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:

The language of second-order arithmetic is expressive enough to allow us to talk about not only natural numbers and subsets of these, but also about integers, rational numbers and real numbers, and even countable sets (i.e. sequences) of real numbers.
 * 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 \(\phi\)).

Subsystems
Harvey Friedman has initiated a program in mathematics, now called, the point of which is to answer what axioms are necessary to prove given theorem, and which axioms aren't strong enough for that.

Z2 is an extremely strong formal theory, and as it turned out, in many cases not all of its strength is required to prove desired results. Because of that, subsystems of second-order arithmetic have been extensively studied to measure strength of theorems. Many "everyday" theorems in mathematics have strength corresponding to one of the following subsystems, often called the "big five".

RCA0
This stands for "recursive comprehension axiom" is a system consisting of axioms of  plus first-order schema of induction limited to \(\Sigma_1^0\) formulas and axiom schema of comprehension limited to \(\Delta_1^0\) formulas (see and ).

This is so called base system, and all other subsystems considered have the axioms which RCA0 has. RCA0 is a relatively weak system, but it can prove the most basic properties of natural numbers and real numbers (e.g. uniqueness of a limit of a sequence).

RCA0 is only strong enough to prove existence of objects which are recursive, meaning that the set existence axioms of this theory are not strong enough to prove existence of, say, uncomputable function.

WKL0
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