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The integers are defined by an axiom that asserts the existence
of a set 
that contains all the integers.
is defined as the set containing
and having the property that if 
is in then 
is in
. To write this compactly we define some
notation. We use the integer 0 to represent the empty set. If we
have some set 
then we can construct a set containing
that is written as 
.
![\begin{displaymath}\exists x \, 0 \in x \wedge [\forall y \, (y \in x) \rightarrow (y \cup \{y\} \in x)]\end{displaymath}](img58.png){kind=small}
This says there exists a set that contains the
empty set 0 and for every set 
that belongs to
the set 
constructed as
also belongs to .
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