The set
[math]\href{/cs2800/wiki/index.php/%E2%84%95}{ℕ}
[/math] of
natural numbers is defined by
[math]\href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} \href{/cs2800/wiki/index.php/Definition}{:=} \href{/cs2800/wiki/index.php/Enumerated_set}{\{0, 1, 2, 3, \dots\}}
[/math].
Later in the semester, we will talk about inductively defined sets; this allows us to define ℕ more formally:
[math]n \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} \href{/cs2800/wiki/index.php/BNF}{::=} \href{/cs2800/wiki/index.php/Z}{Z} \href{/cs2800/wiki/index.php/%5Cmid}{\mid} \href{/cs2800/wiki/index.php/S}{S} ~ n
[/math]
Here Z stands for "zero" and [math]\href{/cs2800/wiki/index.php/S}{S}~n
[/math] stands for "successor" (i.e. the next natural number, [math]n+1
[/math]).
We can then define [math]0 := \href{/cs2800/wiki/index.php/Z}{Z}
[/math], [math]1 := \href{/cs2800/wiki/index.php/S}{S}~\href{/cs2800/wiki/index.php/Z}{Z}
[/math], [math]2 := \href{/cs2800/wiki/index.php/S}{S}~(\href{/cs2800/wiki/index.php/S}{S}~\href{/cs2800/wiki/index.php/Z}{Z})
[/math], and so on.
