RE: [-empyre-] replying to several posts
> Depends if you think humans are limited to operating within formal systems.
Even if a system is not formal, one can examine its properties and generally recast it as a
formal system of equal computational power via proof of the logical equivalence of the two
systems, or by showing that the informal system, in every regard, does not exceed the
computational power of the proposed formal system. For instance, the notion of the Turing
machine has sufficed, currently, as a model of the formal system of all computers.
I am not aware of any accepted proof that the computational power of humans exceeds that of a
Turing machine, or the existence of a formal system (that deals within finitude) that exceeds
that of a Turing machine. Are you?
> 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.
The diagonal argument guarantees the existence of more 'undecidable propositions' regardless of
how many denumerable axioms we add, Simon, like it does in Cantor's arguments concerning
transfinite arithmetic, which is where Godel picked up the technique (again, gorgeous work).
I misspoke somewhat in my last post. I said "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." The size of the set of 'undecidable
propositions' is not denumerable. If we add axioms that render the 'undecidable propositions'
theorems (supposing we can do this, for the moment) for denumerably many 'undecidable
propositions', then there will always be more of them, according to the diagonal argument and
the fact that the union of two denumerable sets is denumerable (ie, we can reapply the diagonal
argument to the larger denumerable set); the size of the set of 'undecidable propositions' is
not denumerable, ie, it is of the order of the size of the set of real numbers (or larger), not
the integers or rationals. There are uncountably many truths that are not provable within any
system of denumerably many axioms, even if we add infinitely many denumerable axioms.
Apologies. Perhaps we are getting too technical for the list? I would be happy to discuss this
with you backchannel further if you like, Simon, or take it further on the list--your call. It's
great to be able to talk about such things, however, in the context of poetry and programming.
Thanks very much; it feels like a good discussion that is relevant to several questions that
have been raised in this discussion, like how to compare human and computer potential 'smarts'
and also the intensity of engagement in language associated with poetry and how that intensity
is currently very active in the theory of computation.
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