Inductive set

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Definition

Let $I$ be a set. We call $I$ an inductive set if both of the following properties hold:

1. $\emptyset\in I$ - often written $0\in I$ as $0$ is represented by the $\emptyset$ - and
2. $\forall n[n\in I\implies$$S(n)$$\in I]$ - often written as "if $n\in I$ then $(n+1)\in I$"
   • $S(x)$ denotes the successor set of $X$

Caveat:Note that this certainly describes the natural numbers as we require $\emptyset\in I$, so they're in there. The problem is that rule 2 seems to require that for every element $n$ that $n\cup\{n\}$ is in there too. - this seems to be intended

See also

• The natural numbers
• The axiom of infinity - positing that an inductive set exists.