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Topic: Godel Incompleteness Theorem
Replies: 172   Last Post: May 2, 2013 2:49 AM

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 tchow@lsa.umich.edu Posts: 1,133 Registered: 12/6/04
Re: Godel Incompleteness Theorem
Posted: Aug 20, 2008 4:54 PM

MoeBlee <jazzmobe@hotmail.com> wrote:
>In other words, I wanted to avoid having to rest on saying
>that a formula of a meta-language "mentions" an object language in the
>sense that the object langugae is itself a member of some domain of
>interpretation of the meta-language; which may be the case, but (1) is
>too complicated, and (2) gets away from the more simple fact that in
>the meta-language we mention object languages simply by DOING it and
>not by a more complicated consideration of yet another layer of
>abstraction of viewing the object language to be an object in some
>domain.

I'm not sure why you think this is a complication.

What does it mean for the language of arithmetic to "mention" a natural
number? Surely we just *do* it. Formally, though, a natural number is
an object in the domain of the language. This isn't normally considered
to be a complicated extra layer of abstraction, but just the usual way
of doing things.

I actually think that the complications with defining a formal notion of
a meta-language don't have to do with the issue you raise here, but with
the issue that in a meta-language, we might potentially want to do
arbitrarily complicated mathematics. That is, the meta-language needs
to be a "general-purpose language" like the language of set theory, that
is capable of discussing all kinds of things in addition to the object
language. So then "M is a meta-language for L" reduces to "M is a language
for mathematics."

But "M is a language for mathematics" is probably not what you are
after. This is why I keep pressing you to say what your goal is, beyond
formalization for its own sake. What restrictions on M do you want? The
usual reason for considering formal languages are to focus on the *limits*
of what we can express in that language---either what we can express, or
what we can prove. So we may be interested in the first-order theory of
graphs or of groups because we want to show that certain graph/group
properties aren't expressible in a first-order language. If you don't
have any restrictions in mind then it's unlikely that you'll be able to
get away from "M is a language for mathematics."
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

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