Second-order arithmetic

In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted Z₂.

Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called “analysis”.

Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo-Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language.

A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z₂). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive.

Definition

Syntax

The language of second-order arithmetic is two-sorted. The first sort of terms and variables, usually denoted by lower case letters, consists of individuals, whose intended interpretation is as natural numbers. The other sort of variables, variously called “set variables,” “class variables,” or even “predicates” are usually denoted by upper case letters. They refer to classes/predicates/properties of individuals, and so can be thought of as sets of natural numbers. Both individuals and set variables can be quantified universally or existentially. A formula with no bound set variables (that is, no quantifiers over set variables) is called arithmetical. An arithmetical formula may have free set variables and bound individual variables.

Individual terms are formed from the constant 0, the unary function S (the successor function), and the binary operations + and · (addition and multiplication). The successor function adds 1 (=S0) to its input. The relations = (equality) and < (comparison of natural numbers) relate two individuals, whereas the relation ∈ (membership) relates an individual and a set (or class). Thus in notation the language of second-order arithmetic is given by the signature $\mathcal{L}=\{0,S,+,\cdot,=,<,\in\}$ .

For example, $\forall n (n\in X \rightarrow Sn \in X)$ , is a well-formed formula of second-order arithmetic that is arithmetical, has one free set variable X and one bound individual variable n (but no bound set variables, as is required of an arithmetical formula)—whereas $\exists X \forall n(n\in X \leftrightarrow n < SSSSSS0\cdot SSSSSSS0)$ is a well-formed formula that is not arithmetical with one bound set variable X and one bound individual variable n.

Semantics

Several different interpretations of the quantifiers are possible. If second-order arithmetic is studied using the full semantics of second-order logic then the set quantifiers range over all subsets of the range of the number variables. If second-order arithmetic is formalized using the semantics of first-order logic then any model includes a domain for the set variables to range over, and this domain may be a proper subset of the full powerset of the domain of number variables.

Although second-order arithmetic was originally studied using full second-order semantics, the vast majority of current research treats second-order arithmetic in first-order predicate calculus. This is because the model theory of subsystems of second-order arithmetic is more interesting in the setting of first-order logic.

Axioms

Basic

The following axioms are known as the basic axioms, or sometimes the Robinson axioms. The resulting first-order theory, known as Robinson arithmetic, is essentially Peano arithmetic without induction. The domain of discourse for the quantified variables is the natural numbers, collectively denoted by N, and including the distinguished member $\ 0$ , called "zero."

The primitive functions are the unary successor function, denoted by prefix $\ S,$ , and two binary operations, addition and multiplication, denoted by infix "+" and " $\cdot$ ", respectively. There is also a primitive binary relation called order, denoted by infix "<".

Axioms governing the successor function and zero:

\forall m [Sm=0 \rightarrow \bot].

(“the successor of a natural number is never zero”)

\forall m \forall n [Sm=Sn \rightarrow m=n].

(“the successor function is injective”)

\forall n [0=n \lor \exists m [Sm=n] ].

(“every natural number is zero or a successor”)

Addition defined recursively:

\forall m [m+0=m].

\forall m \forall n [m+Sn = S(m+n)].

Multiplication defined recursively:

\forall m [m\cdot 0 = 0].

\forall m \forall n [m \cdot Sn = (m\cdot n)+m].

Axioms governing the order relation "<":

\forall m [m<0 \rightarrow \bot].

(“no natural number is smaller than zero”)

\forall n \forall m [m<Sn \leftrightarrow (m<n \lor m=n)].

10.

\forall n [0=n \lor 0<n].

(“every natural number is zero or bigger than zero”)

11.

\forall m \forall n [(Sm<n \lor Sm=n) \leftrightarrow m<n].

These axioms are all first order statements. That is, all variables range over the natural numbers and not sets thereof, a fact even stronger than their being arithmetical. Moreover, there is but one existential quantifier, in axiom 3. Axioms 1 and 2, together with an axiom schema of induction make up the usual Peano-Dedekind definition of N. Adding to these axioms any sort of axiom schema of induction makes redundant the axioms 3, 10, and 11.

Induction and comprehension schema

If φ(n) is a formula of second-order arithmetic with a free number variable n and possible other free number or set variables (written m_• and X_•), the induction axiom for φ is the axiom:

\forall m_\bullet \forall X_\bullet ((\varphi(0) \land \forall n (\varphi(n) \rightarrow \varphi(Sn)) \rightarrow \forall n \varphi(n))

The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas.

One particularly important instance of the induction scheme is when φ is the formula “ $n \in X$ ” expressing the fact that n is a member of X (X being a free set variable): in this case, the induction axiom for φ is

\forall X ((0\in X \land \forall n (n\in X \rightarrow Sn\in X)) \rightarrow \forall n (n\in X))

This sentence is called the second-order induction axiom.

Returning to the case where φ(n) is a formula with a free variable n and possibly other free variables, we define the comprehension axiom for φ to be:

\forall m_\bullet \forall X_\bullet \exists Z \forall n (n\in Z \leftrightarrow \varphi(n))

Essentially, this allows us to form the set $Z = \{ n | \varphi(n) \}$ of natural numbers satisfying φ(n). There is a technical restriction that the formula φ may not contain the variable Z, for otherwise the formula $n \not \in Z$ would lead to the comprehension axiom

\exists Z \forall n ( n \in Z \leftrightarrow n \not \in Z)

which is inconsistent. This convention is assumed in the remainder of this article.

The full system

The formal theory of second-order arithmetic (in the language of second-order arithmetic) consists of the basic axioms, the comprehension axiom for every formula φ, (arithmetic or otherwise) and the second-order induction axiom. This theory is sometimes called full second order arithmetic to distinguish it from its subsystems, defined below. Second-order semantics eliminates the need for the comprehension axiom, because these semantics imply that every possible set exists.

In the presence of the unrestricted comprehension scheme, the single second-order induction axiom implies each instance of the full induction scheme. Subsystems that limit comprehension in some way may offset this limitation by including part of the induction scheme. Examples of such systems are provided below.

Models of second-order arithmetic

Recall that we view second-order arithmetic as a theory in first-order predicate calculus. Thus a model $\mathcal{M}$ of the language of second-order arithmetic consists of a set M (which forms the range of individual variables) together with a constant 0 (an element of M), a function S from M to M, two binary operations + and · on M, a binary relation < on M, and a collection D of subsets of M, which is the range of the set variables. By omitting D we obtain a model of the language of first order arithmetic.

When D is the full powerset of M, the model $\mathcal{M}$ is called a full model. The use of full second-order semantics is equivalent to limiting the models of second-order arithmetic to the full models. In fact, the axioms of second-order arithmetic have only one full model. This follows from the fact that the axioms of Peano arithmetic with the second-order induction axiom have only one model under second-order semantics.

When M is the usual set of natural numbers with its usual operations, $\mathcal{M}$ is called an ω-model. In this case we may identify the model with D, its collection of sets of naturals, because this set is enough to completely determine an ω-model.

The unique full $\omega$ -model, which is the usual set of natural numbers with its usual structure and all its subsets, is called the intended or standard model of second-order arithmetic.

Definable functions of second-order arithmetic

The first-order functions that are provably total in second-order arithmetic are precisely the same as those representable in system F (Girard et al., 1987, pp. 122–123). Almost equivalently, system F is the theory of functionals corresponding to second-order arithmetic in a manner parallel to how Gödel's system T corresponds to first-order arithmetic in the Dialectica interpretation.

Subsystems of second-order arithmetic

Main article: Reverse mathematics

There are many named subsystems of second-order arithmetic.

A subscript 0 in the name of a subsystem indicates that it includes only a restricted portion of the full second-order induction scheme (Friedman 1976). Such a restriction lowers the proof-theoretic strength of the system significantly. For example, the system ACA₀ described below is equiconsistent with Peano arithmetic. The corresponding theory ACA, consisting of ACA₀ plus the full second-order induction scheme, is stronger than Peano arithmetic.

Arithmetical comprehension

Many of the well-studied subsystems are related to closure properties of models. For example, it can be shown that every ω-model of full second-order arithmetic is closed under Turing jump, but not every ω-model closed under Turing jump is a model of full second-order arithmetic. We may ask whether there is a subsystem of second-order arithmetic satisfied by every ω-model that is closed under Turing jump and satisfies some other, more mild, closure conditions. The subsystem just described is called $\mathrm{ACA}_0$ .

$\mathrm{ACA}_0$ is defined as the theory consisting of the basic axioms, the arithmetical comprehension axiom scheme, in other words the comprehension axiom for every arithmetical formula φ, and the ordinary second-order induction axiom; again, we could also choose to include the arithmetical induction axiom scheme, in other words the induction axiom for every arithmetical formula φ, without making a difference.

It can be seen that a collection S of subsets of ω determines an ω-model of $\mathrm{ACA}_0$ if and only if S is closed under Turing jump, Turing reducibility, and Turing join.

The subscript 0 in $\mathrm{ACA}_0$ indicates that we have not included every instance of the induction axiom in this subsystem. This makes no difference when we study only ω-models, which automatically satisfy every instance of the induction axiom. It is of crucial importance, however, when we study models that are not ω-models. The system consisting of $\mathrm{ACA}_0$ plus induction for all formulas is sometimes called $\mathrm{ACA}$ .

The system $\mathrm{ACA}_0$ is a conservative extension of first-order arithmetic (or first-order Peano axioms), defined as the basic axioms, plus the first order induction axiom scheme (for all formulas φ involving no class variables at all, bound or otherwise), in the language of first order arithmetic (which does not permit class variables at all). In particular it has the same proof-theoretic ordinal ε₀ as first-order arithmetic, owing to the limited induction schema.

The arithmetical hierarchy for formulas

Main article: Arithmetical hierarchy

To define a second subsystem, we will need a bit more terminology.

A formula is called bounded arithmetical, or Δ⁰₀, when all its quantifiers are of the form ∀n<t or ∃n<t (where n is the individual variable being quantified and t is an individual term), where

\forall n<t(\cdots)

stands for

\forall n(n<t \rightarrow \cdots)

and

\exists n<t(\cdots)

stands for

\exists n(n<t \land \cdots)

A formula is called Σ⁰₁ (or sometimes Σ₁), respectively Π⁰₁ (or sometimes Π₁) when it of the form ∃m_•(φ), respectively ∀m_•(φ) where φ is a bounded arithmetical formula and m is an individual variable (that is free in φ). More generally, a formula is called Σ⁰_n, respectively Π⁰_n when it is obtained by adding existential, respectively universal, individual quantifiers to a Π⁰_n−1, respectively Σ⁰_n−1 formula (and Σ⁰₀ and Π⁰₀ are all equivalent to Δ⁰₀). Note that by construction all these formulas are arithmetical (no class variables are ever bound) and, in fact, by putting the formula in Skolem prenex form one can see that every arithmetical formula is equivalent to a Σ⁰_n or Π⁰_n formula for all large enough n.

Recursive comprehension

The subsystem $\mathrm{RCA}_0$ is an even weaker system than $\mathrm{ACA}_0$ and is often used as the base system in reverse mathematics. It consists of: the basic axioms, the Σ⁰₁ induction scheme, and the Δ⁰₁ comprehension scheme. The former term is clear: the Σ⁰₁ induction scheme is the induction axiom for every Σ⁰₁ formula φ. The term “Δ⁰₁ comprehension” requires a little more explaining, however: there is no such thing as a Δ⁰₁ formula (the intended meaning is a formula that is both Σ⁰₁ and Π⁰₁), but we are instead postulating the comprehension axiom for every Σ⁰₁ formula subject to the condition that it is equivalent to a Π⁰₁ formula, in other words, for every Σ⁰₁ formula φ and every Π⁰₁ formula ψ we postulate

\forall m \forall X ((\forall n (\varphi(n) \leftrightarrow \psi(n))) \rightarrow \exists Z \forall n (n\in Z \leftrightarrow \varphi(n)))

The set of first-order consequences of $\mathrm{RCA}_0$ is the same as those of the subsystem IΣ₁ of Peano arithmetic in which induction is restricted to Σ⁰₁ formulas. In turn, IΣ₁ is conservative over primitive recursive arithmetic (PRA) for $\Pi^0_2$ sentences. Moreover, the proof-theoretic ordinal of $\mathrm{RCA}_0$ is ω^ω, the same as that of PRA.

It can be seen that a collection S of subsets of ω determines an ω-model of $\mathrm{RCA}_0$ if and only if S is closed under Turing reducibility and Turing join. In particular, the collection of all computable subsets of ω gives an ω-model of $\mathrm{RCA}_0$ . This is the motivation behind the name of this system—if a set can be proved to exist using $\mathrm{RCA}_0$ , then the set is computable (i.e. recursive).

Weaker systems

Sometimes an even weaker system than $\mathrm{RCA}_0$ is desired. One such system is defined as follows: one must first augment the language of arithmetic with an exponential function (in stronger systems the exponential can be defined in terms of addition and multiplication by the usual trick, but when the system becomes too weak this is no longer possible) and the basic axioms by the obvious axioms defining exponentiation inductively from multiplication; then the system consists of the (enriched) basic axioms, plus Δ⁰₁ comprehension plus Δ⁰₀ induction.

Stronger systems

Much as we have defined Σ_n and Π_n (or, more accurately, Σ⁰_n and Π⁰_n) formulae, we can define Σ¹_n and Π¹_n formulae in the following way: a Δ¹₀ (or Σ¹₀ or Π¹₀) formula is just an arithmetical formula, and a Σ¹_n, respectively Π¹_n, formula is obtained by adding existential, respectively universal, class quantifiers in front of a Π¹_n−1, respectively Σ¹_n−1.

It is not too hard to see that over a not too weak system, any formula of second-order arithmetic is equivalent to a Σ¹_n or Π¹_n formula for all large enough n. The system Π¹₁-comprehension is the system consisting of the basic axioms, plus the ordinary second-order induction axiom and the comprehension axiom for every Π¹₁ formula φ. It is an easy exercise to show that this is actually equivalent to Σ¹₁-comprehension (on the other hand, Δ¹₁-comprehension, defined by the same trick as introduced earlier for Δ⁰₁ comprehension, is actually weaker).

Projective Determinacy

Projective determinacy is the assertion that every two-player perfect information game with moves being integers, game length ω and projective payoff set is determined, that is one of the players has a winning strategy. (The first player wins the game if the play belongs to the payoff set; otherwise, the second player wins.) A set is projective iff (as a predicate) it is expressible by a formula in the language of second order arithmetic, allowing real numbers as parameters, so projective determinacy is expressible as a schema in the language of Z₂.

Many natural propositions expressible in the language of second order arithmetic are independent of Z₂ and even ZFC but are provable from projective determinacy. Examples include coanalytic perfect subset property, measurability and the property of Baire for $\Sigma^1_2$ sets, $\Pi^1_3$ uniformization, etc. Over a weak base theory (such as RCA₀), projective determinacy implies comprehension and provides an essentially complete theory of second order arithmetic — natural statements in the language of Z₂ that are independent of Z₂ with projective determinacy are hard to find.^[1]

ZFC + {there are n Woodin cardinals: n is a natural number} is conservative over Z₂ with projective determinacy, that is a statement in the language of second order arithmetic is provable in Z₂ with projective determinacy iff its translation into the language of set theory is provable in ZFC + {there are n Woodin cardinals: n∈N}.

Coding mathematics in second-order arithmetic

Second-order arithmetic allows us to speak directly (without coding) of natural numbers and sets of natural numbers. Pairs of natural numbers can be coded in the usual way as natural numbers, so arbitrary integers or rational numbers are first-class citizens in the same manner as natural numbers. Functions between these sets can be encoded as sets of pairs, and hence as subsets of the natural numbers, without difficulty. Real numbers can be defined as Cauchy sequences of rational numbers, but for technical reasons not discussed here, it is preferable (in the weak axiom systems above) to constrain the convergence rate (say by requiring that the distance between the n-th and (n+1)-th term be less than 2⁻ⁿ). These systems cannot speak of real functions, or subsets of the reals. Nevertheless, continuous real functions are legitimate objects of study, since they are defined by their values on the rationals. Moreover, a related trick makes it possible to speak of open subsets of the reals. Even Borel sets of reals can be coded in the language of second-order arithmetic, although doing so is a bit tricky.

References

Burgess, John P., 2005. Fixing Frege. Princeton University Press.
Buss, S. R., Handbook of proof theory ISBN 0-444-89840-9
Friedman, Harvey. "Systems of second order arithmetic with restricted induction," I, II (Abstracts). Journal of Symbolic Logic, v.41, pp. 557– 559, 1976. JStor
Girard, Lafont and Taylor, 1987. Proofs and Types. Cambridge University Press.
Hilbert, David; Bernays, Paul (1934), Grundlagen der Mathematik, Die Grundlehren der mathematischen Wissenschaften, Band 40, 50, Berlin, New York: Springer-Verlag, MR 0237246
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
Gaisi Takeuti (1975) Proof theory ISBN 0-444-10492-5

↑ W. Hugh Woodin (2001). "The Continuum Hypothesis, Part I". Notices of the American Mathematical Society 48 (6).