At 3:09 PM -0600 1/15/08, Chris Menzel wrote:
>On Tue, 15 Jan 2008, Pat Hayes wrote:
>
>> 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.
>
>Quick clarification. There are of course (as Pat well knows)
>first-order sentences that have ONLY finite models, e.g.,
>
> (forall (x y) (= x y)). (01)
Ah yes. Good catch, I spoke carelessly (as I often do...) (02)
Pat (03)
>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.
>
>-chris
>
>
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