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 FOL

language 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 meaning

established?

Also, as I remarked elsewhere, "x e S' /\ Ay[ y e S' -> y e S]" doesn't

express "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