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**Contents**

Before defining any structure we need an axiom that asserts the existence of the empty set. This axiom uses the existential quantifier (). means there exists some set for which is true. Here is any expression that includes . We also introduce the notation to indicate that is not a member of set .

The axiom of the empty set is as follows.

This says there exists an object that no other set belongs to. contains nothing.

Before we can define the integers we need to give two axioms for constructing finite sets.

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