In mathematical logic , the Peano axioms ( / p i ˈ ɑː n oʊ / , [peˈaːno] ), 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 .
95-569: The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic . The importance of formalizing 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 . In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published
190-453: A + b = b + a {\displaystyle a+b=b+a} by induction on b {\displaystyle b} . The structure ( N , +) is a commutative monoid with identity element 0. ( N , +) is also a cancellative magma , and thus embeddable in a group . The smallest group embedding N is the integers . Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it
285-503: A combination of such coins. Let S ( k ) denote the statement " k dollars can be formed by a combination of 4- and 5-dollar coins". The proof that S ( k ) is true for all k ≥ 12 can then be achieved by induction on k as follows: Base case: Showing that S ( k ) holds for k = 12 is simple: take three 4-dollar coins. Induction step: Given that S ( k ) holds for some value of k ≥ 12 ( induction hypothesis ), prove that S ( k + 1) holds, too. Assume S ( k )
380-583: 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
475-415: A finite chain of deductive reasoning involving the variable n {\displaystyle n} , which can take infinitely many values. The result is a rigorous proof of the statement, not an assertion of its probability. In 370 BC, Plato 's Parmenides may have contained traces of an early example of an implicit inductive proof, however, the earliest implicit proof by mathematical induction
570-424: 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 ε 0 . 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
665-587: A more general version, | sin n x | ≤ n | sin x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} for any real numbers n , x {\displaystyle n,x} , could be proven without induction; but the case n = 1 2 , x = π {\textstyle n={\frac {1}{2}},\,x=\pi } shows it may be false for non-integer values of n {\displaystyle n} . This suggests we examine
760-401: A nonempty X ⊆ N be given and assume X has no least element. Thus, by the strong induction principle, for every n ∈ N , n ∉ X . Thus, X ∩ N = ∅ , which contradicts X being a nonempty subset of N . Thus X has a least element. A model of the Peano axioms is a triple ( N , 0, S ) , where N is a (necessarily infinite) set, 0 ∈ N and S : N → N satisfies
855-524: A set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system. An axiomatic system is said to be consistent if it lacks contradiction . That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven ( principle of explosion ). In an axiomatic system, an axiom
950-451: A simplified version of them as a collection of axioms in his book The 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 about equality ; in modern treatments these are often not taken as part of
1045-555: A single-valued " successor " function S . 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
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#17327725074221140-481: A way such that each new term can be formally eliminated by the priorly introduced terms requires primitive notions (axioms) to avoid infinite regress . This way of doing mathematics is called the axiomatic method . A common attitude towards the axiomatic method is logicism . In their book Principia Mathematica , Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally,
1235-429: Is a method for proving that a statement P ( n ) {\displaystyle P(n)} is true for every natural number n {\displaystyle n} , that is, that the infinitely many cases P ( 0 ) , P ( 1 ) , P ( 2 ) , P ( 3 ) , … {\displaystyle P(0),P(1),P(2),P(3),\dots } all hold. This
1330-399: Is a natural number: This relation is stable under addition and multiplication: for a , b , c ∈ N {\displaystyle a,b,c\in \mathbb {N} } , if a ≤ b , then: Thus, the structure ( 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
1425-408: Is actually a special case of the previous form, because if the statement to be proved is P ( n ) then proving it with these two rules is equivalent with proving P ( n + b ) for all natural numbers n with an induction base case 0 . Assume an infinite supply of 4- and 5-dollar coins. Induction can be used to prove that any whole amount of dollars greater than or equal to 12 can be formed by
1520-461: 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 ≤ {\displaystyle \leq } ) isomorphic to N {\displaystyle \mathbb {N} } . Elements in that segment are called standard elements, while other elements are called nonstandard elements. Finally, Peano arithmetic PA
1615-415: Is an inference rule used in formal proofs , and is the foundation of most correctness proofs for computer programs. Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy , in which the examination of many cases results in a probable conclusion. The mathematical method examines infinitely many cases to prove a general statement, but it does so by
1710-409: Is any set of primitive notions and axioms to logically derive theorems . A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system . A formal theory is an axiomatic system (usually formulated within model theory ) that describes
1805-410: Is arguably finitistic, since the transfinite ordinal ε 0 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
1900-461: Is called categorial (sometimes categorical ). The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by the semantics of the system. As an example, observe the following axiomatic system, based on first-order logic with additional semantics of
1995-456: Is called independent if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent. Unlike consistency, independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the number of axioms in the system. An axiomatic system is called complete if for every statement, either itself or its negation
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#17327725074222090-430: Is commonly abbreviated ZFC , where "C" stands for "choice". Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics . Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing
2185-438: Is defined recursively as: It is easy to see that S ( 0 ) {\displaystyle S(0)} is the multiplicative right identity : To show that S ( 0 ) {\displaystyle S(0)} is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: Therefore, by the induction axiom S ( 0 ) {\displaystyle S(0)}
2280-399: Is derivable from the system's axioms (equivalently, every statement is capable of being proven true or false). Beyond consistency, relative consistency is also the mark of a worthwhile axiom system. This describes the scenario where the undefined terms of a first axiom system are provided definitions from a second, such that the axioms of the first are theorems of the second. A good example
2375-431: Is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis ) and that from each rung we can climb up to
2470-410: 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
2565-547: Is obtained by adding the first-order induction schema. According to Gödel's incompleteness theorems , the theory of PA (if consistent) is incomplete. Consequently, there are sentences of first-order logic (FOL) that are true in the standard model of PA but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as Robinson arithmetic . Axiomatization In mathematics and logic , an axiomatic system
2660-406: Is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema . The term Peano arithmetic is sometimes used for specifically naming this restricted system. When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present
2755-405: 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
2850-399: 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
2945-425: Is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤": 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
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3040-497: Is the earliest extant proof of the sum formula for integral cubes . In India, early implicit proofs by mathematical induction appear in Bhaskara 's " cyclic method ". None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed) was that of Francesco Maurolico in his Arithmeticorum libri duo (1575), who used
3135-437: Is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and distributes over addition: Thus, ( N , + , 0 , ⋅ , S ( 0 ) ) {\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))} is a commutative semiring . The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0
3230-425: Is the real numbers (isomorphic to any other set with the cardinality of the continuum ). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, the property distinguishing these models is their cardinality — a property which cannot be defined within the system. Thus the system is not categorial. However it can be shown to be complete. Stating definitions and propositions in
3325-430: Is the relative consistency of absolute geometry with respect to the theory of the real number system . Lines and points are undefined terms (also called primitive notions ) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems. A model for an axiomatic system is a well-defined set , which assigns meaning for the undefined terms presented in
3420-414: Is the sentence where y ¯ {\displaystyle {\bar {y}}} is an abbreviation for y 1 ,..., 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 formula φ . The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as
3515-617: Is the theory of the natural numbers , which is only partially axiomatized by the Peano axioms (described below). In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals to. For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. the language of the Peano axioms) and a proof might be given that appeals to topology or complex analysis . It might not be immediately clear whether another proof can be found that derives itself solely from
3610-412: Is this initial object, and ( X , 0 X , S X ) is any other object, then the unique map u : ( N , 0, S ) → ( X , 0 X , S X ) is such that This is precisely the recursive definition of 0 X and S X . 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
3705-409: Is true for some arbitrary k ≥ 12 . If there is a solution for k dollars that includes at least one 4-dollar coin, replace it by a 5-dollar coin to make k + 1 dollars. Otherwise, if only 5-dollar coins are used, k must be a multiple of 5 and so at least 15; but then we can replace three 5-dollar coins by four 4-dollar coins to make k + 1 dollars. In each case, S ( k + 1)
3800-414: Is true for some natural number n , it also holds for some strictly smaller natural number m . Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, thereby showing ( by contradiction ) that Q ( n ) cannot be true for any n . The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on
3895-421: Is true. Therefore, by the principle of induction, S ( k ) holds for all k ≥ 12 , and the proof is complete. In this example, although S ( k ) also holds for k ∈ { 4 , 5 , 8 , 9 , 10 } {\textstyle k\in \{4,5,8,9,10\}} , the above proof cannot be modified to replace the minimum amount of 12 dollar to any lower value m . For m = 11 ,
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3990-1832: Is true. Using the angle addition formula and the triangle inequality , we deduce: | sin ( k + 1 ) x | = | sin k x cos x + sin x cos k x | (angle addition) ≤ | sin k x cos x | + | sin x cos k x | (triangle inequality) = | sin k x | | cos x | + | sin x | | cos k x | ≤ | sin k x | + | sin x | ( | cos t | ≤ 1 ) ≤ k | sin x | + | sin x | (induction hypothesis ) = ( k + 1 ) | sin x | . {\displaystyle {\begin{aligned}\left|\sin(k+1)x\right|&=\left|\sin kx\cos x+\sin x\cos kx\right|&&{\text{(angle addition)}}\\&\leq \left|\sin kx\cos x\right|+\left|\sin x\,\cos kx\right|&&{\text{(triangle inequality)}}\\&=\left|\sin kx\right|\left|\cos x\right|+\left|\sin x\right|\left|\cos kx\right|\\&\leq \left|\sin kx\right|+\left|\sin x\right|&&(\left|\cos t\right|\leq 1)\\&\leq k\left|\sin x\right|+\left|\sin x\right|&&{\text{(induction hypothesis}})\\&=(k+1)\left|\sin x\right|.\end{aligned}}} The inequality between
4085-419: The implication P ( k ) ⟹ P ( k + 1 ) {\displaystyle P(k)\implies P(k+1)} for any natural number k {\displaystyle k} . Assume the induction hypothesis: for a given value n = k ≥ 0 {\displaystyle n=k\geq 0} , the single case P ( k ) {\displaystyle P(k)}
4180-412: The proof of commutativity accompanying addition of natural numbers . More complicated arguments involving three or more counters are also possible. The method of infinite descent is a variation of mathematical induction which was used by Pierre de Fermat . It is used to show that some statement Q ( n ) is false for all natural numbers n . Its traditional form consists of showing that if Q ( n )
4275-414: The separation axiom which Felix Hausdorff originally formulated. The Zermelo–Fraenkel set theory , a result of the axiomatic method applied to set theory, allowed the "proper" formulation of set-theory problems and helped avoid the paradoxes of naïve set theory . One such problem was the continuum hypothesis . Zermelo–Fraenkel set theory, with the historically controversial axiom of choice included,
4370-466: The transformation group origins of those studies. Not every consistent body of propositions can be captured by a describable collection of axioms. In recursion theory, a collection of axioms is called recursive if a computer program can recognize whether a given proposition in the language is a theorem. Gödel's first incompleteness theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. Typically,
4465-420: The Peano axioms, but rather as axioms of the "underlying logic". 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, which makes this formulation close to second-order arithmetic . A weaker first-order system
4560-438: The Peano axioms, there is a unique homomorphism f : N A → N B satisfying 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.) 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
4655-503: The Peano axioms. Addition is a function that maps two natural numbers (two elements of N ) to another one. It is defined recursively as: For example: To prove commutativity of addition, first prove 0 + b = b {\displaystyle 0+b=b} and S ( a ) + b = S ( a + b ) {\displaystyle S(a)+b=S(a+b)} , each by induction on b {\displaystyle b} . Using both results, then prove
4750-439: The Peano axioms. Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Philosophers of mathematics sometimes assert that mathematicians choose axioms "arbitrarily", but it is possible that although they may appear arbitrary when viewed only from
4845-439: 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
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#17327725074224940-472: The adjunct must hold for all sets. 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 , US 1 ( C ) as follows: Then C is said to satisfy the Dedekind–Peano axioms if US 1 ( C ) has an initial object; this initial object is known as a natural number object in C . If ( N , 0, S )
5035-439: The axiomatic method. Euclid of Alexandria authored the earliest extant axiomatic presentation of Euclidean geometry and number theory . His idea begins with five undeniable geometric assumptions called axioms . Then, using these axioms, he established the truth of other propositions by proofs , hence the axiomatic method. Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry ,
5130-415: 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) are isomorphic . In particular, given two models ( N A , 0 A , S A ) and ( N B , 0 B , S B ) of
5225-406: 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
5320-530: 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
5415-510: 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
5510-456: The base case is actually false; for m = 10 , the second case in the induction step (replacing three 5- by four 4-dollar coins) will not work; let alone for even lower m . It is sometimes desirable to prove a statement involving two natural numbers, n and m , by iterating the induction process. That is, one proves a base case and an induction step for n , and in each of those proves a base case and an induction step for m . See, for example,
5605-412: The computer can recognize the axioms and logical rules for deriving theorems, and the computer can recognize whether a proof is valid, but to determine whether a proof exists for a statement is only soluble by "waiting" for the proof or disproof to be generated. The result is that one will not know which propositions are theorems and the axiomatic method breaks down. An example of such a body of propositions
5700-554: The exact nature of the property to be proven. All variants of induction are special cases of transfinite induction ; see below . If one wishes to prove a statement, not for all natural numbers, but only for all numbers n greater than or equal to a certain number b , then the proof by induction consists of the following: This can be used, for example, to show that 2 ≥ n + 5 for n ≥ 3 . In this way, one can prove that some statement P ( n ) holds for all n ≥ 1 , or even for all n ≥ −5 . This form of mathematical induction
5795-402: The extreme left hand and right hand sides, we deduce that: 0 + 1 + 2 + ⋯ + k + ( k + 1 ) = ( k + 1 ) ( ( k + 1 ) + 1 ) 2 . {\displaystyle 0+1+2+\cdots +k+(k+1)={\frac {(k+1)((k+1)+1)}{2}}.} That is, the statement P ( k + 1) also holds true, establishing
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#17327725074225890-454: The extreme left-hand and right-hand quantities shows that P ( k + 1 ) {\displaystyle P(k+1)} is true, which completes the induction step. Conclusion: The proposition P ( n ) {\displaystyle P(n)} holds for all natural numbers n . {\displaystyle n.} Q.E.D. In practice, proofs by induction are often structured differently, depending on
5985-469: The following countably infinitely many axioms added (these can be easily formalized as an axiom schema ): Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such a set. The system has at least two different models – one is the natural numbers (isomorphic to any other countably infinite set), and another
6080-1238: The following statement P ( n ) for all natural numbers n . P ( n ) : 0 + 1 + 2 + ⋯ + n = n ( n + 1 ) 2 . {\displaystyle P(n)\!:\ \ 0+1+2+\cdots +n={\frac {n(n+1)}{2}}.} This states a general formula for the sum of the natural numbers less than or equal to a given number; in fact an infinite sequence of statements: 0 = ( 0 ) ( 0 + 1 ) 2 {\displaystyle 0={\tfrac {(0)(0+1)}{2}}} , 0 + 1 = ( 1 ) ( 1 + 1 ) 2 {\displaystyle 0+1={\tfrac {(1)(1+1)}{2}}} , 0 + 1 + 2 = ( 2 ) ( 2 + 1 ) 2 {\displaystyle 0+1+2={\tfrac {(2)(2+1)}{2}}} , etc. Proposition. For every n ∈ N {\displaystyle n\in \mathbb {N} } , 0 + 1 + 2 + ⋯ + n = n ( n + 1 ) 2 . {\displaystyle 0+1+2+\cdots +n={\tfrac {n(n+1)}{2}}.} Proof. Let P ( n ) be
6175-405: The foundations of real analysis , Cantor 's set theory , Frege 's work on foundations, and Hilbert 's 'new' use of axiomatic method as a research tool. For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to
6270-592: The induction hypothesis that for a particular k , the single case n = k holds, meaning P ( k ) is true: 0 + 1 + ⋯ + k = k ( k + 1 ) 2 . {\displaystyle 0+1+\cdots +k={\frac {k(k+1)}{2}}.} It follows that: ( 0 + 1 + 2 + ⋯ + k ) + ( k + 1 ) = k ( k + 1 ) 2 + ( k + 1 ) . {\displaystyle (0+1+2+\cdots +k)+(k+1)={\frac {k(k+1)}{2}}+(k+1).} Algebraically ,
6365-492: The induction step, that the statement holds for a particular n , is called the induction hypothesis or inductive hypothesis . To prove the induction step, one assumes the induction hypothesis for n and then uses this assumption to prove that the statement holds for n + 1 . Authors who prefer to define natural numbers to begin at 0 use that value in the base case; those who define natural numbers to begin at 1 use that value. Mathematical induction can be used to prove
6460-651: The induction step. Conclusion: Since both the base case and the induction step have been proved as true, by mathematical induction the statement P ( n ) holds for every natural number n . Q.E.D. Induction is often used to prove inequalities . As an example, we prove that | sin n x | ≤ n | sin x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} for any real number x {\displaystyle x} and natural number n {\displaystyle n} . At first glance, it may appear that
6555-441: 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: 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: and so on. The set N together with 0 and the successor function s : N → N satisfies
6650-501: The next case n = k + 1 {\displaystyle n=k+1} . These two steps establish that the statement holds for every natural number n {\displaystyle n} . The base case does not necessarily begin with n = 0 {\displaystyle n=0} , but often with n = 1 {\displaystyle n=1} , and possibly with any fixed natural number n = N {\displaystyle n=N} , establishing
6745-414: The next one (the step ). A proof by induction consists of two cases. The first, the base case , proves the statement for n = 0 {\displaystyle n=0} without assuming any knowledge of other cases. The second case, the induction step , proves that if the statement holds for any given case n = k {\displaystyle n=k} , then it must also hold for
6840-427: The order relation can also be defined using first-order axioms. The axiom of induction above is second-order , since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider 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
6935-407: The point of view of the canons of deductive logic, that appearance is due to a limitation on the purposes that deductive logic serves. The mathematical system of natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system first devised by the mathematician Giuseppe Peano in 1889. He chose the axioms, in the language of a single unary function symbol S (short for " successor "), for
7030-627: The reduction of a body of propositions to a particular collection of axioms underlies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra . The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that rings need not be commutative , which differed from Emmy Noether 's original formulation. Mathematicians decided to consider topological spaces more generally without
7125-623: The right hand side simplifies as: k ( k + 1 ) 2 + ( k + 1 ) = k ( k + 1 ) + 2 ( k + 1 ) 2 = ( k + 1 ) ( k + 2 ) 2 = ( k + 1 ) ( ( k + 1 ) + 1 ) 2 . {\displaystyle {\begin{aligned}{\frac {k(k+1)}{2}}+(k+1)&={\frac {k(k+1)+2(k+1)}{2}}\\&={\frac {(k+1)(k+2)}{2}}\\&={\frac {(k+1)((k+1)+1)}{2}}.\end{aligned}}} Equating
7220-409: 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
7315-484: The section § Peano arithmetic as first-order theory below. If we use the second-order induction axiom, it is possible to define addition , multiplication , and total (linear) ordering on N directly using the axioms. However, with first-order induction, this is not possible 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
7410-524: The set of natural numbers to be: In mathematics , axiomatization is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e. axioms ) that relate a number of primitive terms — in order that a consistent body of propositions may be derived deductively from these statements. Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms. Axiom of induction Mathematical induction
7505-619: The statement | sin n x | ≤ n | sin x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} . We induce on n {\displaystyle n} . Base case: The calculation | sin 0 x | = 0 ≤ 0 = 0 | sin x | {\displaystyle \left|\sin 0x\right|=0\leq 0=0\left|\sin x\right|} verifies P ( 0 ) {\displaystyle P(0)} . Induction step: We show
7600-609: The statement 0 + 1 + 2 + ⋯ + n = n ( n + 1 ) 2 . {\displaystyle 0+1+2+\cdots +n={\tfrac {n(n+1)}{2}}.} We give a proof by induction on n . Base case: Show that the statement holds for the smallest natural number n = 0 . P (0) is clearly true: 0 = 0 ( 0 + 1 ) 2 . {\displaystyle 0={\tfrac {0(0+1)}{2}}\,.} Induction step: Show that for every k ≥ 0 , if P ( k ) holds, then P ( k + 1) also holds. Assume
7695-467: The statement P ( n ) defined as " Q ( m ) is false for all natural numbers m less than or equal to n ", it follows that P ( n ) holds for all n , which means that Q ( n ) is false for every natural number n . If one wishes to prove that a property P holds for all natural numbers less than or equal to n , proving P satisfies the following conditions suffices: The most common form of proof by mathematical induction requires proving in
7790-690: The statement specifically for natural values of n {\displaystyle n} , and induction is the readiest tool. Proposition. For any x ∈ R {\displaystyle x\in \mathbb {R} } and n ∈ N {\displaystyle n\in \mathbb {N} } , | sin n x | ≤ n | sin x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} . Proof. Fix an arbitrary real number x {\displaystyle x} , and let P ( n ) {\displaystyle P(n)} be
7885-418: The successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative uses an order relation symbol instead of the successor operation and the language of discretely ordered semirings (axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness): The theory defined by these axioms is known as PA . It
7980-403: The system, in a manner that is correct with the relations defined in the system. The existence of a concrete model proves the consistency of a system . A model is called concrete if the meanings assigned are objects and relations from the real world , as opposed to an abstract model which is based on other axiomatic systems. Models can also be used to show the independence of an axiom in
8075-420: The system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem. Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another
8170-423: The technique to prove that the sum of the first n odd integers is n . The earliest rigorous use of induction was by Gersonides (1288–1344). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665). Another Frenchman, Fermat , made ample use of a related principle: indirect proof by infinite descent . The induction hypothesis
8265-494: The totality of addition and multiplication, but which are still able to prove all true Π 1 {\displaystyle \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"). All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic . The arithmetical operations of addition and multiplication and
8360-447: The truth of the statement for all natural numbers n ≥ N {\displaystyle n\geq N} . The method can be extended to prove statements about more general well-founded structures, such as trees ; this generalization, known as structural induction , is used in mathematical logic and computer science . Mathematical induction in this extended sense is closely related to recursion . Mathematical induction
8455-419: The two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1 ) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri
8550-437: The usual axioms of equality), which contains six of the seven axioms of Robinson arithmetic , is sufficient for this purpose: In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a recursively enumerable and even decidable set of axioms . For each formula φ ( x , y 1 , ..., y k ) in the language of Peano arithmetic, the first-order induction axiom for φ
8645-423: The work of Boole and Schröder . The Peano axioms define the arithmetical properties of natural numbers , usually represented as a set N or N . {\displaystyle \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
8740-590: Was also employed by the Swiss Jakob Bernoulli , and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with George Boole , Augustus De Morgan , Charles Sanders Peirce , Giuseppe Peano , and Richard Dedekind . The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n . The proof consists of two steps: The hypothesis in
8835-485: 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
8930-462: Was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n . He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence
9025-516: Was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle . Whilst the original work was lost, it was later referenced by Al-Samawal al-Maghribi in his treatise al-Bahir fi'l-jabr (The Brilliant in Algebra) in around 1150 AD. Katz says in his history of mathematics Another important idea introduced by al-Karaji and continued by al-Samaw'al and others
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