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 C be the smallest subset of U that contains B and is closed under all operations on some class F.
Any set S that contains B and is closed under all operations in F is called inductive.
For example , consider the set of natural numbers as :
- U = R , B = {0} and F = {S} where S(x) = x+1
My Question is can we define positive Real Numbers as follows :
U = R , B = [0 , 1) and F = {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.
There is something analogue to the induction principle, called transfinite induction. It is not quite as intuitive as simple induction on $\omega$, but you make do with what you have. It works like this: let $P$ be a property, which you want to proof is true for all ordinal (or cardinal) numbers. You want to show that 1)-P(0) is true 2)-For all ordinals $a < b$, if $P(a)$ is true, $P(b)$ is true
Usually, the latter is divided in two stages, the successor ordinal and the limit ordinal: for the successor, you apply induction sort of in the classical way, whereas for the limit ordinal you have to rely on the fact that it is defined as $sup\{b|b > a\}$, and use 2)