[FOM] Q

[Vaughan Pratt]

How about the skeletal subcategory of Set consisting of the countable 
sets, with + as coproduct, * as product, 0 as the empty set, 1 as the 
singleton, Sx as x+1, and = as existence of an isomorphism?

Vaughan Pratt

Andrew Boucher wrote:
>> On Fri, 21 Sep 2007, Bill Taylor wrote:
>>
>>> If we add axioms of additive and multiplicative commutativity to Q,
>>> we get something a little more like regular arithmetic, though still
>>> without the scheme of induction.
>>>
>>> Does this new theory, Q+, have any easy models (other than those  
>>> for PA)?
> 
> Extend N with the element c.  Define:
> 
> 1/  Sc = c
> 2/  x + c = c + x = c
> 3/  0 * c = c * 0 = 0
> 4/  x * c = c * x = c (for x != 0)
> 
> Pf:
> c + Sy = c = Sc = S(c + y)
> x + Sc = x + c = c = Sc = S(x + c)
> 
> c * Sy = c = c * y + c
> 
> 
> 
> 
> 
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