> 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.
If you wish to formalize your model theory you may do so in, say, ZFC. You then get omega axiomatically.
> 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.
A side is a line segment, i.e., a continuum of a certain kind. You, on the other hand, seem to be an isolated point.