
Re: Matheology S 224
Posted:
Apr 18, 2013 9:19 AM


Nam Nguyen wrote: > > On 17/04/2013 8:48 AM, fom wrote: > > On 4/17/2013 9:36 AM, Frederick Williams wrote: > >> Nam Nguyen wrote: > >> > >>> "x is in a nonempty 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 nonempty 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 nonempty 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 nonlogical axioms of that theory. > > Frederick seemed to be confused: what I'm doing here has nothing to > do with formal systems, theories, axioms of formal systems.
You wrote:
'"x is in a nonempty subset of S" could be _expressed_ as a FOL language expression: x e S' /\ Ay[ y e S' > y e S].'
How does the FOL expression express "x is in a nonempty subset of S"? It can only do so if "e" has a particular meaning. How is that meaning established?
Also, as I remarked elsewhere, "x e S' /\ Ay[ y e S' > y e S]" doesn't express "x is in a nonempty subset of S".
> And I _already_ gave clear caveats about that! > > >> > >> 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 nonlogical axioms > >> of that theory. > >> > >> Note that set theory can express its own provability predicate.
 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

