Date: Apr 17, 2013 12:29 PM
Author: Frederick Williams
Subject: Re: Matheology S 224
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?

> I tried to warn Nam that your question would come up.

>

> news://news.giganews.com:119/aoSdnaXPlOO2gfPMnZ2dnUVZ_sKdnZ2d@giganews.com

--

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