Editing Reverse mathematics (section)

== General principles ==
In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems.  For example, to study the theorem “Every bounded [[sequence (mathematics)|sequence]] of [[real number]]s has a [[supremum]]” it is necessary to use a base system that can speak of real numbers and sequences of real numbers.

For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that a system ''S'' is required to prove a theorem ''T'', two proofs are required. The first proof shows ''T'' is provable from ''S''; this is an ordinary mathematical proof along with a justification that it can be carried out in the system ''S''. The second proof, known as a '''reversal''', shows that ''T'' itself implies ''S''; this proof is carried out in the base system.<ref name="Simpson2009" /> The reversal establishes that no axiom system ''S&prime;'' that extends the base system can be weaker than ''S'' while still proving&nbsp;''T''.

=== 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.

=== Use of higher-order arithmetic ===

A recent strand of ''higher-order'' reverse mathematics research, initiated by [[Ulrich Kohlenbach]] in 2005, focuses on subsystems of [[higher-order arithmetic]].{{sfnp|Kohlenbach|2005}}
Due to the richer language of higher-order arithmetic, the use of representations (aka 'codes') common in second-order arithmetic, is greatly reduced.  
For example, a continuous function on the [[Cantor space]] is just a function that maps binary sequences to binary sequences, and that also satisfies the usual 'epsilon-delta'-definition of continuity.

Higher-order reverse mathematics includes higher-order versions of (second-order) comprehension schemes.  Such a higher-order axiom states the existence of a functional that decides the truth or falsity of formulas of a given complexity. In this context, the complexity of formulas is also measured using the [[arithmetical hierarchy]] and [[analytical hierarchy]].  The higher-order counterparts of the major subsystems of second-order arithmetic generally prove the same second-order sentences (or a large subset) as the original second-order systems.<ref name=k05h08>See {{harvtxt|Kohlenbach|2005}} and {{harvtxt|Hunter|2008}}.</ref>  For instance, the base theory of higher-order reverse mathematics, called {{math|RCA{{su|p=''&omega;''|b=0}}}}, proves the same sentences as RCA<sub>0</sub>, up to language.

As noted in the previous paragraph, second-order comprehension axioms easily generalize to the higher-order framework.  However, theorems expressing the ''[[compactness]]'' of basic spaces behave quite differently in second- and higher-order arithmetic: on one hand, when restricted to countable covers/the language of second-order arithmetic, the compactness of the unit interval is provable in WKL<sub>0</sub> from the next section.  On the other hand, given uncountable covers/the language of higher-order arithmetic, the compactness of the unit interval is only provable from (full) second-order arithmetic.{{sfnp|Normann|Sanders|2018}} Other covering lemmas (e.g. due to [[Lindelöf]], [[Giuseppe Vitali|Vitali]], [[Besicovitch]], etc.) exhibit the same behavior, and many basic properties of the [[gauge integral]] are equivalent to the compactness of the underlying space.