Date: Apr 12, 2013 7:20 PM
Author: Frederick Williams
Subject: Re: Matheology § 224

Nam Nguyen wrote:
>
> On 11/04/2013 9:38 AM, Frederick Williams wrote:

> > Nam Nguyen wrote:
> >

> >>
> >> Don't pretend I didn't have a valid reason why I refuse to technically
> >> argue with you on matters related to language structure such as GIT.

> >
> > G\"odels' incompleteness theorem says nothing about language structures,
> > its statement and its proof are wholly syntactical.
> >

>
> So the natural numbers, underlying GIT's arguments, isn't a notion
> of what we'd today refer to (or even define) as a language structure?


No, the natural numbers aren't a language structure. A language
structure looks like this:

<a set U, family of relations of various arities on U, family of
functions of various arities on U, distinguished elements of U>.

The details will depend on the language. U may be the set of natural
numbers, indeed (for first order languages) it can always be chosen to
be N so long as it isn't finite. But that has got nothing to do with
your false claim that G\"odel's incompleteness theorem relates to
language structure.

--
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