I just read about inductively defining a set as follows :

- Take a set of objects
U.- Take a set of starting objects
B⊆U.- Let
Cbe the smallest subset ofUthat containsBand is closed under all operations on some classF.Any set

Sthat containsBand is closed under all operations inFis called inductive.

For example , consider the set of natural numbers as :

U=R,B= {0} andF= {S} where S(x) = x+1

My Question is can we define positive Real Numbers as follows :

U=R,B= [0 , 1) andF= {S} where S(x) = x+1

If so , can we prove property P(x) of positive real numbers inductively as follows :

- P is true in interval [0,1).
- If P(x) is true ,P(x+1) is also true.