. It is possible to create a formal system
that whilst it cannot contain everything at a specific moment can over time
function to expand and adapt infinitely. Perhaps this is closer to
Heisenberg's Theorem? Certainly we have a machine that will happily attempt
to "be" Borges' library.



Simon Biggs

Research Professor
Art and Design Research Centre
Sheffield Hallam University, UK

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