Re: Metamathematically True or False?


george wrote:
> Rupert wrote:
> > george wrote:
> > > 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.
>
> Well, it isn't. Once you have some unambiguous way of specifying
> the language, yes, the existence ceases to be contentious, but
> otherwise, you have to have a name for it, and you have to DEFEND your
> decision to CALL the language by that name.
>

There is an unambiguous way of specifying the first-order language of
arithmetic. See Machover, "Set Theory, Logic, and their Limitaions",
Chapter 10. It's called the first-order language of arithmetic because
it's a first-order language which can be used to describe the structure
which is studied in arithmetic.

> > Perhaps you're talking about existence of the standard
> > semantics for that language?
>
> No, not even perhaps.
> Please, don't do that.
>
> > > 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?
>
> Because if the resulting class is not recursively axiomatizable,
> and the resulting set has the property that neither it nor its
> complement is recursively enumerable, then, by definition, you
> get a lot of relatively short sentences with the property that you
> cannot determine, even after a relatively a long amount of time,
> whether they are or aren't in the set, THAT'S why. "True first-order
> arithmetic" is the classic example of such a theory.

That doesn't mean you don't know what class you're talking about. It
doesn't even mean you don't know which sentences are in the class are
which are not. It means sometimes it's hard to tell whether a given
sentence is in the class or not.


> My point being that rational parlance would NOT call THAT collection
> of sentences (the truths of the standard model) a "theory" in any case
> (DESPITE the fact that it is closed under consequence).
>

Well, you're entitled to your opinion. But if you want to use
nonstandard terminology, you're going to have to explain it to others.

> > But what's that got to
> > do with it? Who mentioned truth?
>
> As you wind up saying along another subthread,
> truth was in AK's definition of sigma-1-complete.
> I'm sorry if I argued ugly and lost potential partial agreement.

.

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