[Doctor Stochastic]

Your comments seem to identify "number" with "integer." The use of the "real line" allows other objects (pi, Sqrt(2), e, etc.) to have the same "existence rights" as the integers or fractions. There are no problems doing so. Actually, arithmetic on the reals (with addition and multiplication) is catagorical; there's only one real line. Arithmetic on the integers (with addition and multiplication) is undecidable.

The above can be taken to mean that, although one can start with the integers, proceed through the rationals, and complete the system by various methods (Caucy sequeces, Dedekind cuts, etc.), and thus obtain the reals; (breath mark); there is no consistent method of starting with the reals and uniquely identifying the integers.

[Hank Kerchief]

...there is no consistent method of starting with the reals and uniquely identifying the integers.

Yes. That's why I start with the integers.

Hank