Peano axioms

Inmathematical logic,thePeano axioms(/piˈɑːnoʊ/,[peˈaːno]), also known as theDedekind–Peano axiomsor thePeano postulates,areaxiomsfor thenatural numberspresented by the 19th-century Italian mathematicianGiuseppe Peano.These axioms have been used nearly unchanged in a number ofmetamathematicalinvestigations, including research into fundamental questions of whethernumber theoryisconsistentandcomplete.

Theaxiomatizationofarithmeticprovided by Peano axioms is commonly calledPeano arithmetic.

The importance of formalizingarithmeticwas not well appreciated until the work ofHermann Grassmann,who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about thesuccessor operationandinduction.In 1881,Charles Sanders Peirceprovided anaxiomatizationof natural-number arithmetic.In 1888,Richard Dedekindproposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his bookThe principles of arithmetic presented by a new method(Latin:Arithmetices principia, nova methodo exposita).

The nine 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 aboutequality;in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".The next three axioms arefirst-orderstatements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is asecond-orderstatement of the principle of mathematical induction over the natural numbers, which makes this formulation close tosecond-order arithmetic.A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing thesecond-order inductionaxiom with a first-orderaxiom schema.The termPeano arithmeticis sometimes used for specifically naming this restricted system.

Historical second-order formulation

When Peano formulated his axioms, the language ofmathematical logicwas 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 forset 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 theBegriffsschriftbyGottlob Frege,published in 1879.Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work ofBooleandSchröder.

The Peano axioms define the arithmetical properties ofnatural numbers,usually represented as asetNor$\mathbb {N}.$Thenon-logical symbolsfor the axioms consist of a constant symbol 0 and a unary function symbolS.

The first axiom states that the constant 0 is a natural number:

1. 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 inFormulario mathematicoinclude zero.

The next four axioms describe theequalityrelation.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.

2. For every natural numberx,x=x.That is, equality isreflexive.
3. For all natural numbersxandy,ifx=y,theny=x.That is, equality issymmetric.
4. For all natural numbersx,yandz,ifx=yandy=z,thenx=z.That is, equality istransitive.
5. For allaandb,ifbis a natural number anda=b,thenais also a natural number. That is, the natural numbers areclosedunder equality.

The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor"functionS.

6. For every natural numbern,S(n) is a natural number. That is, the natural numbers areclosedunderS.
7. For all natural numbersmandn,ifS(m) =S(n),thenm=n.That is,Sis aninjection.
8. For every natural numbern,S(n) = 0is false. That is, there is no natural number whose successor is 0.

The chain of light dominoes on the right, starting with the nearest, can represent the setNof natural numbers.However, axioms 1–8 arealsosatisfied by the set of all dominoes — whether light or dark — taken together.The 9th axiom (induction) limitsNto the chain of light pieces ( "no junk" ) as only light dominoes will fall when the nearest is toppled.

Axioms 1, 6, 7, 8 define aunary representationof the intuitive notion of natural numbers: the number 1 can be defined asS(0), 2 asS(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 applyingsuccessorsufficiently many times to zero requires an additional axiom, which is sometimes called theaxiom of induction.

9. IfKis a set such that:
   • 0 is inK,and
   • for every natural numbern,nbeing inKimplies thatS(n) is inK,
   thenKcontains every natural number.

The induction axiom is sometimes stated in the following form:

9. Ifφis a unarypredicatesuch that:
   • φ(0) is true, and
   • for every natural numbern,φ(n) being true implies thatφ(S(n)) is true,
   thenφ(n) is true for every natural numbern.

In Peano's original formulation, the induction axiom is asecond-order axiom.It is now common to replace this second-order principle with a weakerfirst-orderinduction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section§ Peano arithmetic as first-order theorybelow.

Defining arithmetic operations and relations

If we use the second-order induction axiom, it is possible to defineaddition,multiplication,andtotal (linear) orderingonNdirectly using the axioms. However,with first-order induction, this is not possibleand addition and multiplication are often added as axioms. The respective functions and relations are constructed inset theoryorsecond-order logic,and can be shown to be unique using the Peano axioms.

Addition

Additionis a function thatmapstwo natural numbers (two elements ofN) to another one. It is definedrecursivelyas:

${\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}$

For example:

${\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}$

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$b$. Thestructure(N,+)is acommutativemonoidwith identity element 0.(N,+)is also acancellativemagma,and thusembeddablein agroup.The smallest group embeddingNis theintegers.

Multiplication

Similarly,multiplicationis a function mapping two natural numbers to another one. Given addition, it is defined recursively as:

${\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}$

It is easy to see that$S(0)$is the multiplicativeright 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 shownthat multiplication is commutative anddistributes overaddition:

$a\cdot (b+c)=(a\cdot b)+(a\cdot c)$.

Thus,$(\mathbb {N},+,0,\cdot,S(0))$is a commutativesemiring.

Inequalities

The usualtotal orderrelation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:

For alla,b∈N,a≤bif and only if there exists somec∈Nsuch thata+c=b.

This relation is stable under addition and multiplication: for$a,b,c\in \mathbb {N}$,ifa≤b,then:

• a+c≤b+c,and
• a·c≤b·c.

Thus, the structure(N,+, ·, 1, 0, ≤)is anordered 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 anypredicateφ,if

• φ(0) is true, and
• for everyn∈N,ifφ(k) is true for everyk∈Nsuch thatk≤n,thenφ(S(n)) is true,
• then for everyn∈N,φ(n) is true.

This form of the induction axiom, calledstrong 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 arewell-ordered—everynonemptysubsetofNhas aleast element—one can reason as follows. Let a nonemptyX⊆Nbe given and assumeXhas no least element.

• Because 0 is the least element ofN,it must be that0 ∉X.
• For anyn∈N,suppose for everyk≤n,k∉X.ThenS(n) ∉X,for otherwise it would be the least element ofX.

Thus, by the strong induction principle, for everyn∈N,n∉X.Thus,X∩N= ∅,whichcontradictsXbeing a nonempty subset ofN.ThusXhas a least element.

Models

Amodelof the Peano axioms is a triple(N,0,S),whereNis a (necessarily infinite) set,0 ∈NandS:N→Nsatisfies the axioms above.Dedekindproved in his 1888 book,The Nature and Meaning of Numbers(German:Was sind und was sollen die Zahlen?,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) areisomorphic.In particular, given two models(N_A,0_A,S_A)and(N_B,0_B,S_B)of the Peano axioms, there is a uniquehomomorphismf:N_A→N_Bsatisfying

${\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}$

and it is abijection.This means that the second-order Peano axioms arecategorical.(This is not the case with any first-order reformulation of the Peano axioms, below.)

Set-theoretic models

Main article:Set-theoretic definition of natural numbers

The Peano axioms can be derived fromset theoreticconstructions of thenatural numbersand axioms of set theory such asZF.The standard construction of the naturals, due toJohn von Neumann,starts from a definition of 0 as the empty set, ∅, and an operatorson sets defined as:

$s(a)=a\cup \{a\}$

The set of natural numbersNis defined as the intersection of all setsclosedundersthat contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:

${\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}$

and so on. The setNtogether with 0 and thesuccessor functions:N→Nsatisfies the Peano axioms.

Peano arithmetic isequiconsistentwith several weak systems of set theory.One such system is ZFC with theaxiom of infinityreplaced by its negation. Another such system consists ofgeneral set theory(extensionality,existence of theempty set,and theaxiom 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 usingcategory theory.LetCbe acategorywithterminal object1_C,and define the category ofpointed unary systems,US₁(C) as follows:

• The objects of US₁(C) are triples(X,0_X,S_X)whereXis an object ofC,and0_X:1_C→XandS_X:X→XareC-morphisms.
• A morphismφ:(X,0_X,S_X) → (Y,0_Y,S_Y) is aC-morphismφ:X→Ywithφ0_X= 0_YandφS_X=S_Yφ.

ThenCis said to satisfy the Dedekind–Peano axioms if US₁(C) has an initial object; this initial object is known as anatural number objectinC.If(N,0,S)is this initial object, and(X,0_X,S_X)is any other object, then the unique mapu:(N,0,S) → (X,0_X,S_X)is such that

${\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}$

This is precisely the recursive definition of 0_XandS_X.

Consistency

Further information:Hilbert's second problemandConsistency

When the Peano axioms were first proposed,Bertrand Russelland others agreed that these axioms implicitly defined what we mean by a "natural number".Henri Poincaréwas more cautious, saying they only defined natural numbers if they wereconsistent;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 Hilbertposed the problem of proving their consistency using onlyfinitisticmethods as thesecondof histwenty-three problems.In 1931,Kurt Gödelproved hissecond 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 usingtype theory.In 1936,Gerhard Gentzengavea proof of the consistencyof Peano's axioms, usingtransfinite inductionup to anordinalcalledε₀.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 ε₀can be encoded in terms of finite objects (for example, as aTuring machinedescribing a suitable order on the integers, or more abstractly as consisting of the finitetrees,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 asGentzen's proof.A small number of philosophers and mathematicians, some of whom also advocateultrafinitism,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 betotal.Curiously, there areself-verifying theoriesthat 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

All of the Peano axioms except the ninth axiom (the induction axiom) are statements infirst-order logic.The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above issecond-order,since itquantifiesover predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-orderaxiom schemaof 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.The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).

First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from thesuccessor 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 thesignatureof 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 ofRobinson arithmetic,is sufficient for this purpose:

• $\forall x\ (0\neq S(x))$
• $\forall x,y\ (S(x)=S(y)\Rightarrow x=y)$
• $\forall x\ (x+0=x)$
• $\forall x,y\ (x+S(y)=S(x+y))$
• $\forall x\ (x\cdot 0=0)$
• $\forall x,y\ (x\cdot S(y)=x\cdot y+x)$

In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of arecursively enumerableand even decidable set ofaxioms.For each formulaφ(x,y₁,...,y_k)in the language of Peano arithmetic, thefirst-order induction axiomforφis the sentence

$\forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}$

where${\bar {y}}$is an abbreviation fory₁,...,y_k.The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formulaφ.

Equivalent axiomatizations

The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternativeuses an order relation symbol instead of the successor operation and the language ofdiscretely ordered semirings(axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):

1. $\forall x,y,z\ ((x+y)+z=x+(y+z))$,i.e., addition isassociative.
2. $\forall x,y\ (x+y=y+x)$,i.e., addition iscommutative.
3. $\forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))$,i.e., multiplication is associative.
4. $\forall x,y\ (x\cdot y=y\cdot x)$,i.e., multiplication is commutative.
5. $\forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))$,i.e., multiplicationdistributesover addition.
6. $\forall x\ (x+0=x\land x\cdot 0=0)$,i.e., zero is anidentityfor addition, and anabsorbing elementfor multiplication (actually superfluous).
7. $\forall x\ (x\cdot 1=x)$,i.e., one is anidentityfor multiplication.
8. $\forall x,y,z\ (x<y\land y<z\Rightarrow x<z)$,i.e., the '<' operator istransitive.
9. $\forall x\ (\neg (x<x))$,i.e., the '<' operator isirreflexive.
10. $\forall x,y\ (x<y\lor x=y\lor y<x)$,i.e., the ordering satisfiestrichotomy.
11. $\forall x,y,z\ (x<y\Rightarrow x+z<y+z)$,i.e. the ordering is preserved under addition of the same element.
12. $\forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)$,i.e. the ordering is preserved under multiplication by the same positive element.
13. $\forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))$,i.e. given any two distinct elements, the larger is the smaller plus another element.
14. $0<1\land \forall x\ (x>0\Rightarrow x\geq 1)$,i.e. zero and one are distinct and there is no element between them. In other words, 0 iscoveredby 1, which suggests that these numbers are discrete.
15. $\forall x\ (x\geq 0)$,i.e. zero is the minimum element.

The theory defined by these axioms is known asPA⁻.It is also incomplete and one of its important properties is that any structure$M$satisfying this theory has an initial segment (ordered by$\leq$) isomorphic to$\mathbb {N}$.Elements in that segment are calledstandardelements, while other elements are callednonstandardelements.

Finally, Peano arithmeticPAis obtained by adding the first-order induction schema.

Undecidability and incompleteness

According toGödel's incompleteness theorems,the theory ofPA(if consistent) is incomplete. Consequently, there are sentences offirst-order logic(FOL) that are true in the standard model ofPAbut are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such asRobinson arithmetic.

Closely related to the above incompleteness result (viaGödel's completeness theoremfor FOL) it follows that there is noalgorithmfor deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence,PAis an example of anundecidable theory.Undecidability arises already for the existential sentences ofPA,due to the negative answer toHilbert's tenth problem,whose proof implies that allcomputably enumerablesets arediophantine sets,and thus definable by existentially quantified formulas (with free variables) ofPA.Formulas ofPAwith higherquantifier rank(more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of thearithmetical hierarchy.

Nonstandard models

Main article:Non-standard model of arithmetic

Although the usualnatural numberssatisfy the axioms ofPA,there are other models as well (called "non-standard models"); thecompactness theoremimplies that the existence of nonstandard elements cannot be excluded in first-order logic.The upwardLöwenheim–Skolem theoremshows 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.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-orderset theory,such asZFC,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 asSkolemin 1933 provided an explicit construction of such anonstandard 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 iscomputable.This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possibleorder typeof a countable nonstandard model. Lettingωbe the order type of the natural numbers,ζbe the order type of the integers, andηbe the order type of the rationals, the order type of any countable nonstandard model of PA isω+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

Acutin a nonstandard modelMis a nonempty subsetCofMso thatCis downward closed (x<yandy∈C⇒x∈C) andCis closed under successor. Aproper cutis a cut that is a proper subset ofM.Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.

Overspill lemma—LetMbe a nonstandard model of PA and letCbe a proper cut ofM.Suppose that${\bar {a}}$is a tuple of elements ofMand$\phi (x,{\bar {a}})$is a formula in the language of arithmetic so that

$M\vDash \phi (b,{\bar {a}})$for allb∈C.

Then there is acinMthat is greater than every element ofCsuch that

$M\vDash \phi (c,{\bar {a}}).$

See also

• Foundations of mathematics
• Frege's theorem
• Goodstein's theorem
• Neo-logicism
• Non-standard model of arithmetic
• Paris–Harrington theorem
• Presburger arithmetic
• Skolem arithmetic
• Robinson arithmetic
• Second-order arithmetic
• Typographical Number Theory

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• Shields, Paul (1997)."3. Peirce's Axiomatization of Arithmetic".In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).Studies in the Logic of Charles Sanders Peirce.Indiana University Press. pp. 43–52.ISBN0-253-33020-3.

• Suppes, Patrick(1960).Axiomatic Set Theory.Dover Publications.ISBN0-486-61630-4.Derives the Peano axioms fromZFC

• Tarski, Alfred;Givant, Steven (1987).A Formalization of Set Theory without Variables.AMS Colloquium Publications. Vol. 41.American Mathematical Society.ISBN978-0-8218-1041-5.

• Van Heijenoort, Jean(1967).From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931.Harvard University Press.ISBN978-0-674-32449-7.
  • Contains translations of the following two papers, with valuable commentary:
    • Dedekind, Richard(1890).Letter to Keferstein.On p. 100, he restates and defends his axioms of 1888. pp. 98–103.
    • Peano, Giuseppe(1889).Arithmetices principia, nova methodo exposita[The principles of arithmetic, presented by a new method]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Fratres Bocca. pp. 83–97.

• Van Oosten, Jaap (June 1999)."Introduction to Peano Arithmetic (Gödel Incompleteness and Nonstandard Models)"(PDF).Utrecht University.Retrieved2 September2023.

• Wang, Hao(June 1957). "The Axiomatization of Arithmetic".The Journal of Symbolic Logic.22(2).Association for Symbolic Logic:145–158.doi:10.2307/2964176.JSTOR2964176.S2CID26896458.

• Willard, Dan E.(2001)."Self-verifying axiom systems, the incompleteness theorem and related reflection principles"(PDF).The Journal of Symbolic Logic.66(2): 536–596.doi:10.2307/2695030.JSTOR2695030.MR1833464.S2CID2822314.

Further reading

• Buss, Samuel R. (1998). "Chapter II: First-Order Proof Theory of Arithmetic". In Buss, Samuel R. (ed.).Handbook of Proof Theory.New York: Elsevier Science.ISBN978-0-444-89840-1.

• Mendelson, Elliott(June 2015) [December 1979].Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)(6th ed.). Chapman and Hall/CRC.ISBN978-1-4822-3772-6.

• Smullyan, Raymond M.(December 2013).The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs.Dover Publications.ISBN978-0-486-49705-1.

• Takeuti, Gaisi(2013).Proof theory(Second ed.). Mineola, New York.ISBN978-0-486-49073-1.{{cite book}}:CS1 maint: location missing publisher (link)

External links

• Murzi, Mauro."Henri Poincaré".Internet Encyclopedia of Philosophy.Includes a discussion of Poincaré's critique of the Peano's axioms.
• Podnieks, Karlis (2015-01-25). "3. First Order Arithmetic".What is Mathematics: Gödel's Theorem and Around.pp. 93–121.
• "Peano axioms",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Weisstein, Eric W."Peano's Axioms".MathWorld.
• Burris, Stanley N. (2001)."What are numbers, and what is their meaning?: Dedekind".Commentary on Dedekind's work.

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