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To understand the formal version of these axioms you have to know in what order operations like (AND) and (OR) are performed. This is determined by precedence as described in Table 5.2. Subexpressions involving are evaluated before logical operations like or . Parenthesis and square brackets are used to override standard precedence and to make clearer how an expression is to be evaluated. The portion of an expression inside parenthesis is evaluated first.

The clearest example of how set theory builds on the empty set
is the construction of the integers. The integer
1 is defined as the set containing the empty set. Two is defined as
the set that contains 1 and the empty set (or 0). Not surprisingly
3 is the set that contains 0, 1 and 2. In general
is defined as the union of all the elements of
plus itself. Thus will contain
elements as long as
contains elements.

- Axiom of extensionality
- Axiom of the empty set
- Axiom of unordered pairs
- Axiom of union
- Axiom of infinity

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