[FOM] Q
Vaughan Pratt
pratt at cs.stanford.edu
Wed Sep 26 06:48:31 EDT 2007
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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