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Re: [ontolog-forum] Interesting Category - Chindogu

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Tue, 15 Jan 2008 13:57:32 -0600
Message-id: <p06230919c3b2b94311a1@[]>
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.

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