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| '''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 [[theorem]]s to the [[axiom]]s", in contrast to the ordinary mathematical practice of deriving theorems from axioms. 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.
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| 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]].
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| The program was founded by {{harvs|txt|authorlink=Harvey Friedman|first=Harvey|last= Friedman|year1=1975|year2=1976}}. A standard reference for the subject is {{harv|Simpson|2009}}.
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| == General principles ==
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| 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 number]]s has a [[supremum]]” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.
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| 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''.
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| === Use of second-order arithmetic ===
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| 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 must 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 must be represented as [[Cauchy sequence]]s of [[rational number]]s, each of which can be represented as a set of natural numbers.{{elucidate|date=July 2012}}
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| 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]].
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| 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.
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| 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 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.
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| == The big five subsystems of second order arithmetic ==
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| [[Second order arithmetic]] is a formal theory of the natural numbers and sets of natural numbers. Many mathematical objects, such as [[countable set|countable]] [[ring (mathematics)|rings]], [[group (mathematics)|groups]], and [[field (mathematics)|fields]], as well as points in [[effective Polish space]]s, can be represented as sets of natural numbers, and modulo this representation can be studied in second order arithmetic.
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| 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
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| meaning, this system must not itself be able to prove the mathematical theorem ''T''.
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| {{harvtxt|Simpson|2009}} describes five particular subsystems of second order arithmetic, which he calls the '''Big Five''', that occur frequently in reverse mathematics. In order of increasing strength, these systems are named by the initialisms RCA<sub>0</sub>, WKL<sub>0</sub>, ACA<sub>0</sub>, ATR<sub>0</sub>, and
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| Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-CA<sub>0</sub>.
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| The following table summarizes the "big five" systems {{harvtxt|Simpson|2009|loc=p.42}}
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| {| class="wikitable"
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| !Subsystem
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| !Stands for
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| !Ordinal
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| !Corresponds roughly to
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| !Comments
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| |-
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| |RCA<sub>0</sub>
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| |Recursive comprehension axiom
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| |ω<sup>ω</sup>
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| |Constructive mathematics (Bishop)
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| |The base system for reverse mathematics
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| |-
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| |WKL<sub>0</sub>
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| |Weak König's lemma
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| |ω<sup>ω</sup>
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| |Finitistic reductionism (Hilbert)
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| | Conservative over [[Primitive recursive arithmetic|PRA]] for {{nowrap|Π{{su|p=0|b=2}}}} sentences. Conservative over RCA<sub>0</sub> for {{nowrap|Π{{su|p=1|b=1}}}} sentences.
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| |-
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| |ACA<sub>0</sub>
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| |Arithmetical comprehension axiom
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| |[[epsilon zero|ε<sub>0</sub>]]
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| |Predicativism (Weyl, Feferman)
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| |Conservative over Peano arithmetic for arithmetical sentences
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| |-
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| |ATR<sub>0</sub>
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| |Arithmetical transfinite recursion
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| |[[Feferman–Schütte ordinal|Γ<sub>0</sub>]]
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| |Predicative reductionism (Friedman, Simpson)
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| |Conservative over Feferman's system IR for {{nowrap|Π{{su|p=1|b=1}}}} sentences
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| |-
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| |Π{{su|p=1|b=1}}-CA<sub>0</sub>
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| |Π{{su|p=1|b=1}} comprehension axiom
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| |[[Psi0(Omega omega)|Ψ<sub>0</sub>(Ω<sub>ω</sub>)]]
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| |Impredicativism
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| |}
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| The subscript <sub>0</sub> in these names means that the induction scheme has been restricted from the full second-order induction scheme {{harv|Simpson|2009|loc=p. 6}}. For example, ACA<sub>0</sub> includes the induction axiom (0∈X ∧ ∀n(n∈X → n+1∈X)) → ∀n n∈X. This together with the full comprehension axiom of second order arithmetic implies the full second-order induction scheme given by the universal closure of (φ(0) ∧ ∀n(φ(n) → φ(n+1))) → ∀n φ(n) for any second order formula φ. However ACA<sub>0</sub> does not have the full comprehension axiom, and the subscript <sub>0</sub> 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.
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| === The base system RCA<sub>0</sub>===
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| RCA<sub>0</sub> is the fragment of second-order arithmetic whose axioms are the axioms of [[Robinson arithmetic]], induction for Σ{{su|p=0|b=1}} formulas, and comprehension for Δ{{su|p=0|b=1}} formulas.
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| The subsystem RCA<sub>0</sub> is the one most commonly used as a base system for reverse mathematics. The initials "RCA" stand for "recursive comprehension axiom", where "recursive" means "computable", as in [[computable function|recursive function]]. This name is used because RCA<sub>0</sub> corresponds informally to "computable mathematics". In particular, any set of natural numbers that can be proven to exist in RCA<sub>0</sub> is computable, and thus any theorem which implies that noncomputable sets exist is not provable in RCA<sub>0</sub>. To this extent, RCA<sub>0</sub> is a constructive system, although it does not meet the requirements of the program of [[constructivism (mathematics)|constructivism]] because it is a theory in classical logic including the excluded middle.
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| Despite its seeming weakness (of not proving any noncomputable sets exist), RCA<sub>0</sub> is sufficient to prove a number of classical theorems which, therefore, require only minimal logical strength. These theorems are, in a sense, below the reach of the reverse mathematics enterprise because they are already provable in the base system. The classical theorems provable in RCA<sub>0</sub> include:
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| * Basic properties of the natural numbers, integers, and rational numbers (for example, that the latter form an [[ordered field]]).
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| * Basic properties of the real numbers (the real numbers are an [[Archimedean property|Archimedean]] ordered field; any [[nested sequence of closed intervals]] whose lengths tend to zero has a single point in its intersection; the real numbers are not countable).
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| * The [[Baire category theorem]] for a [[complete metric space|complete]] [[separable space|separable]] [[metric space]] (the separability condition is necessary to even state the theorem in the language of second-order arithmetic).
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| * The [[intermediate value theorem]] on continuous real functions.
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| * The [[Banach–Steinhaus theorem]] for a sequence of continuous linear operators on separable Banach spaces.
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| * A weak version of [[Gödel's completeness theorem]] (for a set of sentences, in a countable language, that is already closed under consequence).
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| * The existence of an [[algebraic closure]] for a countable field (but not its uniqueness).
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| * The existence and uniqueness of the [[Real closed field|real closure]] of a countable ordered field.
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| The first-order part of RCA<sub>0</sub> (the theorems of the system that do not involve any set variables) is the set of theorems of first-order Peano arithmetic with induction limited to Σ<sup>0</sup><sub style="margin-left:-0.6em">1</sub> formulas. It is provably consistent, as is RCA<sub>0</sub>, in full first-order Peano arithmetic.
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| === Weak König's lemma WKL<sub>0</sub>===
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| The subsystem WKL<sub>0</sub> consists of RCA<sub>0</sub> plus a weak form of [[König's lemma]], namely the statement that every infinite subtree of the full binary tree (the tree of all finite sequences of 0's and 1's) has an infinite path. This proposition, which is known as ''weak König's lemma'', is easy to state in the language of second-order arithmetic. WKL<sub>0</sub> can also be defined as the principle of Σ<sup>0</sup><sub style="margin-left:-0.6em">1</sub> separation (given two Σ<sup>0</sup><sub style="margin-left:-0.6em">1</sub> formulas of a free variable ''n'' which are exclusive, there is a class containing all ''n'' satisfying the one and no ''n'' satisfying the other).
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| The following remark on terminology is in order. The term “weak König's lemma” refers to the sentence which says that any infinite subtree of the binary tree has an infinite path. When this axiom is added to RCA<sub>0</sub>, the resulting subsystem is called WKL<sub>0</sub>. A similar distinction between particular axioms, on the one hand, and subsystems including the basic axioms and induction, on the other hand, is made for the stronger subsystems described below.
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| In a sense, weak König's lemma is a form of the [[axiom of choice]] (although, as stated, it can be proven in classical Zermelo–Fraenkel set theory without the axiom of choice). It is not constructively valid in some senses of the word constructive.
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| To show that WKL<sub>0</sub> is actually stronger than (not provable in) RCA<sub>0</sub>, it is sufficient to exhibit a theorem of WKL<sub>0</sub> which implies that noncomputable sets exist. This is not difficult; WKL<sub>0</sub> implies the existence of separating sets for effectively inseparable recursively enumerable sets.
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| It turns out that RCA<sub>0</sub> and WKL<sub>0</sub> have the same first-order part, meaning that they prove the same first-order sentences. WKL<sub>0</sub> can prove a good number of classical mathematical results which do not follow from RCA<sub>0</sub>, however. These results are not expressible as first order statements but can be expressed as second-order statements.
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| The following results are equivalent to weak König's lemma and thus to WKL<sub>0</sub> over RCA<sub>0</sub>:
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| * The [[Heine–Borel theorem]] for the closed unit real interval, in the following sense: every covering by a sequence of open intervals has a finite subcovering.
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| * The Heine–Borel theorem for complete totally bounded separable metric spaces (where covering is by a sequence of open balls).
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| * A continuous real function on the closed unit interval (or on any compact separable metric space, as above) is bounded (or: bounded and reaches its bounds).
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| * A continuous real function on the closed unit interval can be uniformly approximated by polynomials (with rational coefficients).
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| * A continuous real function on the closed unit interval is uniformly continuous.
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| * A continuous real function on the closed unit interval is [[Riemann integral|Riemann]] integrable.
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| * The [[Brouwer fixed point theorem]] (for continuous functions on a finite product of copies of the closed unit interval).
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| * The separable [[Hahn–Banach theorem]] in the form: a bounded linear form on a subspace of a separable Banach space extends to a bounded linear form on the whole space.
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| * The [[Jordan curve theorem]]
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| * Gödel's completeness theorem (for a countable language).
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| * Determinacy for open (or even clopen) games on {0,1} of length ω.
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| * Every countable [[commutative ring]] has a [[prime ideal]].
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| * Every countable formally real field is orderable.
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| * Uniqueness of algebraic closure (for a countable field).
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| === Arithmetical comprehension ACA<sub>0</sub>===
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| ACA<sub>0</sub> is RCA<sub>0</sub> plus the comprehension scheme for arithmetical formulas (which is sometimes called the "arithmetical comprehension axiom"). That is, ACA<sub>0</sub> allows us to form the set of natural numbers satisfying an arbitrary arithmetical formula (one with no bound set variables, although possibly containing set parameters). Actually, it suffices to add to RCA<sub>0</sub> the comprehension scheme for Σ<sub>1</sub> formulas in order to obtain full arithmetical comprehension.
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| The first-order part of ACA<sub>0</sub> is exactly first-order Peano arithmetic; ACA<sub>0</sub> is a ''conservative'' extension of first-order Peano arithmetic. The two systems are provably (in a weak system) equiconsistent. ACA<sub>0</sub> can be thought of as a framework of [[Impredicativity|predicative]] mathematics, although there are predicatively provable theorems that are not provable in ACA<sub>0</sub>. Most of the fundamental results about the natural numbers, and many other mathematical theorems, can be proven in this system.
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| One way of seeing that ACA<sub>0</sub> is stronger than WKL<sub>0</sub> is to exhibit a model of WKL<sub>0</sub> that doesn't contain all arithmetical sets. In fact, it is possible to build a model of WKL<sub>0</sub> consisting entirely of [[Low (computability)|low sets]] using the [[low basis theorem]], since low sets relative to low sets are low.
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| The following assertions are equivalent to ACA<sub>0</sub>
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| over RCA<sub>0</sub>:
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| * The sequential completeness of the real numbers (every bounded increasing sequence of real numbers has a limit).
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| * The [[Bolzano–Weierstrass theorem]].
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| * [[Ascoli's theorem]]: every bounded equicontinuous sequence of real functions on the unit interval has a uniformly convergent subsequence.
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| * Every countable commutative ring has a [[maximal ideal]].
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| * Every countable vector space over the rationals (or over any countable field) has a basis.
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| * Every countable field has a [[transcendence basis]].
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| * König's lemma (for arbitrary finitely branching trees, as opposed to the weak version described above).
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| * Various theorems in combinatorics, such as certain forms of Ramsey's theorem.
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| === Arithmetical transfinite recursion ATR<sub>0</sub>===
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| The system ATR<sub>0</sub> adds to ACA<sub>0</sub> an axiom which states, informally, that any arithmetical functional (meaning any arithmetical formula with a free number variable ''n'' and a free class variable ''X'', seen as the operator taking ''X'' to the set of ''n'' satisfying the formula) can be iterated transfinitely along any countable [[well ordering]] starting with any set. ATR<sub>0</sub> is equivalent over ACA<sub>0</sub> to the principle of Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub> separation. ATR<sub>0</sub> is impredicative, and has the proof-theoretic ordinal <math>\Gamma_0</math>, the supremum of that of predicative systems.
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| ATR<sub>0</sub> proves the consistency of ACA<sub>0</sub>, and thus by [[Gödel's incompleteness theorems|Gödel's theorem]] it is strictly stronger.
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| The following assertions are equivalent to
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| ATR<sub>0</sub> over RCA<sub>0</sub>:
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| * Any two countable well orderings are comparable. That is, they are isomorphic or one is isomorphic to a proper initial segment of the other.
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| * [[Ulm's theorem]] for countable reduced Abelian groups.
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| * The [[perfect set property|perfect set theorem]], which states that every uncountable closed subset of a complete separable metric space contains a perfect closed set.
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| * [[Lusin's separation theorem]] (essentially Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub> separation).
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| * [[Determinacy]] for [[open set]]s in the [[Baire space]].
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| === Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub> comprehension Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-CA<sub>0</sub>===
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| Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-CA<sub>0</sub> is stronger than arithmetical transfinite recursion and is fully impredicative. It consists of RCA<sub>0</sub> plus the comprehension scheme for Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub> formulas.
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| In a sense, Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-CA<sub>0</sub> comprehension is to arithmetical transfinite recursion (Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub> separation) as ACA<sub>0</sub> is to weak König's lemma (Σ<sup>0</sup><sub style="margin-left:-0.6em">1</sub> separation). It is equivalent to several statements of descriptive set theory whose proofs make use of strongly impredicative arguments; this equivalence shows that these impredicative arguments cannot be removed.
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| The following theorems are equivalent to Π<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-CA<sub>0</sub> over RCA<sub>0</sub>:
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| * The [[Cantor–Bendixson theorem]] (every closed set of reals is the union of a perfect set and a countable set).
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| * Every countable abelian group is the direct sum of a divisible group and a reduced group.
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| == Additional systems ==
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| * Weaker systems than recursive comprehension can be defined. The weak system RCA{{su|p=*|b=0}} consists of [[elementary function arithmetic]] EFA (the basic axioms plus Δ<sup>0</sup><sub style="margin-left:-0.6em">0</sub> induction in the enriched language with an exponential operation) plus Δ<sup>0</sup><sub style="margin-left:-0.6em">1</sub> comprehension. Over RCA{{su|p=*|b=0}}, recursive comprehension as defined earlier (that is, with Σ<sup>0</sup><sub style="margin-left:-0.6em">1</sub> induction) is equivalent to the statement that a polynomial (over a countable field) has only finitely many roots and to the classification theorem for finitely generated Abelian groups. The system RCA{{su|p=*|b=0}} has the same proof theoretic ordinal ω<sup>3</sup> as EFA and is conservative over EFA for Π{{su|p=0|b=2}} sentences.
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| * Weak Weak König's Lemma is the statement that a subtree of the infinite binary tree having no infinite paths has an asymptotically vanishing proportion of the leaves at length ''n'' (with a uniform estimate as to how many leaves of length ''n'' exist). An equivalent formulation is that any subset of Cantor space that has positive measure is nonempty (this is not provable in RCA<sub>0</sub>). WWKL<sub>0</sub> is obtained by adjoining this axiom to RCA<sub>0</sub>. It is equivalent to the statement that if the unit real interval is covered by a sequence of intervals then the sum of their lengths is at least one. The model theory of WWKL<sub>0</sub> is closely connected to the theory of [[algorithmically random sequence]]s. In particular, an ω-model of RCA<sub>0</sub> satisfies weak weak König's lemma if and only if for every set ''X'' there is a set ''Y'' which is 1-random relative to ''X''.
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| * DNR (short for "diagonally non-recursive") adds to RCA<sub>0</sub> an axiom asserting the existence of a [[Kleene's_recursion_theorem#Fixed_point_free_functions|diagonally non-recursive]] function relative to every set. That is, DNR states that, for any set ''A'', there exists a total function ''f'' such that for all ''e'' the ''e''th partial recursive function with oracle ''A'' is not equal to ''f''. DNR is strictly weaker than WWKL (Lempp ''et al.'', 2004).
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| * Δ<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-comprehension is in certain ways analogous to arithmetical transfinite recursion as recursive comprehension is to weak König's lemma. It has the hyperarithmetical sets as minimal ω-model. Arithmetical transfinite recursion proves Δ<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-comprehension but not the other way around.
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| * Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-choice is the statement that if η(''n'',''X'') is a Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub> formula such that for each ''n'' there exists an ''X'' satisfying η then there is a sequence of sets ''X<sub>n</sub>'' such that η(''n'',''X<sub>n</sub>'') holds for each ''n''. Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-choice also has the hyperarithmetical sets as minimal ω-model. Arithmetical transfinite recursion proves Σ<sup>1</sup><sub style="margin-left:-0.6em">1</sub>-choice but not the other way around.
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| ==ω-models and β-models==
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| The ω in ω-model stands for the set of non-negative integers (or finite ordinals). An ω-model is a model for a fragment of second-order arithmetic whose first-order part is the standard model of Peano arithmetic, but whose second-order part may be non-standard. More precisely, an ω-model is given by a choice ''S''⊆2<sup>ω</sup> of subsets of ω. The first order variables are interpreted in the usual way as elements of ω, and +, × have their usual meanings, while second order variables are interpreted as elements of ''S''. There is a standard ω model where one just takes ''S'' to consist of all subsets of the integers. However there are also other ω-models; for example, RCA<sub>0</sub> has a minimal ω-model where ''S'' consists of the recursive subsets of ω.
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| A β model is an ω model that is equivalent to the standard ω-model for Π{{su|p=1|b=1}} and Σ{{su|p=1|b=1}} sentences (with parameters).
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| Non-ω models are also useful, especially in the proofs of conservation theorems.
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| == References ==
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| * {{Citation
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| | author = Ambos-Spies, K.
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| | coauthors = Kjos-Hanssen, B.; Lempp, S.; Slaman, T.A.
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| | year = 2004
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| | title = Comparing DNR and WWKL
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| | journal = Journal of Symbolic Logic
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| | volume = 69
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| | issue = 4
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| | doi = 10.2178/jsl/1102022212
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| | pages = 1089
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| | postscript = .
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| }}
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| * {{Citation | last1=Friedman | first1=Harvey | title=Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 | publisher=Canad. Math. Congress, Montreal, Que. | id={{MathSciNet | id = 0429508}} | year=1975 | chapter=Some systems of second order arithmetic and their use | pages=235–242}}
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| *{{Citation | last1=Friedman | first1=Harvey | title=Meeting of the Association for Symbolic Logic | publisher=Association for Symbolic Logic | year=1976 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=41 | issue=2 | chapter=Systems of second order arithmetic with restricted induction, I, II | pages=557–559 | jstor=2272259 | last2=Martin | first2=D. A. | last3=Soare | first3=R. I. | last4=Tait | first4=W. W.}}
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| * {{Citation | last1=Simpson | first1=Stephen G. | title=Subsystems of second order arithmetic | url=http://www.math.psu.edu/simpson/sosoa/ | publisher=[[Cambridge University Press]] | edition=2nd | series=Perspectives in Logic | isbn=978-0-521-88439-6 | id={{MathSciNet | id = 2517689}} | year=2009}}
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| *{{Citation | doi=10.2307/421140 | last1=Solomon | first1=Reed | title=Ordered groups: a case study in reverse mathematics | id={{MathSciNet | id = 1681895}} | year=1999 | journal=The Bulletin of Symbolic Logic | issn=1079-8986 | volume=5 | issue=1 | pages=45–58 | jstor=421140}}
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| == External links ==
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| * [http://www.math.ohio-state.edu/~friedman/ Harvey Friedman's home page]
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| * [http://www.math.psu.edu/simpson/ Stephen G. Simpson's home page]
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| [[Category:Mathematical logic]]
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| [[Category:Computability theory]]
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| [[Category:Proof theory]]
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