Inductive set (axiom of infinity)

In the context of the axiom of infinity, an inductive set (also known as a successor set) is a set X with the property that, for every $x \in X$, the successor $x' = x \cup \{x\}$ of x is also an element of X and the set X contains the empty set $\varnothing$.

More formally, X is inductive if

$\varnothing \in X \land (\forall x) (x\in X \Rightarrow x\cup\{x\} \in X)$

An example of an inductive set is the set of natural numbers.

See also

• Natural number
• Peano axioms

External links

• Mathworld: Inductive set

References

• Hrbacek, Karel; Jech, Thomas (1999). Introduction to Set Theory (3 ed.). Marcel Dekker. ISBN 0824779150. 

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