fom wrote: > > On 4/17/2013 9:36 AM, Frederick Williams wrote: > > Nam Nguyen wrote: > > > >> "x is in a non-empty subset of S" could be _expressed_ as a FOL language > >> expression: x e S' /\ Ay[ y e S' -> y e S]. > >> > >> On the other hand, in "x is proven to be in a non-empty subset of S", > >> the _meta phrase_ "is proven" can not be expressed by a FOL language: > >> "is proven" pertains to a meta truth, which in turns can't be equated > >> to a language expression: truth and semantics aren't the same. > > > > "x is in a non-empty subset of S" can be expressed in the language of a > > first order theory with a binary predicate e. The intended meaning of e > > is given by the non-logical axioms of that theory. > > > > What reason is there to suppose that "x is proven" cannot be expressed > > in the language of a first order theory with a unary predicate p (say)? > > The intended meaning of p would then be given by the non-logical axioms > > of that theory. > > > > Note that set theory can express its own provability predicate. > > > > Really? Are you referring to, say, Kunen's discussion of > Tarski's undefinability of truth by representing formulas > with their Goedel numbers?
Yes, that is what I had in mind. Have I misrepresented Kunen?
> I tried to warn Nam that your question would come up. > > news://news.giganews.com:119/aoSdnaXPlOO2gfPMnZ2dnUVZ_sKdnZ2d@giganews.com
-- 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