Nam Nguyen wrote: > > On 12/04/2013 3:29 AM, Alan Smaill wrote: > > Nam Nguyen <email@example.com> writes: > > > >> But what is "meta-logic of the meta-language", in the context of FOL > >> structure? Or is that at best just intuition and at worst just a > >> buzzword? > > > > You tell us that it is possible to reason about language structures. > > What logic are you using to do that -- or is that at best just intuition? > > I've used FOL ( _First Order Logic_ ) definitions that one should be > familiar with.
First order logic is useless in this respect. What is needed is a higher order logic, or first order logic with non-logical axioms governing the non-logical symbols. For example, higher order logic, or a theory of sets, is needed to do formal model theory. That is what Tarski did in his 'Concept of truth in formalized languages' paper.
> If anything, notation like "this" is defined in term of FOL > terminologies.
> So I don't see all that historical context of "meta-logic" would > have anything to do with the issue of, say, whether or not it's > impossible to construct the naturals as a language model. > > If you could construct it, as I did construct Mg, M1, ..., then present > the construction, otherwise at least for the time being admit you > couldn't do it. > > Why would that be such a difficult task for one to do?
Your Ms cannot be defined outside of a (formal or informal) language of sets.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting