On 12/04/2013 7:50 AM, Alan Smaill wrote: > Nam Nguyen <email@example.com> writes: > >> On 12/04/2013 3:29 AM, Alan Smaill wrote: >>> Nam Nguyen <firstname.lastname@example.org> 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.
OK. Why don't we just call it First Order Logic as everyone is supposed to understand already? > >> 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?
So you're talking the strength (and weakness) of First Order Logic. > >> 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.
"Suppose" is hypothetical but sure, I could hypothesize such. > > Is there a way of characterising the element of Z correponding to "0"? > Or is that a matter of opinion? > But, FOL definition of language structure already provides that, e.g.:
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