Re: Metamathematically True or False?


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.

.

Relevant Pages

  • Re: Metamathematically True or False?
    ... > It's easy for a theory in the first-order language of arithmetic not to ... Standard parlance in that arena is unreasonable. ... Theories deserve to be called that BECAUSE they contain ... Theorems deserve to be called that because they are PROVABLE, ...
    (sci.logic)
  • Re: Torkel Franzen on truth
    ... truth with his 1950s definition of the different notion of truth-in-a- ... It is almost impossible to confuse ... If you start with a signature and a set of axioms phrased ...
    (sci.logic)
  • Re: Sorry Godel - All Truths are Provable
    ... same as its provability-value. ... the first-order language of arithmetic and provability in PA. ... The incompletness theorems do ... every truth of first-order arithmetic is provable in PA. ...
    (sci.logic)