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In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and sets thereof. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. more...

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Second order arithmetic is stronger than Peano arithmetic because it allows quantification over sets of numbers as well as numbers themselves. It can also be seen as a weak version of set theory in which the set elements are either natural numbers or sets of natural numbers. Although much weaker than Zermelo-Fraenkel set theory, second-order arithmetic is much stronger than what is required to do essentially all of classical mathematics expressible in its language. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second order arithmetic allows quantification over such sets (precisely because it is second order), it is possible to formalize the real numbers in second-order arithmetic. Hence second-order arithmetic is sometimes called “analysis�.

A subsystem of second-order arithmetic L is a theory in the language of L whose axioms are theorems of L. Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived from second order extensions of Peano arithmetic, of varying strength. Much of core mathematics can be formalized in such subsystems, some of which are defined below. Reverse mathematics also reveals the extent to which classical mathematics is nonconstructive.

Definitional

Syntax

The language of second-order arithmetic is two-sorted. The first sort of terms and variables, usually denoted by lower case letters, consists of individuals, whose intended interpretation is as natural numbers. The other sort of variables, variously called “set variables,� “class variables,� or even “predicates� are usually denoted by upper case letters. They refer to classes/predicates/properties of individuals, and so can be thought of as sets of natural numbers. Both individuals and set variables can be quantified universally or existentially. A formula which has no bound set variables (that is, no quantifiers over set variables) is called arithmetical. An arithmetical formula may have free set variables and bound individual variables.

Individual terms are formed from the constant 0, the unary function S (the successor function), and the binary operations + and · (addition and multiplication). The successor function adds 1 (=S0)to its input. The relations = (equality) and < (comparison of natural numbers) relate two individuals, whereas the relation ∈ (membership) relates an individual and a set (or class).

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