```Date: Apr 18, 2013 9:19 AM
Author: Frederick Williams
Subject: Re: Matheology S 224

Nam Nguyen wrote:> > On 17/04/2013 8:48 AM, 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.> > Frederick seemed to be confused: what I'm doing here has nothing to> do with formal systems, theories, axioms of formal systems.You wrote:'"x is in a non-empty subset of S" could be _expressed_ as a FOLlanguage expression: x e S' /\ Ay[ y e S' -> y e S].'How does the FOL expression express "x is in a non-empty subset of S"? It can only do so if "e" has a particular meaning.  How is that meaningestablished?Also, as I remarked elsewhere, "x e S' /\ Ay[ y e S' -> y e S]" doesn'texpress "x is in a non-empty subset of S".> And I _already_ gave clear caveats about that!> > >>> >> 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.-- 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
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