From: Simon Gordon (sim_dizzy@yahoo.com)
Date: Thu May 01 2003 - 19:14:48 MDT
Sorry to bring up the same thread but i do believe
that metamathematics, particularly Godels theorems (if
you are to take Penrose's suggestions with any
seriousness), do have at least some level of relevance
to AGI. I also see SIs themselves being particularly
occupied with the intricacies of metamathematics (what
else is there for them to do with all that excessive
intelligence!!)
-------------------------------------------------------
> > All formal systems are expressable as finite
strings
> > of symbols (by
> > definition), and thus trivially mappable to
> > integers. QED.
>
> Thanks Perry, thats all i need to know. Clearly i
> want my systems to be a little more generic than
> just formal systems
Well, sadly the Church-Turing Thesis has yet to be
disproven, so I'm afraid that for now your wants may
remain unfulfilled.
> because of set-theory-envy, if set
> theory has uncountable sets why cant the set of all
> primitive entities be uncountable?
> Or even infinite to
> the same extent as the set of all sets is infinite
What is the "set of all sets"? Many kinds of set
theory can't express such a thing. Again, you keep
throwing out ideas about sets as though they somehow
had a life outside a particular flavor of set
theory. They don't, any more than the integers have a
life outside of theories like Peano's axioms.
> > It is literally the set of all formal systems,
> > because it is the set
> > of all consistent mathematical systems.
>
> I consider formal systems to be a subset of the set
of
> consistent mathematical systems but you have equated
> them.
What other sort of mathematics is there outside of
formal systems?
-------------------------------------------------------
It seems the above discussion has already occured
albeit in a much more fleshed out form on the
everything-list last year between Hal Finney, Russel
Standish and others. From a subjective point of view
these rantings are more than just interesting because
these knowledgeable people have clearly read from the
same source material as i have, namely : Rudy Rucker's
excellent book "Infinity and the Mind". BTW Perry
where i said "set of all sets" clearly i meant "class
of all sets" ;)
everything-list@eskimo.com/msg03883.html">http://www.mail-archive.com/everything-list@eskimo.com/msg03883.html
See also the other sections of the everything thread.
Simon.
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