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Next: Infinity Up: Axioms of Set Theory Previous: Axiom of union   Contents

Axiom of infinity

The integers are defined by an axiom that asserts the existence of a set $\omega$ that contains all the integers. $\omega$ is defined as the set containing $0$ and having the property that if $n$ is in $\omega$ then $n+1$ is in $\omega$. Writing this compactly requires some notation. $\emptyset$ represents the empty set. From any set $x$ then one can construct a set containing $x$. This set is written as $\{x\}$.


\begin{displaymath}\exists x \, \emptyset \in x \wedge [\forall y \, (y \in x) \rightarrow (y \cup \{y\} \in x)]\end{displaymath}

This says there exists a set $x$ that contains the empty set $\emptyset$ and for every set $y$ that belongs to $x$ the set $y+1$ constructed as $y \cup \{y\}$ also belongs to $x$.

The remaining axioms are developed in the next chapter starting in Section 6.3. The discussion of the infinite at the end of this chapter and the start of the next lays the groundwork for those axioms.


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