FOM: Nonstandard models of N

[Todd Wilson]

Dear FOMers:

As we know, every countable non-standard model of arithmetic has order
type

    NN = omega + (Z * Q) ,

where Z and Q are the order types of the integers and rationals, and
where the successor function s : NN -> NN is obvious.  Are there any
explicit definitions of + and * on NN that make (NN, 0, s, +, *) into
a model of arithmetic?  If not, is there a proof that such definitions
are impossible (for example, a model of set theory in which all models
of arithmetic are standard)?  In either case, where was this first
established?  Thanks,

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh