Second-order arithmetic

Second-order arithmetic or Z2 is a theory of arithmetic allowing quantification over the natural numbers as well as sets of natural numbers.

Language
The language of second-order arithmetic allows us to discuss two types of objects, namely number themselves, or sets of numbers. It is an extension of predicate calculus with the following:


 * Infinitely many numeric variables \(x_0\), \(x_1\), ... and infinitely many set variables \(X_0\), \(X_1\), ...
 * Constant symbol \(0\), called zero
 * Three binary relation symbols \(=,<,\in\), called equality, comparison and set membership. The first two are relations between numbers, and the last one is a relation between a number and a set
 * Unary function symbol \(S(x)\), called successor, mapping numbers to numbers
 * Two binary function symbols, \(+(a,b),\cdot(a,b)\), called addition and multiplication respectively, which map pairs of numbers to a single number. They are often denoted by \(a+b,a\cdot b\).

Axioms
The theory of second-order arithmetic consists of the axioms of Peano arithmetic, plus:


 * \(\forall X (0 \in X \wedge \forall x (x \in X \Rightarrow Sx \in X) \Rightarrow \forall x : x \in X)\), the second-order induction axiom. It states that if a set contains 0 and contains the successor of each of its own members, then that set contains every natural numbers.
 * For every formula \(\phi\) in which \(X\) doesn't appear free: \(\exists X \forall x (x \in X \Leftrightarrow \phi(x))\), the axiom schema of comprehension. It tells us that every predicate over the natural numbers defines a subset of the natural numbers (namely the set of these number which satisfy \(\phi\)).