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Peano axioms
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{{short description|axioms for the natural numbers}}In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.{{sfn|Grassmann|1861}} In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.{{harvnb|Peirce|1881}}, {{harvnb|Shields|1997}} In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method ().The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".{{sfn|van Heijenoort|1967|page=94}} The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.

Formulation

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) and implication (⊃, which comes from Peano's reversed 'C'.) 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.{{sfn|van Heijenoort|1967|page=2}} Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.{{sfn|van Heijenoort|1967|page=83}}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:{{ordered list|start=1| 1= 0 is a natural number.}}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.{{sfn|van Heijenoort|1967|page=83}}{{ordered list|start=2< y land y < z Rightarrow x < z ), i.e., the '<' operator is transitive.
  1. forall x in mathbf{N} ( neg (x < x) ), i.e., the '<' operator is irreflexive.
  2. forall x, y in mathbf{N} ( x < y lor x = y lor y < x ), i.e., the ordering satisfies trichotomy.
  3. forall x, y, z in mathbf{N} ( x < y Rightarrow x + z < y + z ), i.e. the ordering is preserved under addition of the same element.
  4. forall x, y, z in mathbf{N} ( 0 < z land x < y Rightarrow x cdot z < y cdot z ), i.e. the ordering is preserved under multiplication by the same positive element.
  5. forall x, y in mathbf{N} ( x < y Rightarrow exists z in mathbf{N} ( x + z = y ) ), i.e. given any two distinct elements, the larger is the smaller plus another element.
  6. 0 < 1 land forall x in mathbf{N} ( x > 0 Rightarrow x ge 1 ), i.e. zero and one are distinct and there is no element between them.
  7. forall x in mathbf{N} ( x ge 0 ), i.e. zero is the minimum element.
The theory defined by these axioms is known as PA−; the theory PA is obtained by adding the first-order induction schema. An important property of PA− is that any structure M satisfying this theory has an initial segment (ordered by le) isomorphic to mathbf{N}. Elements in that segment are called standard elements, while other elements are called nonstandard elements.

Models

A model of the Peano axioms is a triple {{nowrap|(N, 0, S)}}, where N is a (necessarily infinite) set, {{nowrap|0 ∈ N}} and {{nowrap|S : N → N}} 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 {{nowrap|(NA, 0A, S'A)}} and {{nowrap|(NB, 0B, S'B)}} of the Peano axioms, there is a unique homomorphism {{nowrap|f : NA → NB}} 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, however.

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.{{harvnb|Suppes|1960}}, {{harvnb|Hatcher|1982}} 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}
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 {{nowrap|s : N → N}} satisfies the Peano axioms.Peano arithmetic is equiconsistent with several weak systems of set theory.{{sfn|Tarski|Givant|1987|loc=Section 7.6}} 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 1C, and define the category of pointed unary systems, US1(C) as follows:
  • The objects of US1(C) are triples {{nowrap|(X, 0X, S'X)}} where X is an object of C, and {{nowrap|0X : 1C → X}} and {{nowrap|S'X : X → X}} are C-morphisms.
  • A morphism φ : (X, 0X, S'X) → (Y, 0Y, S'Y) is a C-morphism {{nowrap|φ : X → Y}} with {{nowrap|1=φ 0X = 0Y}} and {{nowrap|1=φ S'X = S'Y φ}}.
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 {{nowrap|(N, 0, S)}} is this initial object, and {{nowrap|(X, 0X, S'X)}} is any other object, then the unique map {{nowrap|u : (N, 0, S) → (X, 0X, S'X)}} is such that
begin{align}
u 0 &= 0_X, u (S x) &= S_X (u x).end{align}This is precisely the recursive definition of 0X and SX.

Nonstandard models

Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.{{sfn|Hermes|1973|loc=VI.4.3|ps=, presenting a theorem of Thoralf Skolem}} The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism.{{sfn|Hermes|1973|loc=VI.3.1}} This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as Skolem in 1933 provided an explicit construction of such a nonstandard model. On the other hand, Tennenbaum's theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable.{{sfn|Kaye|1991|loc=Section 11.3}} This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. However, there is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is {{nowrap|ω + ζ·η}}, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.

Overspill

A cut in a nonstandard model M is a nonempty subset I of M so that I is downward closed ( x

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x, {{nowrap>1=x = x}}. That is, equality is reflexive.x and y, if {{nowrap>1=x = y}}, then {{nowrapy = x}}. That is, equality is symmetric relation>symmetric.x, y and z, if x = y and y = z, then {{nowrap>1=x = z}}. That is, equality is transitive.a and b, if b is a natural number and {{nowrap>1=a = b}}, then a is also a natural number. That is, the natural numbers are closed under equality.}}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.{{ordered list|start=6| 6=For every natural number n, S(n) is a natural number.m and n, {{nowrap>1=m = n}} if and only if {{nowrapS(m) = S(n)}}. That is, S is an injective function>injection.n, {{nowrap>1=S(n) = 0}} is false. That is, there is no natural number whose successor is 0.}}Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number.{{sfn|Peano|1889|page=1}} This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. 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. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.{{ordered list|start=9| 9=If K is a set such that:
  • 0 is in K, and
  • for every natural number n, n being in K implies that S(n) is in K,
then K contains every natural number.}}The induction axiom is sometimes stated in the following form:{{ordered list|start=9
φ is a unary predicate (mathematics)>predicate such that:
  • φ(0) is true, and
  • for every natural number n, φ(n) being true implies that φ(S(n)) is true,
then φ(n) is true for every natural number n.}}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 {{section link||Models}} below.

Arithmetic

The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. 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}The structure {{nowrap|(N, +)}} is a commutative monoid with identity element 0. {{nowrap|(N, +)}} 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) (or "1", in the familiar language of decimal representation) is the multiplicative right identity:
a · S(0) = a + (a · 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) · 0 = 0.
  • If S(0) is the left identity of a (that is S(0) · a = a), then S(0) is also the left identity of S(a): S(0) · S(a) = S(0) + S(0) · a = S(0) + a = a + S(0) = S(a + 0) = S(a).
Therefore by the induction axiom S(0) is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication distributes over addition:
a · (b + c) = (a · b) + (a · c).
Thus, {{nowrap|(N, +, 0, ·, 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 {{nowrap|a, b ∈ N}}, {{nowrap|a ≤ b}} if and only if there exists some {{nowrap|c ∈ N}} such that {{nowrap|1=a + c = b}}.
This relation is stable under addition and multiplication: for a, b, c in mathbf{N} , if {{nowrap|a ≤ b}}, then:
  • a + c ≤ b + c, and
  • a · c ≤ b · c.
Thus, the structure {{nowrap|(N, +, ·, 1, 0, ≤)}} 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 {{nowrap|n, k ∈ N}}, if {{nowrap|k ≤ n}} implies that φ(k) is true, then φ(S(n)) is true, then for every {{nowrap|n ∈ N}}, 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 {{nowrap|X ⊆ N}} be given and assume X has no least element.
  • Because 0 is the least element of N, it must be that {{nowrap|0 ∉ X}}.
  • For any {{nowrap|n ∈ N}}, suppose for every {{nowrap|k ≤ n}}, {{nowrap|k ∉ X}}. Then {{nowrap|S(n) ∉ X}}, for otherwise it would be the least element of X.
Thus, by the strong induction principle, for every {{nowrap|n ∈ N}}, {{nowrap|n ∉ X}}. Thus, {{nowrap|1=X ∩ N = ∅}}, which contradicts X being a nonempty subset of N. Thus X has a least element.

First-order theory of arithmetic

All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.{{sfn|Partee|Ter Meulen|Wall|2012|page=215}} The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction is in second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.{{sfnp|Harsanyi|1983}}First-order axiomatizations of Peano arithmetic have an important limitation, however. In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of Robinson arithmetic, is sufficient for this purpose:{{sfn|Mendelson|1997|page=155}}
  • forall x in mathbf{N} (0 neq S ( x ))
  • forall x, y in mathbf{N} (S( x ) = S( y ) Rightarrow x = y)
  • forall x in mathbf{N} (x + 0 = x )
  • forall x, y in mathbf{N} (x + S( y ) = S( x + y ))
  • forall x in mathbf{N} (x cdot 0 = 0)
  • forall x, y in mathbf{N} (x cdot S ( y ) = x cdot y + x )
In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a countably infinite set of axioms. For each formula {{nowrap|φ(x, y1, ..., y'k)}} in the language of Peano arithmetic, the first-order induction axiom' for φ'' is the sentence
forall bar{y} (varphi(0,bar{y}) land forall x ( varphi(x,bar{y})Rightarrowvarphi(S(x),bar{y})) Rightarrow forall x varphi(x,bar{y}))
where bar{y} is an abbreviation for y1,...,yk. The first-order induction schema includes every instance of the first-order induction axiom, that is, it includes the induction axiom for every formula φ.

Equivalent axiomatizations

There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, use a signature that only has symbols for 0 and the successor, addition, and multiplications operations, other axiomatizations use the language of ordered semirings, including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.{{sfn|Kaye|1991|pages=16–18}}
  1. forall x, y, z in mathbf{N} ( (x + y) + z = x + (y + z) ), i.e., addition is associative.
  2. forall x, y in mathbf{N} ( x + y = y + x ), i.e., addition is commutative.
  3. forall x, y, z in mathbf{N} ( (x cdot y) cdot z = x cdot (y cdot z) ), i.e., multiplication is associative.
  4. forall x, y in mathbf{N} ( x cdot y = y cdot x ), i.e., multiplication is commutative.
  5. forall x, y, z in mathbf{N} ( x cdot (y + z) = (x cdot y) + (x cdot z) ), i.e., multiplication distributes over addition.
  6. forall x in mathbf{N} ( x + 0 = x land x cdot 0 = 0 ), i.e., zero is an identity for addition, and an absorbing element for multiplication (actually superfluous" forall x in mathbf{N} ( x cdot 0 = 0 ) " can be proven from the other axioms (in first-order logic) as follows. Firstly, x cdot 0 + x cdot 0 = x cdot (0+0) = x cdot 0 = x cdot 0 + 0 by distributivity and additive identity. Secondly, x cdot 0 = 0 lor x cdot 0 > 0 by Axiom 15. If x cdot 0 > 0 then x cdot 0 + x cdot 0 > x cdot 0 + 0 by addition of the same element and commutativity, and hence x cdot 0 + 0 > x cdot 0 + 0 by substitution, contradicting irreflexivity. Therefore it must be that x cdot 0 = 0 .).
  7. forall x in mathbf{N} ( x cdot 1 = x ), i.e., one is an identity for multiplication.
  8. forall x, y, z in mathbf{N} ( x