Re: Robot Evolution

Phil Roberts, Jr. wrote:

Godel demonstrated (1931) that for any formal
(logical, mathematical, rule driven, etc.) system
capable of simple arithmetic, there is at least
one well-formed sentence or theorem, usually
referred to as the Godel or G sentence, that
cannot be proven in the system. Interestingly
enough, because of our ability to attach meaning
to the symbols employed in such a proof, the G
sentence is one that we humans can quite easily
"see" to be true in that its semantic
interpretation is simply: 'This sentence can not
be proven in this system'.

But the fallacy in your, and your antecedents'
thinking, is right there and very obvious.

Goedel proved that there are well formed sentences
stating theorems that a computer cannot _prove_ to
be true _or_ to be false, within the same axiomatic
system of arithmetic that the theorem concerns.

In the _same terms in which Goedel worked_, NEITHER
CAN A HUMAN _prove_ that the theorem is true, or is
false, using only the axioms of the system of
arithmetic that the theorem describes.

That the human can "see" the truth or falsehood of
the theorem is an unrelated topic;

[and arguably a fantasy as well; thinking
you know something is not the same thing as
knowing something; like a theisim, "knowing"
something you cannot prove isn't "knowledge"
at all, it's merely _faith_, the same trap
into which theists so consistently fall]

the human is merely working in some other demesne
than the one in which Goedel's machine was working.

In particular, the human is not working in the
demesne of accomplishing that proof as Goedel
described that the proof must be accomplished.

Goedel was more than willing to admit that some
theorem unprovable in one system of arithmetic might
well be provable under a stronger set of axioms, but
he then showed that the stronger set of axioms would
form a system for which exactly the same sort of
unprovable sentence could again be written.

So, all you've proved is that the human mind _may_
employ a stronger set of axioms, not that it is
somehow different in kind.

FWIW

xanthian.


.

Relevant Pages

  • Re: Goedel - interesting problem?
    ... Let's refer to the "Masterpiece of explanatory text" aka "Effect of the ... Goedel Theorem" as "the article" for convenience. ... >say exactly what particular axioms of arithmetic one has in mind. ... distinction changes lots of editing criteria for the article. ...
    (sci.logic)
  • Re: Help writing a paper on Godels Incompleteness Thorem
    ... Are you refuting Godel or not? ... That's not necessarily about the most axiom-like or examplar of axioms, ... Gregory Chaitin has taken the methodogy of Goedel even ... Boucher posits a maximal element of sorts of the natural integers. ...
    (sci.logic)
  • A comment from an interested onlooker
    ... artificial superintelligences, called a sophotechs, is beyond human ... persuade another about in error in its moral reasoning. ... But Mr. Novak is not free to interpret Goedel any way he wishes. ... metaphysical axioms, does not say anything about the axioms in one ...
    (rec.arts.sf.written)
  • Null-Axiom Set Theory
    ... The idea of having no axioms is for several reasons. ... empty, or an ur-element or minimal element, exist, that from anything ... Goedel sentence G_0 and the infinite chain of ghost axioms that each ... and applied mathematics. ...
    (sci.logic)