[FOM] Q

[Gyorgy Sereny]

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)?


The finite theory PA^- defined in Richard Kaye: Models of Peano
arithmetic (Clarendon Press, Oxford, 1991) is stronger than Q and
contains the axioms of additive and multiplicative commutativity
(cf. pp. 16-18). (Kaye calls PA^- the theory of the non-negative parts
of discretely ordered rings. This name make it possible to guess
what kind of axioms it contains.)

The language of PA^- is (0,1,+,x,<).  One can find a short and easy
proof on p.18 showing that if Z[X]^+ is the structure of all
non-negative polynomials in the ring of polynomials in one variable X
with coefficients from Z (with an appropriately chosen order <), then
Z[X]^+ is a model of PA^- in which the sentence that `every number
is either even or odd' is false.


Gyorgy Sereny