[FOM] Dual of Dedekind-infiniteness.

[Larry Stout]

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
>
>
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