On 4/17/2013 11:29 AM, Frederick Williams wrote: > 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?
No. I had to think about what you said and how it applied.
But, here is a strange thought motivated by Kunen's remarks and Kleene's discussion on the eliminability of descriptions:
"So, mathematics has reached this curious place where the meaning of statements is given by semantic truth conditions based upon an indeterminable ontology presumed through purported denotation in relation to the descriptive introduction of names eliminable through representation within the uninterpreted syntax by means of arithmetization of syntax in terms of statements whose meanings are purported to be that of individual natural numbers given by semantic truth conditions based upon...."
The person to whom I made that remark never responded.