Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. In higher-order reverse mathematics, the focus is on subsystems of higher-order arithmetic, and the associated richer languages.
Richer languages are generated through analysis of the axiomatic strength of mathematical theorems utilizing the language defined as all finite types. This richer language allows for a more natural direct formalization of core concepts of modern analysis, topology and functionals.
The program was founded by Harvey FriedmanH. Friedman, Some systems of second-order arithmetic and their use (1974), Proceedings of the International Congress of Mathematicians and brought forward by Steve Simpson.
Constructive reverse mathematics is related program which is applied to constructive mathematics.
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 of real numbers 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. The reversal establishes that no axiom system S′ that extends the base system can be weaker than S while still proving 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 numbers or sets of natural numbers. For example, in order to prove theorems about real numbers, the real numbers can be represented as Cauchy sequences of rational numbers, 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 schemes 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 algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces. 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 RCA0, 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. Due to the richer language of higher-order arithmetic, the use of representations (also known as '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.See and . For instance, the base theory of higher-order reverse mathematics, called RCA, proves the same sentences as RCA0, 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 WKL0 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. Other covering lemmas (e.g. due to Lindelöf, Vitali, Besicovitch, etc.) exhibit the same behavior, and many basic properties of the gauge integral are equivalent to the compactness of the underlying space.
The big five subsystems of second-order arithmetic
Second-order arithmetic is a formal theory of the natural numbers and sets of natural numbers. Many mathematical objects, such as countable rings, groups, and fields, as well as points in effective Polish spaces, can be represented as sets of natural numbers, and modulo this representation can be studied in second-order arithmetic.
Reverse mathematics makes use of several subsystems of second-order arithmetic. A typical reverse mathematics theorem shows that a particular mathematical theorem T is equivalent to a particular subsystem S of second-order arithmetic over a weaker subsystem B. This weaker system B is known as the base system for the result; in order for the reverse mathematics result to have meaning, this system must not itself be able to prove the mathematical theorem T.
Steve Simpson describes five particular subsystems of second-order arithmetic, which he calls the Big Five, that occur frequently in reverse mathematics.Simpson claims to have not invented the term. [] In order of increasing strength, these systems are named by the initialisms RCA0, WKL0, ACA0, ATR0, and Π-CA0.
The following table summarizes the "big five" systems and lists the counterpart systems in higher-order arithmetic. The latter generally prove the same second-order sentences (or a large subset) as the original second-order systems. SubsystemStands forOrdinalCorresponds roughly toCommentsHigher-order counterpartRCA0Recursive comprehension axiomωωConstructive mathematics (Bishop)The base theory RCA; proves the same second-order sentences as RCA0WKL0Weak Kőnig's lemmaωωFinitistic reductionism (Hilbert) Conservative over PRA (resp. RCA0) for Π (resp. Π) sentencesFan functional; computes modulus of uniform continuity on for continuous functionsACA0Arithmetical comprehension axiomε0Predicativism (Weyl, Feferman)Conservative over Peano arithmetic for arithmetical sentencesThe 'Turing jump' functional ∃2 expresses the existence of a discontinuous function on ATR0Arithmetical transfinite recursionΓ0Predicative reductionism (Friedman, Simpson)Conservative over Feferman's system IR for Π sentences The 'transfinite recursion' functional outputs the set claimed to exist by ATR0. Π-CA0Π comprehension axiomΨ0(Ωω)Impredicativism The Suslin functional S2 decides Π-formulas (restricted to second-order parameters).
The subscript 0 in these names means that the induction scheme has been restricted from the full second-order induction scheme. For example, ACA0 includes the induction axiom . This together with the full comprehension axiom of second-order arithmetic implies the full second-order induction scheme given by the universal closure of for any second-order formula φ. However ACA0 does not have the full comprehension axiom, and the subscript 0 is a reminder that it does not have the full second-order induction scheme either. This restriction is important: systems with restricted induction have significantly lower proof-theoretical ordinals than systems with the full second-order induction scheme.
Base system RCA0 RCA0 is the fragment of second-order arithmetic whose axioms are the axioms of Robinson arithmetic, induction axiom scheme for Σ formulas, and comprehension axiom scheme for formulas (also called recursive comprehension).
The -induction axiom scheme statesfor all -formulas that do not quantify over set variables. More explicitly, these are formulas of the form where is a -formula that can include set variables. In words, is obtained by starting with a quantifier-free formula that can involve both first-order and second-order variables, then adding bounded quantifiers over the first-order variables, and finally adding existential quantifiers over the first-order variables.
The -comprehension axiom scheme states that for all -formulas and all -formulas .