Movatterモバイル変換

Jump to content

Peano axioms

From Wikipedia, the free encyclopedia

(Redirected fromPeano arithmetic)

Axioms for the natural numbers

Inmathematical logic, thePeano axioms (/piˈɑːnoʊ/,^[1][peˈaːno]), also known as theDedekind–Peano axioms or thePeano postulates, areaxioms for thenatural numbers presented by the 19th-century Italian mathematicianGiuseppe Peano. These axioms have been used nearly unchanged in a number ofmetamathematical investigations, including research into fundamental questions of whethernumber theory isconsistent andcomplete.

Theaxiomatization ofarithmetic provided by Peano axioms is commonly calledPeano arithmetic.

The importance of formalizingarithmetic was 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 operation andinduction.^[2]^[3] In 1881,Charles Sanders Peirce provided anaxiomatization of natural-number arithmetic.^[4]^[5] 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 bookThe principles of arithmetic presented by a new method (Latin:Arithmetices principia, nova methodo exposita).

Peano’s axioms can be divided into groups according to their subject matter. 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".^[6] The next three axioms arefirst-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final, axiom is asecond-order statement 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 induction axiom with a first-orderaxiom schema. The termPeano arithmetic is sometimes used for specifically naming this restricted system.

Historical second-order formulation

This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Peano axioms" – news ·newspapers ·books ·scholar ·JSTOR(May 2024) (Learn how and when to remove this message)

When Peano formulated his axioms, the language ofmathematical 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 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 theBegriffsschrift byGottlob Frege, published in 1879.^[7] Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work ofBoole andSchröder.^[8]

The Peano axioms define the arithmetical properties ofnatural numbers, usually represented as asetN or $\mathbb {N} .$ Thenon-logical symbols for 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:

0 is a natural number.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number,^[9] while the axioms inFormulario mathematico include zero.^[10]

The next four axioms describe theequality 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.^[8]

For every natural numberx,x =x. That is, equality isreflexive.
For all natural numbersx andy, ifx =y, theny =x. That is, equality issymmetric.
For all natural numbersx,y andz, ifx =y andy =z, thenx =z. That is, equality istransitive.
For alla andb, ifb is a natural number anda =b, thena is also a natural number. That is, the natural numbers areclosed 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"functionS.

For every natural numbern,S(n) is a natural number. That is, the natural numbers areclosed underS.
For all natural numbersm andn, ifS(m) =S(n), thenm =n. That is,S is aninjection.
For every natural numbern,S(n) = 0 is 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 setN of natural numbers.^{[note 1]}^[11]^[12] However, axioms 1–8 arealso satisfied by the set of all dominoes—whether light or dark—taken together.^{[note 2]} The 9th axiom (induction) limitsN to the chain of light pieces ("no junk") as only light dominoes will fall when the nearest is toppled.^[13]

Axioms 1, 6, 7, 8 define aunary representation of 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 applyingsuccessor sufficiently many times to zero requires an additional axiom, which is sometimes called theaxiom of induction.

IfK is a set such that:
- 0 is inK, and
- for every natural numbern,n being inK implies thatS(n) is inK,
thenK contains every natural number.

The induction axiom is sometimes stated in the following form:

Ifφ is a unarypredicate such 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-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section§ Peano arithmetic as first-order theory below.

Defining arithmetic operations and relations

If we use the second-order induction axiom, it is possible to defineaddition,multiplication, andtotal (linear) ordering onN directly using the axioms. However,with first-order induction, this is not possible^{[citation needed]} and addition and multiplication are often added as axioms. The respective functions and relations are constructed inset theory orsecond-order logic, and can be shown to be unique using the Peano axioms.

Addition

Addition is a function thatmaps two natural numbers (two elements ofN) to another one. It is definedrecursively as:

{\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 $b {\displaystyle b}$ . Using both results, then prove $a+b=b+a$ by induction on $b {\displaystyle b}$ .Thestructure(N, +) is acommutative monoid with identity element 0.(N, +) is also acancellative magma, and thusembeddable in agroup. The smallest group embeddingN is theintegers.^{[citation needed]}

Multiplication

Similarly,multiplication is 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 $a {\displaystyle a}$ (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^[14] that multiplication is commutative anddistributes over addition:

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

.

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

Inequalities

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

For alla,b ∈N,a ≤b if and only if there exists somec ∈N such 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 ∈N such 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—everynonempty subset ofN has aleast element—one can reason as follows. Let a nonemptyX ⊆N be given and assumeX has 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 = ∅, whichcontradictsX being a nonempty subset ofN. ThusX has a least element.

Models

Amodel of the Peano axioms is a triple(N, 0,S), whereN is a (necessarily infinite) set,0 ∈N andS:N →N satisfies the axioms above.Dedekind proved 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_B satisfying

{\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 theoretic constructions of thenatural numbers and axioms of set theory such asZF.^[15] The standard construction of the naturals, due toJohn von Neumann, starts from a definition of 0 as the empty set, ∅, and an operators on sets defined as:

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

The set of natural numbersN is defined as the intersection of all setsclosed unders that 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 setN together with 0 and thesuccessor functions :N →N satisfies the Peano axioms.

Peano arithmetic isequiconsistent with several weak systems of set theory.^[16] One such system is ZFC with theaxiom of infinity replaced 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. LetC be acategory withterminal object 1_C, and define the category ofpointed unary systems, US₁(C) as follows:

The objects of US₁(C) are triples(X, 0_X,S_X) whereX is an object ofC, and0_X : 1_C →X andS_X :X →X areC-morphisms.
A morphismφ : (X, 0_X,S_X) → (Y, 0_Y,S_Y) is aC-morphismφ :X →Y withφ 0_X = 0_Y andφS_X =S_Yφ.

ThenC is said to satisfy the Dedekind–Peano axioms if US₁(C) has an initial object; this initial object is known as anatural number object inC. 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_X andS_X.

Consistency

Further information:Hilbert's second problem andConsistency

When the Peano axioms were first proposed,Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number".^[17]Henri Poincaré was 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.^[18] In 1900,David Hilbert posed the problem of proving their consistency using onlyfinitistic methods as thesecond of histwenty-three problems.^[19] In 1931,Kurt Gödel proved hissecond incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself, if Peano arithmetic is consistent.^[20]

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.^[21] In 1936,Gerhard Gentzen gavea proof of the consistency of Peano's axioms, usingtransfinite induction up to anordinal calledε₀.^[22] 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 aTuring machine describing 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 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").^[23]

Peano arithmetic as first-order theory

All of the Peano axioms except the ninth axiom (the induction axiom) are statements infirst-order logic.^[24] 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 itquantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-orderaxiom 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.^[25] 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 thesignature 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 ofRobinson arithmetic, is sufficient for this purpose:^[26]

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

In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of arecursively enumerable and even decidable set ofaxioms. For each formulaφ(x,y₁, ...,y_k) in the language of Peano arithmetic, thefirst-order induction axiom forφ is 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 alternative^[27] uses 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):

$\forall x,y,z\ ((x+y)+z=x+(y+z))$ , i.e., addition isassociative.
$\forall x,y\ (x+y=y+x)$ , i.e., addition iscommutative.
$\forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))$ , i.e., multiplication is associative.
$\forall x,y\ (x\cdot y=y\cdot x)$ , i.e., multiplication is commutative.
$\forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))$ , i.e., multiplicationdistributes over addition.
$\forall x\ (x+0=x\land x\cdot 0=0)$ , i.e., zero is anidentity for addition, and anabsorbing element for multiplication (actually superfluous^{[note 3]}).
$\forall x\ (x\cdot 1=x)$ , i.e., one is anidentity for multiplication.
$\forall x,y,z\ (x<y\land y<z\Rightarrow x<z)$ , i.e., the '<' operator istransitive.
$\forall x\ (\neg (x<x))$ , i.e., the '<' operator isirreflexive.
$\forall x,y\ (x<y\lor x=y\lor y<x)$ , i.e., the ordering satisfiestrichotomy.
$\forall x,y,z\ (x<y\Rightarrow x+z<y+z)$ , i.e. the ordering is preserved under addition of the same element.
$\forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)$ , i.e. the ordering is preserved under multiplication by the same positive element.
$\forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))$ , i.e. given any two distinct elements, the larger is the smaller plus another element.
$0<1\land \forall x\ (x>0\Rightarrow x\geq 1)$ , i.e. zero and one are distinct and there is no element between them. In other words, 0 iscovered by 1, which suggests that these numbers are discrete.
$\forall x\ (x\geq 0)$ , 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 $M {\displaystyle M}$ satisfying this theory has an initial segment (ordered by $\leq$ ) isomorphic to $\mathbb {N}$ . Elements in that segment are calledstandard elements, while other elements are callednonstandard elements.

Finally, Peano arithmeticPA is 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 ofPA but 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 theorem for FOL) it follows that there is noalgorithm for deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence,PA is 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 enumerable sets arediophantine sets, and thus definable by existentially quantified formulas (with free variables) ofPA. Formulas ofPA with 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 numbers satisfy the axioms ofPA, there are other models as well (called "non-standard models"); thecompactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.^[28] The upwardLöwenheim–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.^[29] 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 asSkolem in 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.^[30] 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 type of a countable nonstandard model. Lettingω be the order type of the natural numbers,ζ be the order type of the integers, andη be 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

Acut in a nonstandard modelM is a nonempty subsetC ofM so thatC is downward closed (x <y andy ∈C ⇒x ∈C) andC is closed under successor. Aproper cut is 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^[31]—LetM be a nonstandard model of PA and letC be a proper cut ofM. Suppose that ${\bar {a}}$ is a tuple of elements ofM and $\varphi (x,{\bar {a}})$ is a formula in the language of arithmetic so that

M\vDash \varphi (b,{\bar {a}})

for allb ∈C.

Then there is ac inM that is greater than every element ofC such that

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

See also

Notes

^the nearest light piece corresponding to 0, and a neighbor piece corresponding to successor
^The non-contiguous set satisfies axiom 1 as it has a 0 element, 2–5 as it doesn't affect equality relations, 6 as all pieces have a successor, axiom 7 as no two dominos topple, or are toppled by, the same piece, and axiom 8 as every domino bar the first light domino is toppled by another domino.
^" $\forall x\ (x\cdot 0=0)$ " 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 $x\cdot 0=0$ .

References

Citations

^"Peano".Random House Webster's Unabridged Dictionary.
^Grassmann 1861.
^Wang 1957, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an orderedintegral domain in which each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.".
^Peirce 1881.
^Shields 1997.
^Van Heijenoort 1967, p. 94.
^Van Heijenoort 1967, p. 2.
^^a ^bVan Heijenoort 1967, p. 83.
^Peano 1889, p. 1.
^Peano 1908, p. 27.
^Matt DeVos,Mathematical Induction,Simon Fraser University
^Gerardo con Diaz,Mathematical Induction Archived 2 May 2013 at theWayback Machine,Harvard University
^Meseguer & Goguen 1986, sections 2.3 (p. 464) and 4.1 (p. 471).
^For formal proofs, see e.g.File:Inductive proofs of properties of add, mult from recursive definitions.pdf.
^Suppes 1960 harvnb error: no target: CITEREFSuppes1960 (help),Hatcher 2014
^Tarski & Givant 1987, Section 7.6.
^Fritz 1952,p. 137
An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form $x_{0},x_{1},x_{2},\ldots ,x_{n},\ldots$ of which the series of the natural numbers is one instance.
^Gray 2013,p. 133
So Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834).
^Hilbert 1902.
^Gödel 1931.
^Gödel 1958
^Gentzen 1936
^Willard 2001.
^Partee, Ter Meulen & Wall 2012, p. 215.
^Harsanyi (1983).
^Mendelson 1997, p. 155.
^Kaye 1991, pp. 16–18.
^Hermes 1973, VI.4.3, presenting a theorem ofThoralf Skolem
^Hermes 1973, VI.3.1.
^Kaye 1991, Section 11.3.
^Kaye 1991, pp. 70ff..

Sources

Davis, Martin (1974).Computability. Notes by Barry Jacobs.Courant Institute of Mathematical Sciences,New York University.

Dedekind, Richard (1888).Was sind und was sollen die Zahlen? [What are and what should the numbers be?](PDF). Vieweg. Archived fromthe original(PDF) on 20 October 2016. Retrieved4 July 2016.
- Two English translations:
  - Beman, Wooster Woodruff (1901).Essays on the Theory of Numbers(PDF).Dover Publications.
  - Ewald, William B. (1996).From Kant to Hilbert: A Source Book in the Foundations of Mathematics.Oxford University Press. pp. 787–832.ISBN 978-0-19-853271-2.

Fritz, Charles A. Jr. (1952).Bertrand Russell's construction of the external world. New York, Humanities Press.

Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie".Mathematische Annalen.112. Reprinted in English translation in his 1969Collected works, M. E. Szabo, ed.:132–213.doi:10.1007/bf01565428.S2CID 122719892.

Gödel, Kurt (1931)."Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"(PDF).Monatshefte für Mathematik.38. SeeOn Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.:173–198.doi:10.1007/bf01700692.S2CID 197663120. Archived fromthe original(PDF) on 2018-04-11. Retrieved2013-10-31.

Gödel, Kurt (1958)."Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes".Dialectica.12 (3–4). Reprinted in English translation in 1990. Gödel'sCollected Works, Vol II.Solomon Feferman et al., eds.Oxford University Press:280–287.doi:10.1111/j.1746-8361.1958.tb01464.x.

Grassmann, Hermann Günther (1861).Lehrbuch der Arithmetik für höhere Lehranstalten. Verlag von Theod. Chr. Fr. Enslin.

Gray, Jeremy (2013)."The Essayist".Henri Poincaré: A scientific biography.Princeton University Press. p. 133.ISBN 978-0-691-15271-4.

Harsanyi, John C. (1983). "Mathematics, the Empirical Facts, and Logical Necessity". In Hempel, Carl G.; Putnam, Hilary; Essler, Wilhelm K. (eds.).Methodology, Epistemology, and Philosophy of Science. pp. 167–192.doi:10.1007/978-94-015-7676-5_8.ISBN 978-90-481-8389-0.S2CID 121297669.

Hatcher, William S. (2014) [1982].The Logical Foundations of Mathematics. Elsevier.ISBN 978-1-4831-8963-5. Derives the Peano axioms (calledS) from severalaxiomatic set theories and fromcategory theory.

Hermes, Hans (1973).Introduction to Mathematical Logic. Hochschultext. Springer.ISBN 3-540-05819-2.ISSN 1431-4657.

Hilbert, David (1902)."Mathematische Probleme" [Mathematical Problems].Bulletin of the American Mathematical Society.8 (10). Translated by Winton, Maby:437–479.doi:10.1090/s0002-9904-1902-00923-3.

Kaye, Richard (1991).Models of Peano arithmetic.Oxford University Press.ISBN 0-19-853213-X.

Landau, Edmund (1965).Grundlagen Der Analysis. Derives the basic number systems from the Peano axioms. English/German vocabulary included.AMS Chelsea Publishing.ISBN 978-0-8284-0141-8.

Mendelson, Elliott (December 1997) [December 1979].Introduction to Mathematical Logic (Discrete Mathematics and Its Applications) (4th ed.). Springer.ISBN 978-0-412-80830-2.

Meseguer, José; Goguen, Joseph A. (Dec 1986). "Initiality, induction, and computability". In Maurice Nivat and John C. Reynolds (ed.).Algebraic Methods in Semantics(PDF). Cambridge: Cambridge University Press. pp. 459–541.ISBN 978-0-521-26793-9.

Partee, Barbara;Ter Meulen, Alice; Wall, Robert (2012).Mathematical Methods in Linguistics. Springer.ISBN 978-94-009-2213-6.

Peano, Giuseppe (1908).Formulario Mathematico (V ed.). Turin, Bocca frères, Ch. Clausen. p. 27.

Peirce, C. S. (1881)."On the Logic of Number".American Journal of Mathematics.4 (1):85–95.doi:10.2307/2369151.JSTOR 2369151.MR 1507856.

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.ISBN 0-253-33020-3.

Suppes, Patrick (1970) [1960].Axiomatic Set Theory.Dover Publications.ISBN 0-486-61630-4. 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.ISBN 978-0-8218-1041-5.

Van Heijenoort, Jean (1967).From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press.ISBN 978-0-674-32449-7.
- 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 September 2023.

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.JSTOR 2964176.S2CID 26896458.

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.JSTOR 2695030.MR 1833464.S2CID 2822314. Archived fromthe original(PDF) on 2020-11-09. Retrieved2018-06-30.

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.ISBN 978-0-444-89840-1.

Mendelson, Elliott (June 2015) [December 1979].Introduction to Mathematical Logic (Discrete Mathematics and Its Applications) (6th ed.). Chapman and Hall/CRC.ISBN 978-1-4822-3772-6.

Smullyan, Raymond M. (December 2013).The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs. Dover Publications.ISBN 978-0-486-49705-1.

Takeuti, Gaisi (2013).Proof theory (Second ed.). Mineola, New York: Dover Publications.ISBN 978-0-486-49073-1.

External links

Murzi, Mauro."Henri Poincaré". In Fieser, James; Dowden, Bradley (eds.).Internet Encyclopedia of Philosophy.ISSN 2161-0002.OCLC 37741658. 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". Archived fromthe original on 2022-10-26. Retrieved2007-03-23. Commentary on Dedekind's work.

This article incorporates material fromPA onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

v
t
e

Mathematical logic

General

Theorems (list)
and paradoxes

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

Formal systems (list),
language and syntax

Example axiomatic
systems (list)

Principia Mathematica

Computability theory

Related

Mathematics portal

Retrieved from "https://en.wikipedia.org/w/index.php?title=Peano_axioms&oldid=1317454370"

Hidden categories:

[8]ページ先頭

©2009-2025 Movatter.jp