Date: Apr 17, 2013 12:29 PM
Author: Frederick Williams
Subject: Re: Matheology S 224
> 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?
> I tried to warn Nam that your question would come up.
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