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george wrote:
Tbe first incompleteness theorem is usually expressed something like the
following
For any extension T of Robinson arithmetic we can effectively find a
sentence G_T, s.t. G_T is true and undecidable in T iff T is
(1-)consistent.
Not *any* extension.
Obviously, if you extend all the way to something negation complete
(is there a canonical complete NON-standard model, the way N is
the canonical complete standard one?) then it won't be diagonalizable.
There is no canonical non-standard model. Your general point is correct,
though: the above should include the qualifier "axiomatizable" or "true
and arithemetically definable".
But that, despite the fact that it is exactly what one means,
is the LAST thing one could say! It is the LEAST permissible
way of phrasing it -- BECAUSE the
Whole REASON why the theorem is
significant in the first place is that formal extensions
MUST be r.e.-but-not-recursive;
that is the conclusion the theorem is demonstrating.
Infinitely many consistent recursively - and, equivalently, recursively
enumerably - axiomatizable theories are extensions of Robinson arithmetic.
As to the rest of your points, as usual, I haven't anything interesting
to say about them.
--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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