[FOM] Dual of Dedekind-infiniteness.

[Alex Blum]

I assumed that you maintained that there is a core concept of finitude 
but that the concept wasn't clear. Now I'm not sure, since you speak of 
and appeal to a multitude, e.g., "the common definitions of finitude". 
If they are definitions, that is one thing and if they are explications 
of a core concept of finitude or finite number then it is another thing. 
I'm assuming it is the latter since you think that the notion of a 
finite set  is more basic than the notion of a natural number. I wonder 
if the latter is true for if one were to posit that a given set is 
finite and yet is unfamiliar with the concept of a finite number of 
objects, we might think that he failed to grasp what finite means. The 
converse however is clearly untrue. Following through with the 
assumption that there is  a core concept, would it not be true that one 
who asks if the number of stars is infinite or finite is asking in other 
words if the number of stars is either infinite or is a(standard) integer?

Larry Stout wrote:

>You've assumed that "finite" means isomorphic to a finite cardinal.   
>That only makes sense if you have a pre-existing notion of natural  
>number.  In the absence of the axiom of choice several of the common  
>definitions of finite do not agree with the notion of cardinal  
>finite.  Certainly if the underlying logic you work in is not  
>classical (for instance, if you are working in a topos where the  
>logic is intuitionistic and you usually do not have the axiom of  
>choice) there are even more different notions of finite.
>
>I would think that the notion of finite set is more basic than the  
>notion of natural number-- the natural numbers form the skeleton of  
>the category of finite sets, so different notions of finite should  
>give rise to different notions of natural number.
>
>L.N. Stout
>
>On Oct 8, 2007, at 11:40 AM, Alex Blum wrote:
>
>  
>
>>Lawrence Stout wrote:
>>
>>    
>>
>>>In general, nailing down what "finite" means can be very difficult.
>>>
>>>
>>>
>>>      
>>>
>>No doubt if one wants to include 'finite being', ' finite universe'  
>>etc.
>>But why would there be a problem with  'finite number'? For isn' t a
>>number finite iff it is an integer? And  a number of items is  
>>finite iff
>>its number is an integer.
>>
>>Alex Blum
>>    
>>
  Alex Blum