The language of secondorder arithmetic is twosorted. The first sort of terms and in particular 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 uppercase 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 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 secondorder arithmetic is given by the signature
L
=
{
0
,
S
,
+
,
⋅
,
=
,
<
,
∈
}
.
For example,
∀
n
(
n
∈
X
→
S
n
∈
X
)
, is a wellformed formula of secondorder 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
∃
X
∀
n
(
n
∈
X
↔
n
<
S
S
S
S
S
S
0
⋅
S
S
S
S
S
S
S
0
)
is a wellformed formula that is not arithmetical, having one bound set variable X and one bound individual variable n.
Several different interpretations of the quantifiers are possible. If secondorder arithmetic is studied using the full semantics of secondorder logic then the set quantifiers range over all subsets of the range of the number variables. If secondorder arithmetic is formalized using the semantics of firstorder 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 (Shapiro 1991, pp. 74–75).
The following axioms are known as the basic axioms, or sometimes the Robinson axioms. The resulting firstorder 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 "
⋅
", respectively. There is also a primitive binary relation called order, denoted by infix "<".
Axioms governing the successor function and zero:
1.
∀
m
[
S
m
=
0
→
⊥
]
.
(“the successor of a natural number is never zero”)
2.
∀
m
∀
n
[
S
m
=
S
n
→
m
=
n
]
.
(“the successor function is injective”)
3.
∀
n
[
0
=
n
∨
∃
m
[
S
m
=
n
]
]
.
(“every natural number is zero or a successor”)
Addition defined recursively:
4.
∀
m
[
m
+
0
=
m
]
.
5.
∀
m
∀
n
[
m
+
S
n
=
S
(
m
+
n
)
]
.
Multiplication defined recursively:
6.
∀
m
[
m
⋅
0
=
0
]
.
7.
∀
m
∀
n
[
m
⋅
S
n
=
(
m
⋅
n
)
+
m
]
.
Axioms governing the order relation "<":
8.
∀
m
[
m
<
0
→
⊥
]
.
(“no natural number is smaller than zero”)
9.
∀
n
∀
m
[
m
<
S
n
↔
(
m
<
n
∨
m
=
n
)
]
.
10.
∀
n
[
0
=
n
∨
0
<
n
]
.
(“every natural number is zero or bigger than zero”)
11.
∀
m
∀
n
[
(
S
m
<
n
∨
S
m
=
n
)
↔
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 PeanoDedekind definition of N. Adding to these axioms any sort of axiom schema of induction makes redundant the axioms 3, 10, and 11.
If φ(n) is a formula of secondorder 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:
∀
m
∀
X
(
(
φ
(
0
)
∧
∀
n
(
φ
(
n
)
→
φ
(
S
n
)
)
→
∀
n
φ
(
n
)
)
The (full) secondorder induction scheme consists of all instances of this axiom, over all secondorder formulas.
One particularly important instance of the induction scheme is when φ is the formula “
n
∈
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
∀
X
(
(
0
∈
X
∧
∀
n
(
n
∈
X
→
S
n
∈
X
)
)
→
∀
n
(
n
∈
X
)
)
This sentence is called the secondorder induction axiom.
If φ(n) is a formula with a free variable n and possibly other free variables, but not the variable Z, the comprehension axiom for φ is the formula
∀
m
∀
X
∃
Z
∀
n
(
n
∈
Z
↔
φ
(
n
)
)
This axiom makes it possible to form the set
Z
=
{
n

φ
(
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
∉
Z
would lead to the comprehension axiom
∃
Z
∀
n
(
n
∈
Z
↔
n
∉
Z
)
,
which is inconsistent. This convention is assumed in the remainder of this article.
The formal theory of secondorder arithmetic (in the language of secondorder arithmetic) consists of the basic axioms, the comprehension axiom for every formula φ (arithmetic or otherwise), and the secondorder induction axiom. This theory is sometimes called full second order arithmetic to distinguish it from its subsystems, defined below. Because full secondorder semantics imply that every possible set exists, the comprehension axioms may be taken to be part of the deductive system when these semantics are employed (Shapiro 1991, p. 66).
This section describes secondorder arithmetic with firstorder semantics. Thus a model
M
of the language of secondorder 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. Omitting D produces a model of the language of first order arithmetic.
When D is the full powerset of M, the model
M
is called a full model. The use of full secondorder semantics is equivalent to limiting the models of secondorder arithmetic to the full models. In fact, the axioms of secondorder arithmetic have only one full model. This follows from the fact that the axioms of Peano arithmetic with the secondorder induction axiom have only one model under secondorder semantics.
When M is the usual set of natural numbers with its usual operations,
M
is called an ωmodel. In this case, the model may be identified with D, its collection of sets of naturals, because this set is enough to completely determine an ωmodel.
The unique full
ω
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 secondorder arithmetic.
The firstorder functions that are provably total in secondorder arithmetic are precisely the same as those representable in system F (Girard and Taylor 1987, pp. 122–123). Almost equivalently, system F is the theory of functionals corresponding to secondorder arithmetic in a manner parallel to how Gödel's system T corresponds to firstorder arithmetic in the Dialectica interpretation.
There are many named subsystems of secondorder arithmetic.
A subscript 0 in the name of a subsystem indicates that it includes only a restricted portion of the full secondorder induction scheme (Friedman 1976). Such a restriction lowers the prooftheoretic strength of the system significantly. For example, the system ACA_{0} described below is equiconsistent with Peano arithmetic. The corresponding theory ACA, consisting of ACA_{0} plus the full secondorder induction scheme, is stronger than Peano arithmetic.
Many of the wellstudied subsystems are related to closure properties of models. For example, it can be shown that every ωmodel of full secondorder arithmetic is closed under Turing jump, but not every ωmodel closed under Turing jump is a model of full secondorder arithmetic. The subsystem ACA_{0} includes just enough axioms to capture the notion of closure under Turing jump.
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 secondorder induction axiom. It would be equivalent to include the entire arithmetical induction axiom scheme, in other words to include the induction axiom for every arithmetical formula φ.
It can be shown that a collection S of subsets of ω determines an ωmodel of ACA_{0} if and only if S is closed under Turing jump, Turing reducibility, and Turing join (Simpson 2009, pp. 311–313).
The subscript 0 in ACA_{0} indicates that not every instance of the induction axiom scheme is included this subsystem. This makes no difference for ωmodels, which automatically satisfy every instance of the induction axiom. It is of importance, however, in the study of nonωmodels. The system consisting of ACA_{0} plus induction for all formulas is sometimes called ACA with no subscript.
The system ACA_{0} is a conservative extension of firstorder arithmetic (or firstorder 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 prooftheoretic ordinal ε_{0} as firstorder arithmetic, owing to the limited induction schema.
A formula is called bounded arithmetical, or Δ^{0}_{0}, 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
∀
n
<
t
(
⋯
)
stands for
∀
n
(
n
<
t
→
⋯
)
and
∃
n
<
t
(
⋯
)
stands for
∃
n
(
n
<
t
∧
⋯
)
.
A formula is called Σ^{0}_{1} (or sometimes Σ_{1}), respectively Π^{0}_{1} (or sometimes Π_{1}) 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 Σ^{0}_{n}, respectively Π^{0}_{n} when it is obtained by adding existential, respectively universal, individual quantifiers to a Π^{0}_{n−1}, respectively Σ^{0}_{n−1} formula (and Σ^{0}_{0} and Π^{0}_{0} are all equivalent to Δ^{0}_{0}). 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 Σ^{0}_{n} or Π^{0}_{n} formula for all large enough n.
The subsystem RCA_{0} is a weaker system than ACA_{0} and is often used as the base system in reverse mathematics. It consists of: the basic axioms, the Σ^{0}_{1} induction scheme, and the Δ^{0}_{1} comprehension scheme. The former term is clear: the Σ^{0}_{1} induction scheme is the induction axiom for every Σ^{0}_{1} formula φ. The term “Δ^{0}_{1} comprehension” is more complex, because there is no such thing as a Δ^{0}_{1} formula. The Δ^{0}_{1} comprehension scheme instead asserts the comprehension axiom for every Σ^{0}_{1} formula which is equivalent to a Π^{0}_{1} formula. This scheme this includes, for every Σ^{0}_{1} formula φ and every Π^{0}_{1} formula ψ, the axiom:
∀
m
∀
X
(
(
∀
n
(
φ
(
n
)
↔
ψ
(
n
)
)
)
→
∃
Z
∀
n
(
n
∈
Z
↔
φ
(
n
)
)
)
The set of firstorder consequences of RCA_{0} is the same as those of the subsystem IΣ_{1} of Peano arithmetic in which induction is restricted to Σ^{0}_{1} formulas. In turn, IΣ_{1} is conservative over primitive recursive arithmetic (PRA) for
Π
2
0
sentences. Moreover, the prooftheoretic ordinal of
R
C
A
0
is ω^{ω}, the same as that of PRA.
It can be seen that a collection S of subsets of ω determines an ωmodel of 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 ωmodel of RCA_{0}. This is the motivation behind the name of this system—if a set can be proved to exist using RCA_{0}, then the set is recursive (i.e. computable).
Sometimes an even weaker system than 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 Δ^{0}_{1} comprehension, plus Δ^{0}_{0} induction.
Over ACA_{0}, each formula of secondorder arithmetic is equivalent to a Σ^{1}_{n} or Π^{1}_{n} formula for all large enough n. The system Π^{1}_{1}comprehension is the system consisting of the basic axioms, plus the ordinary secondorder induction axiom and the comprehension axiom for every Π^{1}_{1} formula φ. This is equivalent to Σ^{1}_{1}comprehension (on the other hand, Δ^{1}_{1}comprehension.
Projective determinacy is the assertion that every twoplayer 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 Z_{2}.
Many natural propositions expressible in the language of second order arithmetic are independent of Z_{2} and even ZFC but are provable from projective determinacy. Examples include coanalytic perfect subset property, measurability and the property of Baire for
Σ
2
1
sets,
Π
3
1
uniformization, etc. Over a weak base theory (such as RCA_{0}), projective determinacy implies comprehension and provides an essentially complete theory of second order arithmetic — natural statements in the language of Z_{2} that are independent of Z_{2} with projective determinacy are hard to find (Woodin 2001).
ZFC + {there are n Woodin cardinals: n is a natural number} is conservative over Z_{2} with projective determinacy, that is a statement in the language of second order arithmetic is provable in Z_{2} with projective determinacy iff its translation into the language of set theory is provable in ZFC + {there are n Woodin cardinals: n∈N}.
Secondorder arithmetic directly formalizes natural numbers and sets of natural numbers. However, it is able to formalize other mathematical objects indirectly via coding techniques, a fact which was first noticed by Weyl (Simpson 2009, p. 16). The integers, rational numbers, and real numbers can all be formalized in the subsystem RCA_{0}, along with complete separable metric spaces and continuous functions between them (Simpson 2009, Chapter II).
The research program of Reverse Mathematics uses these formalizations of mathematics in secondorder arithmetic to study the setexistence axioms required to prove mathematical theorems (Simpson 2009, p. 32). For example, the intermediate value theorem for functions from the reals to the reals is provable in RCA_{0} (Simpson 2009, p. 87), while the Bolzano/Weirstrass theorem is equivalent to ACA_{0} over RCA_{0} (Simpson 2009, p. 34).