 From the semantic point of view, to give meaning to the new variables we need new universes whose elements are sets and relations over the universe of individuals, say \(\mathbf{A}.\) In the so-called standard semantics, a set variable ranges over the power set of the individual’s universe, \(\wp (\mathbf{A})\), and a n-ary relation variable ranges over the power set of the n-ary Cartesian product of the individual’s universe, \(\wp (\mathbf{A}^{n})\). To put an example of the overwhelming expressive power of second order logic with standard semantics, Arithmetical induction can be expressed