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To understand 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 3.2.
Subexpressions involving 
are
evaluated before logical operations like or
. Parenthesis 
and square brackets ![$[]$](img35.png){kind=small}
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.
Completed
second draft of this book
PDF version
of this book
Next: Axiom of extensionality Up: Mathematical
structure Previous: Formal mathematics
Contents
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