In article <64702ee8-65a2-4622-9fc6-7514ca8347cf@r15g2000prh.googlegroups.com>, 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
