On 12/04/2013 6:59 PM, fom wrote: > On 4/12/2013 8:45 AM, Nam Nguyen wrote: >> 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. >> > > Nam, > > I told you before. The textbooks are very bad about > all of these things.
> ========================================== > > With the first two definitions technically being > axioms because the symbols may not be eliminated > through substitution. That is why the signature > is formally introduced initially as > > <<M, |M|>, <c, 2>, <e, 2>> > > Obviously, I cannot specify an infinite domain.
Exactly, fom. You, I, et al. don't seem to have a disagreement here.
What we seem to have is a difference in understanding in where we _can_ go from here, i.e., from one "cannot specify an infinite domain"!
My presentation over the years is that it does _not_ matter what, say, Nam, fom, Frederick, Peter, ... would do to "specify an infinite domain", including IP (Induction Principle), a cost will be exacted on the ability to claim we know, verify, or otherwise prove, in FOL level or in metalogic level.
The opponents of the presentation seem to believe that with IP we could go as far as proving/disproving anything assertion, except it would be just a matter of time. Which sounds like Hilbert's false paradigm of a different kind.
That's the difference on the two sides.
> And, the circular definitions constrain the > interpretation of the primitive relations, > so any set model of the sentences must be a > partial model since the defining relations > cannot be represented as elements of any class > to which the language symbols refer. > > On the other hand, this would be the point at > which you begin specifying the model of > constructive objects that you wish to serve > as your domain.
I already did at length: individuals comprising of XML tags. > > Your use of a "this" operator occurs at this > point since it is how you are defining your > domain. It is a metalinguistic usage.
It's just a _notation_ for _set_ . Everyone is entitled to have, to define notations, defineable symbols within FOL reasoning framework. No extra metalogic is necessary. > > If you try to use it in a proof concerning the > theory of the natural numbers written in > the object language, then you have to explain > it in the signature and your theory is no > longer a standard theory.
I've never said what I try to prove about cGC is a FOL. On the contrary, I've always claimed it as a meta-proof about a _meta statement_ . But that should constitute that I use knowledge outside the understanding of FOL as a reasoning framework.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.