[FOM] Q

[Andrew Boucher]

>
> 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