> 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. > > If anything, notation like "this" is defined in term of FOL > terminologies.
ie, you use a first-order meta-logic.
> 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.
The strength of the meta-logic does have a bearing. Do you allow proof by induction over the syntax of formulas, for example?
> 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?
That's not what's at issue here.
Suppose you have a language structure for the language of Peano Arithmetic where all the axioms of PA are true, and suppose that the underlying set is X, and "0" is the constant used for the number zero.
Is there a way of characterising the element of Z correponding to "0"? Or is that a matter of opinion?