On 10 Jun 2009, at 04:14, Jesse Mazer wrote:
> I think I remember reading in one of Roger Penrose's books that
> there is a difference between an ordinary consistency condition
> (which just means that no two propositions explicitly contradict
> each other) and "omega-consistency"--see http://en.wikipedia.org/wiki/Omega-consistent_theory
> . I can't quite follow the details, but I'm guessing the condition
> means (or at least includes) something like the idea that if you
> have a statement of the form "there exists a number (or set of
> numbers) with property X" then there must actually be some other
> proposition describing a particular number (or set of numbers) does
> in fact have this property. The fact that you can add either a Godel
> statement or its negation to the Peano axioms without creating a
> contradiction (as long as the Peano axioms are not inconsistent) may
> not mean you can add either one and still have an omega-consistent
> theory; if that's true, would there be a unique omega-consistent way
> to set the truth value of all well-formed propositions about
> arithmetic which are undecidable by the Peano axioms? Again, Bruno
> might know...
The notion of omega-consistency is a red-herring. The notion exists
only for technical reason. Gödel did not succeed in proving the
undecidability of its "Gödel-sentences" without using it, but this
will be done succesfully by Rosser. Smullyan introduced better notion
than omega-consistency, like his notion of stability, but personaly I
prefer to use the (non effective, ok) notion of soundness, and use the
notion of stability only latter in more advanced course. The notion of
arithmetical soundness was not well seen at the time of Gödel, due to
historical circumstances. That's all.
But the answer is "no". There are non unique omega-consistent
extension of PA. Omega-consistency is just a bit more powerful than
consistency for proviong undecidability, but Rosser has been able to
replace omega-consistency by consistency in the proof of the existence
of undecidable statements. Would Gödel have seen Rosser point before
Rosser, the notion of omega-consistency could have not appeared at all.
I will probably come back on stability, consistency and soundness when
we arrive at the AUDA part. This is not for tomorrow. I can give
references, well see my URL.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Jun 10 2009 - 18:21:54 PDT