### Inductive sets

Inductive sets are sets such that

1. $\emptyset\in A$, where $A$ is the inductive set under consideration.
2. $x\in A\implies x^+\in A$, where $x^+=x\cup \{x\}$

Natural numbers (here, natural numbers include $0$) are those sets that are present in every inductive set. Let us explore this strand of thought more.

Clearly, $\emptyset$ belongs to every inductive set. Hence, it also belongs to the set of natural numbers (or $\Bbb{N}$). Also, seeing as $0^+,0^{++},\dots$ belong to every inductive set, all these elements make up the set of natural numbers.

An inductive set that is a proper superset of $\Bbb{N}$ would be $\{0,0.5,1,1.5,2,2.5,\dots\}$. The successor of every element is there in this set, and it also contains $0$. Hence, it is an inductive set.

EDIT: I forgot to mention the motivation behind writing this post. To prove a property for all elements in an inductive set, we have to prove it for all elements without predecessors, and then show that if $a$ satisfies the property, then so does $a^+$. For example, in the inductive set $\{0,0.5,1,1.5,2,2.5,\dots\}$, we have to prove the property for $0$ and $0.5$, and then also prove that $a$ satisfies the property $\implies$ $a^+$ satisfies the property. Because the only starting point of $\Bbb{N}$ (element without predecessor) is $0$, now you  know how and why induction works in natural numbers.