
Re: Matheology S 224
Posted:
Apr 19, 2013 7:55 AM


Nam Nguyen wrote: > > On 18/04/2013 7:19 AM, Frederick Williams wrote: > > 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? > > By establishing the interpretation that "e" would mean "is a member of".
How is that done? Note that "is a member of" does not have one meaning.
> > 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". > > Why?
It says that x is in S' and S' is a subset of S.
 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

