
Re: Matheology S 224
Posted:
Apr 19, 2013 9:08 PM


On 19/04/2013 8:42 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 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 Def1, Def2_ .
> > 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.
NYOGEN SENZAKI 

