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Re: [ontolog-forum] Interesting Category - Chindogu

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Chris Menzel <cmenzel@xxxxxxxx>
Date: Tue, 15 Jan 2008 15:09:28 -0600 (CST)
Message-id: <alpine.DEB.1.00.0801151452080.10841@xxxxxxxxxxxxxxxxx>
On Tue, 15 Jan 2008, Pat Hayes wrote:    (01)

> 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.    (02)

Quick clarification.  There are of course (as Pat well knows)
first-order sentences that have ONLY finite models, e.g.,    (03)

   (forall (x y) (= x y)).    (04)

The more accurate claim here is that it's impossible to axiomatize
finiteness in FOL in the sense of a sentence being true in ALL and only
finite models.  More generally, there is no *set* of first-order
sentences, recursive or not, with arbitrarily large finite models but no
infinite model.    (05)

-chris    (06)

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