Editing Reverse mathematics (section)

=== Use of second-order arithmetic ===

Most reverse mathematics research focuses on subsystems of [[second-order arithmetic]]. The body of research in reverse mathematics has established that weak subsystems of second-order arithmetic suffice to formalize almost all undergraduate-level mathematics. In second-order arithmetic, all objects can be represented as either [[natural number]]s or sets of natural numbers.  For example, in order to prove theorems about real numbers, the real numbers can be represented as [[Cauchy sequence]]s of [[rational number]]s, each of which sequence can be represented as a set of natural numbers.

The axiom systems most often considered in reverse mathematics are defined using [[axiom scheme]]s called '''comprehension schemes'''.  Such a scheme states that any set of natural numbers definable by a formula of a given complexity exists. In this context, the complexity of formulas is measured using the [[arithmetical hierarchy]] and [[analytical hierarchy]].

The reason that reverse mathematics is not carried out using set theory as a base system is that the language of set theory is too expressive.  Extremely complex sets of natural numbers can be defined by simple formulas in the language of set theory (which can quantify over arbitrary sets). In the context of second-order arithmetic, results such as [[Post's theorem]] establish a close link between the complexity of a formula and the (non)computability of the set it defines.

Another effect of using second-order arithmetic is the need to restrict general mathematical theorems to forms that can be expressed within arithmetic. For example, second-order arithmetic can express the principle "Every countable [[vector space]] has a basis" but it cannot express the principle "Every vector space has a basis". In practical terms, this means that theorems of [[abstract algebra|algebra]] and [[combinatorics]] are restricted to countable structures, while theorems of [[analysis (mathematics)|analysis]] and [[topology]] are restricted to [[separable space]]s. Many principles that imply the [[axiom of choice]] in their general form (such as "Every vector space has a basis") become provable in weak subsystems of second-order arithmetic when they are restricted. For example, "every field has an algebraic closure" is not provable in ZF set theory, but the restricted form "every countable field has an algebraic closure" is provable in RCA<sub>0</sub>, the weakest system typically employed in reverse mathematics.