New! Recursive Definitions of Sets: The set \(S\) (pick a name) is defined by: \[\begin{array}{ll} \textrm{Basis Step: } & \textrm{Specify finitely many elements of } S\\ \textrm{Recursive Step: } & \textrm{Give rule(s) for creating a new element of } S \textrm{ from known values existing in } S, \\ & \textrm{and potentially other values}. \\ \end{array}\] The set \(S\) then consists of all and only elements that are put in \(S\) by finitely many (a nonnegative integer number) of applications of the recursive step after the basis step.