Re: Recursivity vs. Provability


> Charlie-Boo <chvol@xxxxxxx> wrote:
> >Assume that the set of provable statements is
> >recursively enumerable.
> >For any given statement, whether it is provable
> >or not is the same as
> >whether it is in this r.e. set.
> >Now what does that tell us about the
> >statements that we can prove?
[ in particular, is there...]
> >Or a statement that is not provable and which is true?


H. Enderton wrote:
> True where?

I just had to quote that for emphasis.
This is one of the core terminological points
we sort of have to pound through the heads of
normal people: here in THIS weird room, there
is NO SUCH THING as "true", just PLAIN true.
You only get to be true (or false) UNDER AN
INTERPRETATION, or IN A MODEL.

Statements that are "just plain" true are being
talked about in the context of 1 particular STANDARD
model, and what is actually being alleged is that they
are true in THAT model. They are likely to be false
in another one (because if they were true in all models,
we would be saying that they were provable, AS OPPOSED to
saying that they were true).

Sometimes it is well-known in advance that you have
one intended model in mind, and that the theory's value
is only in helping you prove things about that model.
Other times, the theory itself is important and all its
various models are more or less created equal.

For theories of the kind that C-B was talking about,
that are r.e. but not recursive, yes, there must be
some statements that the theory does not decide, and
yes, there must also be some models in which these
statements are true (and others in which their denials
are true).

.

Relevant Pages

  • Re: Shabbos candles (Was Re: Whats wrong with Conservative Judaism)
    ... only slaves and animals are included. ... Seems to me that this quote supports the opposite conclusion - it's saying ... authoritative interpretation. ... Is it the "real" interpretation? ...
    (soc.culture.jewish.moderated)
  • Re: Is sentience an emergent brain behavior?
    ... Allan's argument for dualism as you would like it to be. ... best arguments for your opponent's interpretation and not the worst. ... He didn't want to acknowledge what he was really saying either. ... Can it animate matter? ...
    (comp.ai.philosophy)
  • Re: I have been CHALLENGED. . .
    ... The wife and I were married by a woman priest in the Anglican ... but you guys have a more fairinterpretation afaik. ... You didn't excommunicate someone for thinking the earth goes around the ... I started out saying liberal. ...
    (alt.sysadmin.recovery)
  • 7.18.1p1 uxxx_t <=> xxx_t
    ... What is the interpretation of 7.18.1p1, ... When typedef names differing only in the absence or presence ... Defect Report 269 has a parenthetical comment saying ...
    (comp.std.c)
  • Re: Running Check Disk and Disk Defrag
    ... You're saying that the parts in the drive won't wear down ... The physical HD has no awareness of defrag as a process different from ... anything else that swings the heads and writes through them. ... the heads around a lot for a certain amount of time, ...
    (microsoft.public.windowsxp.general)