When Peano formulated his axioms, the language of
mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for
set membership (∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the
Begriffsschrift by
Gottlob Frege, published in 1879. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of
Boole and
Schröder. The Peano axioms define the arithmetical properties of
natural numbers, usually represented as a
set N or \mathbb{N}. The
non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol
S. The first axiom states that the constant 0 is a natural number: Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in
Formulario mathematico include zero. The next four axioms describe the
equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments. The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "
successor"
function S. ) limits
N to the chain of light pieces ("no junk") as only light dominoes will fall when the nearest is toppled. Axioms 1, 6, 7, 8 define a
unary representation of the intuitive notion of natural numbers: the number 1 can be defined as
S(0), 2 as
S(
S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. The intuitive notion that each natural number can be obtained by applying
successor sufficiently many times to zero requires an additional axiom, which is sometimes called the
axiom of induction. The induction axiom is sometimes stated in the following form: In Peano's original formulation, the induction axiom is a
second-order axiom. It is now common to replace this second-order principle with a weaker
first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section below.
Defining arithmetic operations and relations If the second-order induction axiom is used, it is possible to define
addition,
multiplication, and
total (linear) ordering on
N directly using the axioms. However, and addition and multiplication are often added as axioms. The respective functions and relations are constructed in
set theory or
second-order logic, and can be shown to be unique using the Peano axioms.
Addition Addition is a function that
maps two natural numbers (two elements of
N) to another one. It is defined
recursively as: : \begin{align} a + 0 &= a , & \textrm{(1)}\\ a + S (b) &= S (a + b). & \textrm{(2)} \end{align} For example: : \begin{align} a + 1 &= a + S(0) & \mbox{by definition} \\ &= S(a + 0) & \mbox{using (2)} \\ &= S(a), & \mbox{using (1)} \\ \\ a + 2 &= a + S(1) & \mbox{by definition} \\ &= S(a + 1) & \mbox{using (2)} \\ &= S(S(a)) & \mbox{using } a + 1 = S(a) \\ \\ a + 3 &= a + S(2) & \mbox{by definition} \\ &= S(a + 2) & \mbox{using (2)} \\ &= S(S(S(a))) & \mbox{using } a + 2 = S(S(a)) \\ \text{etc.} & \\ \end{align} To prove commutativity of addition, first prove 0+b=b and S(a)+b=S(a+b), each by induction on b. Using both results, then prove a+b=b+a by induction on b. The
structure is a
commutative monoid with identity element 0. is also a
cancellative magma, and thus
embeddable in a
group. The smallest group embedding
N is the
integers.
Multiplication Similarly,
multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as: : \begin{align} a \cdot 0 &= 0, \\ a \cdot S (b) &= a + (a \cdot b). \end{align} It is easy to see that S(0) is the multiplicative
right identity: :a\cdot S(0) = a + (a\cdot 0) = a + 0 = a To show that S(0) is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: • S(0) is the left identity of 0: S(0)\cdot 0 = 0. • If S(0) is the left identity of a (that is S(0)\cdot a = a), then S(0) is also the left identity of S(a): S(0)\cdot S(a) = S(0) + S(0)\cdot a = S(0) + a = a + S(0) = S(a + 0) = S(a), using commutativity of addition. Therefore, by the induction axiom S(0) is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and
distributes over addition: : a \cdot (b + c) = (a\cdot b) + (a\cdot c). Thus, (\N, +, 0, \cdot, S(0)) is a commutative
semiring.
Inequalities The usual
total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number: : For all , if and only if there exists some such that . This relation is stable under addition and multiplication: for a, b, c \in \N , if , then: •
a +
c ≤
b +
c, and •
a ·
c ≤
b ·
c. Thus, the structure is an
ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤": : For any
predicate φ, if :*
φ(0) is true, and :* for every , if
φ(
k) is true for every such that , then
φ(
S(
n)) is true, :* then for every ,
φ(
n) is true. This form of the induction axiom, called
strong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are
well-ordered—every
nonempty subset of
N has a
least element—one can reason as follows. Let a nonempty be given and assume
X has no least element. • Because 0 is the least element of
N, it must be that . • For any , suppose for every , . Then , for otherwise it would be the least element of
X. Thus, by the strong induction principle, for every , . Thus, , which
contradicts X being a nonempty subset of
N. Thus
X has a least element.
Models A
model of the Peano axioms is a triple , where
N is a (necessarily infinite) set, and satisfies the axioms above.
Dedekind proved in his 1888 book,
The Nature and Meaning of Numbers (, i.e., "What are the numbers and what are they good for?") that any two models of the Peano axioms (including the second-order induction axiom) are
isomorphic. In particular, given two models and of the Peano axioms, there is a unique
homomorphism satisfying : \begin{align} f(0_A) &= 0_B \\ f(S_A (n)) &= S_B (f (n)) \end{align} and it is a
bijection. This means that the second-order Peano axioms are
categorical. (This is not the case with any first-order reformulation of the Peano axioms, below.)
Set-theoretic models The Peano axioms can be derived from
set theoretic constructions of the
natural numbers and axioms of set theory such as
ZF. The standard construction of the naturals, due to
John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator
s on sets defined as: : s(a) = a \cup \{a\} The set of natural numbers
N is defined as the intersection of all sets
closed under
s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it: : \begin{align} 0 &= \emptyset \\ 1 &= s(0) = s(\emptyset) = \emptyset \cup \{ \emptyset \} = \{ \emptyset \} = \{ 0 \} \\ 2 &= s(1) = s(\{ 0 \}) = \{ 0 \} \cup \{ \{ 0 \} \} = \{ 0 , \{ 0 \} \} = \{ 0, 1 \} \\ 3 &= s(2) = s(\{ 0, 1 \}) = \{ 0, 1 \} \cup \{ \{ 0, 1 \} \} = \{ 0, 1, \{ 0, 1 \} \} = \{ 0, 1, 2 \} \end{align} and so on. The set
N together with 0 and the
successor function satisfies the Peano axioms. Peano arithmetic is
equiconsistent with several weak systems of set theory. One such system is ZFC with the
axiom of infinity replaced by its negation. Another such system consists of
general set theory (
extensionality, existence of the
empty set, and the
axiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
Interpretation in category theory The Peano axioms can also be understood using
category theory. Let
C be a
category with
terminal object 1
C, and define the category of
pointed unary systems, US1(
C) as follows: • The objects of US1(
C) are triples where
X is an object of
C, and and are
C-morphisms. • A morphism
φ : (
X, 0
X,
SX) → (
Y, 0
Y,
SY) is a
C-morphism with and . Then
C is said to satisfy the Dedekind–Peano axioms if US1(
C) has an initial object; this initial object is known as a
natural number object in
C. If is this initial object, and is any other object, then the unique map is such that : \begin{align} u (0) &= 0_X, \\ u (S) &= S_X (u). \end{align} This is precisely the recursive definition of 0
X and
SX.
Consistency When the Peano axioms were first proposed,
Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number".
Henri Poincaré was more cautious, saying they only defined natural numbers if they were
consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900,
David Hilbert posed the problem of proving their consistency using only
finitistic methods as the
second of his
twenty-three problems. In 1931,
Kurt Gödel proved his
second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself, if Peano arithmetic is consistent. Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using
type theory. In 1936,
Gerhard Gentzen gave
a proof of the consistency of Peano's axioms, using
transfinite induction up to an
ordinal called
ε0. Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a
Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite
trees, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition. The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as
Gentzen's proof. A small number of philosophers and mathematicians, some of whom also advocate
ultrafinitism, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be
total. Curiously, there are
self-verifying theories that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true \Pi_1 theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a Hilbert-style proof of "0=1"). == Peano arithmetic as first-order theory ==