Inductive function I want to understand what is meant by an inductive set.  I've found it to be defined as: if z is in a set K then $$z\cup \{z\} $$ is in the set.  How is this possible since the reunion is between sets in general?  Thank you. 
 A: This definition is incomplete. We say that $I$ is inductive set when $\emptyset\in I$ and $(\forall z\in I)\ z\cup\{z\}\in I$. 
Inductive sets appear, for example, in ZFC, where Axiom of infinity postulates existence of inductive set, and consequently the existence of the set of natural numbers. What seems to be your misunderstanding is that you are not treating $z$ as a set, but in ZFC everything is a set. Thus, $z\cup\{z\}$ is a set that contains all elements of $z$ and $z$ itself.
Let $z = \emptyset\in I$. Then $z\cup\{z\} = \{\emptyset\}\in I$, and by iterating the process, $\{\emptyset,\{\emptyset\}\}\in I$, $\{\emptyset, \{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$ etc. We usually denote these sets by $0$, $1$, $2$, $3\ldots$, respectively.
A: You may be used to sets of numbers but with set theory you have to think of sets containing sets that contain sets ...  The notation $z\cup\{z\}$ looks a bit odd at first. This creates a new set from z containing all the elements of $z$ and the set z as a member of the new set. If z = $\{a,b\}$ then $z\cup\{z\}$ = $\{a,b,\{a,b\}\}$
With induction we define a set $I$ by starting with one or more base elements in the first step, often $∅$, and then provide a rule for building ever larger supersets from prior elements. These new supersets will also be members of the inductive set $I$.
