Reverse mathematics

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.

The program was founded by Harvey Friedman (1975, 1976). A standard reference for the subject is (Simpson 2009).

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 which 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 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.

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.

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 RCA₀, WKL₀, ACA₀, ATR₀, and Π¹₁-CA₀.

The following table summarizes the "big five" systems Simpson (2009, p.42)

The subscript ₀ in these names means that the induction scheme has been restricted from the full second-order induction scheme (Simpson 2009, p. 6). For example, ACA₀ 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 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.

The base system RCA₀

RCA₀ is the fragment of second-order arithmetic whose axioms are the axioms of Robinson arithmetic, induction for Σ0
1 formulas, and comprehension for Δ0
1 formulas.

The subsystem RCA₀ 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 recursive function. This name is used because RCA₀ corresponds informally to "computable mathematics". In particular, any set of natural numbers that can be proven to exist in RCA₀ is computable, and thus any theorem which implies that noncomputable sets exist is not provable in RCA₀. To this extent, RCA₀ is a constructive system, although it does not meet the requirements of the program of constructivism because it is a theory in classical logic including the excluded middle.

Despite its seeming weakness (of not proving any noncomputable sets exist), RCA₀ 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 RCA₀ include:

• Basic properties of the natural numbers, integers, and rational numbers (for example, that the latter form an ordered field).
• Basic properties of the real numbers (the real numbers are an 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).
• The Baire category theorem for a complete separable metric space (the separability condition is necessary to even state the theorem in the language of second-order arithmetic).
• The intermediate value theorem on continuous real functions.
• The Banach–Steinhaus theorem for a sequence of continuous linear operators on separable Banach spaces.
• A weak version of Gödel's completeness theorem (for a set of sentences, in a countable language, that is already closed under consequence).
• The existence of an algebraic closure for a countable field (but not its uniqueness).
• The existence and uniqueness of the real closure of a countable ordered field.

The first-order part of RCA₀ (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 Σ⁰₁ formulas. It is provably consistent, as is RCA₀, in full first-order Peano arithmetic.

Weak König's lemma WKL₀

The subsystem WKL₀ consists of RCA₀ 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. WKL₀ can also be defined as the principle of Σ⁰₁ separation (given two Σ⁰₁ 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).

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 RCA₀, the resulting subsystem is called WKL₀. 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.

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.

To show that WKL₀ is actually stronger than (not provable in) RCA₀, it is sufficient to exhibit a theorem of WKL₀ which implies that noncomputable sets exist. This is not difficult; WKL₀ implies the existence of separating sets for effectively inseparable recursively enumerable sets.

It turns out that RCA₀ and WKL₀ have the same first-order part, meaning that they prove the same first-order sentences. WKL₀ can prove a good number of classical mathematical results which do not follow from RCA₀, however. These results are not expressible as first order statements but can be expressed as second-order statements.

The following results are equivalent to weak König's lemma and thus to WKL₀ over RCA₀:

• 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.
• The Heine–Borel theorem for complete totally bounded separable metric spaces (where covering is by a sequence of open balls).
• 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).
• A continuous real function on the closed unit interval can be uniformly approximated by polynomials (with rational coefficients).
• A continuous real function on the closed unit interval is uniformly continuous.
• A continuous real function on the closed unit interval is Riemann integrable.
• The Brouwer fixed point theorem (for continuous functions on a finite product of copies of the closed unit interval).
• 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.
• The Jordan curve theorem
• Gödel's completeness theorem (for a countable language).
• Determinacy for open (or even clopen) games on {0,1} of length ω.
• Every countable commutative ring has a prime ideal.
• Every countable formally real field is orderable.
• Uniqueness of algebraic closure (for a countable field).

Arithmetical comprehension ACA₀

ACA₀ is RCA₀ plus the comprehension scheme for arithmetical formulas (which is sometimes called the "arithmetical comprehension axiom"). That is, ACA₀ 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 RCA₀ the comprehension scheme for Σ₁ formulas in order to obtain full arithmetical comprehension.

The first-order part of ACA₀ is exactly first-order Peano arithmetic; ACA₀ is a conservative extension of first-order Peano arithmetic. The two systems are provably (in a weak system) equiconsistent. ACA₀ can be thought of as a framework of predicative mathematics, although there are predicatively provable theorems that are not provable in ACA₀. Most of the fundamental results about the natural numbers, and many other mathematical theorems, can be proven in this system.

One way of seeing that ACA₀ is stronger than WKL₀ is to exhibit a model of WKL₀ that doesn't contain all arithmetical sets. In fact, it is possible to build a model of WKL₀ consisting entirely of low sets using the low basis theorem, since low sets relative to low sets are low.

The following assertions are equivalent to ACA₀ over RCA₀:

• The sequential completeness of the real numbers (every bounded increasing sequence of real numbers has a limit).
• The Bolzano–Weierstrass theorem.
• Ascoli's theorem: every bounded equicontinuous sequence of real functions on the unit interval has a uniformly convergent subsequence.
• Every countable commutative ring has a maximal ideal.
• Every countable vector space over the rationals (or over any countable field) has a basis.
• Every countable field has a transcendence basis.
• König's lemma (for arbitrary finitely branching trees, as opposed to the weak version described above).
• Various theorems in combinatorics, such as certain forms of Ramsey's theorem.

Arithmetical transfinite recursion ATR₀

The system ATR₀ adds to ACA₀ 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. ATR₀ is equivalent over ACA₀ to the principle of Σ¹₁ separation. ATR₀ is impredicative, and has the proof-theoretic ordinal , the supremum of that of predicative systems.

ATR₀ proves the consistency of ACA₀, and thus by Gödel's theorem it is strictly stronger.

The following assertions are equivalent to ATR₀ over RCA₀:

• Any two countable well orderings are comparable. That is, they are isomorphic or one is isomorphic to a proper initial segment of the other.
• Ulm's theorem for countable reduced Abelian groups.
• The perfect set theorem, which states that every uncountable closed subset of a complete separable metric space contains a perfect closed set.
• Lusin's separation theorem (essentially Σ¹₁ separation).
• Determinacy for open sets in the Baire space.

Π¹₁ comprehension Π¹₁-CA₀

Π¹₁-CA₀ is stronger than arithmetical transfinite recursion and is fully impredicative. It consists of RCA₀ plus the comprehension scheme for Π¹₁ formulas.

In a sense, Π¹₁-CA₀ comprehension is to arithmetical transfinite recursion (Σ¹₁ separation) as ACA₀ is to weak König's lemma (Σ⁰₁ 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.

The following theorems are equivalent to Π¹₁-CA₀ over RCA₀:

• The Cantor–Bendixson theorem (every closed set of reals is the union of a perfect set and a countable set).
• Every countable abelian group is the direct sum of a divisible group and a reduced group.

Additional systems

• Weaker systems than recursive comprehension can be defined. The weak system RCA*
  0 consists of elementary function arithmetic EFA (the basic axioms plus Δ⁰₀ induction in the enriched language with an exponential operation) plus Δ⁰₁ comprehension. Over RCA*
  0, recursive comprehension as defined earlier (that is, with Σ⁰₁ 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*
  0 has the same proof theoretic ordinal ω³ as EFA and is conservative over EFA for Π0
  2 sentences.

• 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 RCA₀). WWKL₀ is obtained by adjoining this axiom to RCA₀. 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 WWKL₀ is closely connected to the theory of algorithmically random sequences. In particular, an ω-model of RCA₀ 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.

• DNR (short for "diagonally non-recursive") adds to RCA₀ an axiom asserting the existence of a 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 eth partial recursive function with oracle A is not equal to f. DNR is strictly weaker than WWKL (Lempp et al., 2004).

• Δ¹₁-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 Δ¹₁-comprehension but not the other way around.

• Σ¹₁-choice is the statement that if η(n,X) is a Σ¹₁ formula such that for each n there exists an X satisfying η then there is a sequence of sets X_n such that η(n,X_n) holds for each n. Σ¹₁-choice also has the hyperarithmetical sets as minimal ω-model. Arithmetical transfinite recursion proves Σ¹₁-choice but not the other way around.

ω-models and β-models

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^ω 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, RCA₀ has a minimal ω-model where S consists of the recursive subsets of ω.

A β model is an ω model that is equivalent to the standard ω-model for Π1
1 and Σ1
1 sentences (with parameters).

Non-ω models are also useful, especially in the proofs of conservation theorems.

References

• Ambos-Spies, K.; Kjos-Hanssen, B.; Lempp, S.; Slaman, T.A. (2004), "Comparing DNR and WWKL", Journal of Symbolic Logic 69 (4): 1089, doi:10.2178/jsl/1102022212.  Cite uses deprecated parameter |coauthors= (help)

• Friedman, Harvey (1975), "Some systems of second order arithmetic and their use", Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., pp. 235–242, MR 0429508 
• Friedman, Harvey; Martin, D. A.; Soare, R. I.; Tait, W. W. (1976), "Meeting of the Association for Symbolic Logic: Systems of second order arithmetic with restricted induction, I, II", The Journal of Symbolic Logic (Association for Symbolic Logic) 41 (2): 557–559, doi:10.2307/2272259 
• Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, MR 2517689 
• Solomon, Reed (1999), "Ordered groups: a case study in reverse mathematics", The Bulletin of Symbolic Logic 5 (1): 45–58, doi:10.2307/421140, ISSN 1079-8986, JSTOR 421140, MR 1681895 

External links

• Harvey Friedman's home page
• Stephen G. Simpson's home page