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Axioms of Set Theory

To understand these axioms you have to know in what order operations like $\wedge$ (AND) and $\vee$ (OR) are performed. This is determined by precedence as described in Table 3.2. Subexpressions involving $\in$ are evaluated before logical operations like $\wedge$ or $\rightarrow$. 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 $n$ is defined as the union of all the elements of $n$ plus $n$ itself. Thus $n+1$ will contain $n+1$ elements as long as $n$ contains $n$ elements.

Completed second draft of this book

PDF version of this book