Lester Zick wrote: > Technically set theory only assumes the set of real numbers by > assuming a natural number and the suc( ) axiom.
In Z set theory, from just the axiom schema of separation and the axiom of extensionality, we prove the existence of a unique empty set, which then fits the definition of the least natural number. And for any given natural number, we can prove its existence, even without the axiom of infinity. And we prove the existence of the set of natural numbers from the axioms but now including the axiom of infinity. And then the existence of the set of real numbers is proven (and we can construct different structures that are complete ordered fields, all of which are isomorphic).
So I don't know what assumption of the existence of a (particular) natural number you have in mind and then what axiom you mean by the "suc( )" axiom.
As to circles, we can define them in set theory:
Let X stand for Cartesian product (which is definable in set theory). Let R stand for the set of real numbers (which is definable in set theory). Let d stand for the standard distance function on RxR (which is definable in set theory). Let > stand for the standard ordering on R (which is definiable in set theory). Let 0 stand for the real number zero (which is definable in set theory).
c is a circle <-> EpEe(p in RXR & e>0 & Az(zec -> (z in RXR & d(p z) = e)))
I.e., a circle c is a set of points in RXR such that there is a given point p and a given non-zero distance e such that each member of c is e distance from p.
And this definition can be adjusted to define 'is a circle' in other n-dimensional spaces.