Re: Metamathematically True or False?


Rupert wrote:
> george wrote:
> > Rupert wrote:
> > > It's easy for a theory in the first-order language of arithmetic not to
> > > be Sigma-1 complete.
> >
> > Sure,
> > if you concede the existence of a first-order language of
> > arithmetic.
> >
>
> Existence of the language seems to me to be a pretty uncontentious sort
> of proposition. Perhaps you're talking about existence of the standard
> semantics for that language? But that would have no bearing on what I
> said.
>
> > Standard parlance in that arena is unreasonable.
> >
> > If you define a theory (as is standardly and wrongly done)
> > as just any old consequence-closed class of sentences, then,
> > yes, you can get a whole lot of basically worthless junk.
> > Those definitions of those
> > classes are problematic for the simple reason that it is
> > too hard to say that you ever know what class you are talking
> > about; you don't know which sentences are in the class and which
> > are not.
> >
>
> Why not?
>
> > Standard parlance usually uses "formal theory" to mean
> > what "theory" ought to mean, but even that is not sufficient.
> >
> > Theories deserve to be called that BECAUSE they contain
> > THEOREMS, NOT merely sentences.
> > Theorems deserve to be called that because they are PROVABLE,
> > NOT "true".
> >
>
> I think I could agree with this last sentence. But what's that got to
> do with it? Who mentioned truth?
>
> > > The completeness theorem has nothing to do with it.
> >
> > It would be more accurate to say that "truth" has nothing to do with
> > it,
> > although the separate/AK front of the battle.
>
> What does AK stand for?
>
> Yes, I quite agree, truth has nothing to do with it.

Actually, this is not quite right. Truth in the standard model for
arithmetic does come up in the definition of Sigma-1-complete.

.

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