On 2008年1月15日, 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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