|
|
Re: Godel Incompleteness Theorem
Posted:
Aug 19, 2008 6:35 PM
|
|
On Aug 19, 2:31 pm, tc...@lsa.umich.edu wrote:
> In article <df6e5a98-7203-4cb4-b8e7-c073affe1...@p31g2000prf.googlegroups.com>,
>
> MoeBlee <jazzm...@hotmail.com> wrote:
> >M is a metalanguage for L <-> [fill in here some formula P (with only
> >free variables being 'M' and 'L') in said extended language of set
> >theory]
>
> >I don't know about you, but for me, it's not at all clear how to
> >devise such a formula P.
>
> Perhaps the problem is that there are *too many* ways to devise a P.
>
> A somewhat trivial way would be something like, "M is Quine's protosyntax
> and L is a language that protosyntax is equipped to talk about."
I trust then that 'is equipped to talk about' is something that can be
expressed in a formula? Anyway, I'll review Quine's protosyntax. (It's
in his book 'Mathematical Logic', if I recall.) But I'm not sure this
is the way I want to go, since what I have in mind is not expressing
that some PARTICULAR method provides for a meta-language, but rather a
definition that allows for various methods, all of which have the
property of being a meta-language for another language.
> This might not satisfy you because you might be interested in
> meta-languages other than protosyntax.
Right.
> So you could define some class
> C of languages for syntax and then say, "M belongs to C, and L belongs
> to the class of languages that C talks about."
Two points: (1) Not only syntax but semantics is something that a
metalanguage may handle. (2) It is the very phrase "talks about" that
I am wondering how to formalize.
> Meta-languages don't necessarily even have to be languages of syntax.
> Often one identifies syntactical entities with natural numbers, so that
> the language of arithmetic can be treated as a meta-language.
Yes.
> At some point one has to ask the question of why you want to write down
> an explicit P.
I've answered that question for myself. Granted, I don't expect to get
other people enthusiastic about the question unless I give some reason
for them to be enthusiastic. I asked you only because the context of
this discussion reminded me of the question and because you are quite
knowledgable. But if the question does not interest you enough, then I
surely do not mean to press you any further for help beyond what our
exchanges now bring up.
> Is it because you feel you don't understand exactly what
> you mean by "M is a meta-language for L" unless you can formalize it?
No, I don't feel that something is necessarily not adequately
understood merely for lack of formalization. Rather, in this case, the
question occurred to me, and then it seemed interesting to me at least
in the sense that no obvious answer came to me. Also, I think such a
rigorous definition would be helpful (to me at least) in dealing with
the very kind of question that arose in this thread: whether symbols
may be used as variables in both a meta-language and one of its object
languages. By having a rigorous definition of 'metalanguage for an
object language' (and 'meta-theory' too), we might be able to give a
rigorous determination of such matter. Moreover, since the notion of
meta-language is so basic and pervasive, it strikes me of interest to
have a rigorous mathematical definition, just as we formalize other
notions, such as computability, proof, truth, etc.
We have all kinds of rigorous definitions and theorems about
languages, theories, structures etc. being sub___, extensions,
conservative extensions, elementary embedded, interpretable,
isomorphic, definability in, etc. It doesn't strike me as unreasonable
to wonder whether we might also give a rigorous definition for a
language being a meta-language of another language.
> Or are you trying to show that some specific meta-mathematical argument
> can be carried out on the basis of some weak set of axioms, so that you
> need to be more formal about what the meta-language is than is usually
> the case?
No, that hasn't come up at this point for me.
> If you're just trying to do it "for fun" then I think the
> problem is that there are too many ways to proceed and it's not clear
> what choice to make unless you have some idea of what you're trying to
> accomplish.
I think I was as specific as I could be about what I want to
accomplish. It is to find a suitable 'P' to be a definiens for 'M is a
metalanguage for L'.
> One of the skills one needs to develop as a mathematician is to learn
> how to reason at the level of formality appropriate to the situation.
> Excessive formality can be an impediment both to clarity and creativity
> if it is pursued when there is no clear need for it.
I have my own standards formalization and understanding. Formalization
itself is one of my interests in mathematics (though not my exclusive
interest). That is my personal choice based on my own inclinations as
one who studies mathematics recreationally and to pursue certain
matters of intellectual curiousity. I am not too worried about
excessiveness. Of course, if I bring up certain questions about
formalization in a conversation, then anyone is free to say that he or
she is not interested in that particular point of formalizing.
So, I can understand that you might not be interested very much in the
question I asked, but I don't think the question is somehow
intrinsically excessive, uninteresting, or unworthy of effort to
answer.
MoeBlee
|
|
