Re: [ontolog-forum] Interesting Category - Chindogu

[Pat Hayes <phayes@xxxxxxx>]

At 10:26 AM -0800 1/15/08, Randall R Schulz wrote:

> On Tuesday 15 January 2008 08:00, Pat
> Hayes wrote:
> > At 3:57 PM -0800 1/14/08, Duane Nickull wrote:
> > >...
> > >
> > >As a long time Kawakami fan, I have to both admire his work
> but find
> > > this list raises bigger issues.  Our universe, for
> example, is a
> > > paradox of logic.  According to most logic, nothing can
> go on
> > > eternally
> >
> > Wha? Where did you get that notion from? Most
> > logics say nothing at all about eternity or
> > otherwise, but those that do address temporal
> > continuity allow for eternal entities.
>
> Is it not impossible in FOL to finitely axiomatize finiteness?
> (Does
> that count as a paradox??)

Its impossible to axiomatize finiteness (in the sense of a sentence having only finite models) in FOL, true. No, that does not count as a paradox. It is possible to axiomatize finiteness (in the same technical sense) in logics of higher order. Here's an axiomatization of finiteness in Common Logic:

(= (length) 0)

(forall (x ...)(= (length x ...)(s (length ...))))

(forall (x)(iff (Num x)(exists (...)(= x (length ...)))))

Num is true of all and only the natural numbers in any CL model of these axioms. This uses sequence markers, of course, which take CL beyond first-order expressivity.

Pat

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