On 19/04/2013 8:42 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 a first order > language which has a binary relation symbol 'e' interpreted as the 'is > an element of' relation between sets (though not in the way you have > written).
Which specific way that I've used the symbol 'e' contrary to its usual interpretation?
> What makes you think that there is no first order language > with a unary predicate (say) 'p' with 'px' interpreted as 'x is proven' > among formulae? I refer you to provability logic.
If "provability logic" isn't First Order Logic, then it's not relevant to the context I'm talking _here about Def-1, Def-2_ .
> > When you wrote 'truth and semantics aren't the same' did you mean 'truth > and syntax aren't the same'? Or, perhaps, syntax and semantics aren't > the same; though G\"odel and Carnap taught us that a lot of semantics > can be reduced to syntax (but not all, pace Tarski).
I'm talking about what I already said above: "truth and semantics aren't the same".
The formula "m=n" has the semantics that the individual m equals to the individual n. Yet, what's the truth value of this formula?
Do you understand now that "truth and semantics aren't the same"?
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.