From: Lee Corbin (lcorbin@tsoft.com)
Date: Sun Sep 01 2002 - 12:34:39 MDT
Giu1i0 Pri5c0 writes
> New on Transhumanity (www.transhumanism.com)
> Interview with Rudy Rucker, by G.P., September 1, 2002.
> ...Broadly speaking, transrealism is writing about your
> immediate perceptions in a fantastic way. Working day
> to day reality into your SFictional constructions ...As
> I understood him, Gödel said that his theorems prove
> that you can't in fact specify a formal system whose
> power is equal to your mind...
No, I think that this exaggerates what Gödel's proof
implies. We, because of our realism, or our belief in
the existence of numbers, or by going to a higher
formal system than the system in which some axioms
are written, can see that certain things are true
even though those things *cannot* be proven from
those axioms (i.e., in the original formal system).
Perhaps soon, (2040?), there will be a formal system
based on a finite number of axioms, which will none-
theless proceed to make conjectures, and study our
universe and us, and be altogether vastly wiser than
we are. It too will understand Gödel's theorem, and
it will have a proof, as do we, of the theorem
using metamathematics (like we do) to prove its
results. So a formal system can be just as powerful
as we are.
But even for *that* formal system---say a specific one
which is very smart in the year 2041 and is telling you
all about everything---there will be true statements
about what it can do (i.e. results that it can achieve)
that it cannot prove. (Of course, we couldn't either,
but only because we're not smart enough to "see" that
truth.) We'll have to wait until 2042 for a new program
that can write a formal string which "Gödelizes" the first
axiom system (the 2041 machine). This formal string of
logical symbols will be written entirely in terms used
by the axioms of the 2041 machine, and the string will
be true, yet the 2041 machine cannot possibly prove it.
Only the 2042 machine can prove it, referring to concepts
that it can "see": true relationships among integers that
are apparent to it; and even formally prove it (if it
really worked very very hard) using new axioms not in
the 2041 system.
Uh, when I write "true relationships among the integers",
I'm falling back on my mathematical platonism. Even
though the non-platonists are very smart and knowledgable,
I don't know how they would explain all this. Neither
did Gödel.
Lee
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