Re: Godel's theorem is uninteresting?

Charlie-Boo wrote:
>
> We can construct (express) the assertion "I am unprovable." which is
> not a theorem in S1 given that S1 is sound (a requirement of the weaker
> form of Godel's 1st. Incompleteness theorem.) There is a theorem in S2
> that states that "I am unprovable." is not provable in S1.
>

OK, theoremhood => provability A new definition of theorem for me
anyway. In any event, using this new terminology, I'm hypothesizing
about an S1 in which only interesting true statements are provable
in S1 or any supersystem containing S1, say S2, in which itself only
interesting true statements are provable. And, any interesting true
statement in this (supposedlly) minimal S1 is provable in some S2
containing S1. In fact, I'm wondering whether there is possible some
infinite sequence of such S(i)'s. I do NOT want any S(i) to be
constructed from S(i-1) by simply adding as an axiom some interesting
statement in S(i-1) that we can somehow prove that, if it is provable,

it is only provable by a proof dependent on the axioms and interesting
theorems of S(i-1). This means interestingness and provability are
both derivable (or not) by logico-mathematical derivations from axioms
and theorems - at least in my imaginary world.

What I'm really talking about sounds suspiciously to me like trying to
revive Russell's imaginary world in which all true statements were
theorems (at least I think that's what he thought). Only just tweaked a
little (or a lot) here and there. A reconception of the fundamental
axioms of set theory. FOL however seems to me untouchable. That may
be a mistake. I assume there are alternatives to FOL out there, at
approximately the same level of expressiveness?

I apologize for the morass-like nature of the above. I simply don't
have the requisite sophistication here to be cogent.

> >
> All theorems are provable by defnition. Do you mean all interesting
> true statements?
>

See above. It may be that a statement in S1 which is interesting and
true can only be proved in some S2 (as restricted above - no cheating
by just adding the damned theorem as an axiom).

> Do you feel that Godel's Theorems contain self-reference, or is it just
> the proofs?
>

The statement "I am unprovable" you mentioned above seems to me
to clearly contain self-reference. I'm not sure whether the
self-reference is in its 'native language', or S1, as I think we both
used above, or in S2. I suppose the latter.

>
> Define interesting.
>

Interesting is like FLT, or the Mean Value Theorem. Substantive
statements about the subject matter of, in our case here, S1.

> Very interesting . . .
>

Very funny, but I'm not sure my sentence(s) you're referring to are
really interesting, unless you tell me how YOU define interesting. Did
you mean you know this HAS been covered before?

> C-B
>
> According to the supreme court, pornography is recursive but not
> recursively enumerable.
>
>

If I understood the above I'd be happy, for awhile anyway.

Thanks.

Ken

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