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74-419: A simple example is N {\displaystyle \mathbb {N} } , the set of natural numbers . From Galileo's paradox , there exists a bijection that maps every natural number n to its square n . Since the set of squares is a proper subset of N {\displaystyle \mathbb {N} } , N {\displaystyle \mathbb {N} } is Dedekind-infinite. Until

148-675: A and b with b ≠ 0 there are natural numbers q and r such that The number q is called the quotient and r is called the remainder of the division of a by  b . The numbers q and r are uniquely determined by a and  b . This Euclidean division is key to the several other properties ( divisibility ), algorithms (such as the Euclidean algorithm ), and ideas in number theory. The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: Two important generalizations of natural numbers arise from

222-425: A + c = b . This order is compatible with the arithmetical operations in the following sense: if a , b and c are natural numbers and a ≤ b , then a + c ≤ b + c and ac ≤ bc . An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number ; for

296-466: A + 1 = S ( a ) and a × 1 = a . Furthermore, ( N ∗ , + ) {\displaystyle (\mathbb {N^{*}} ,+)} has no identity element. In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where

370-497: A one-to-one correspondence between them. He invoked similarity to give the first precise definition of an infinite set : a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets . Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N , ( N → N ): Dedekind's work in this area anticipated that of Georg Cantor , who

444-588: A tally mark for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set. The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak , dating back from around 1500 BCE and now at

518-401: A × ( b + c ) = ( a × b ) + ( a × c ) . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring . Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N {\displaystyle \mathbb {N} }

592-404: A × 0 = 0 and a × S( b ) = ( a × b ) + a . This turns ( N ∗ , × ) {\displaystyle (\mathbb {N} ^{*},\times )} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers . Addition and multiplication are compatible, which is expressed in the distribution law :

666-421: A bold N or blackboard bold ⁠ N {\displaystyle \mathbb {N} } ⁠ . Many other number sets are built from the natural numbers. For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and the real numbers add infinite decimals. Complex numbers add the square root of −1 . This chain of extensions canonically embeds

740-766: A complete English phrase is in 1763. The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article. Starting at 0 or 1 has long been a matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889, Giuseppe Peano used N for the positive integers and started at 1, but he later changed to using N 0 and N 1 . Historically, most definitions have excluded 0, but many mathematicians such as George A. Wentworth , Bertrand Russell , Nicolas Bourbaki , Paul Halmos , Stephen Cole Kleene , and John Horton Conway have preferred to include 0. Mathematicians have noted tendencies in which definition

814-460: A natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox . To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements. In 1881, Charles Sanders Peirce provided

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888-526: A need to improve upon the logical rigor in the foundations of mathematics . In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications. Set-theoretical definitions of natural numbers were initiated by Frege . He initially defined

962-470: A numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals do not have a symbol for 0; instead, nulla (or the genitive form nullae ) from nullus ,

1036-509: A set (because of Russell's paradox ). The standard solution is to define a particular set with n elements that will be called the natural number n . The following definition was first published by John von Neumann , although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set as a definition of ordinal number , the sets considered below are sometimes called von Neumann ordinals . The definition proceeds as follows: It follows that

1110-447: A set A is Dedekind-finite if in the category of sets , every monomorphism f  : A → A is an isomorphism . A von Neumann regular ring R has the analogous property in the category of (left or right) R - modules if and only if in R , xy = 1 implies yx = 1 . More generally, a Dedekind-finite ring is any ring that satisfies the latter condition. Beware that a ring may be Dedekind-finite even if its underlying set

1184-409: A set of the form {0, 1, 2, ..., n −1} for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection. During the latter half of the 19th century, most mathematicians simply assumed that a set is infinite if and only if it is Dedekind-infinite. However, this equivalence cannot be proved with the axioms of Zermelo–Fraenkel set theory without

1258-574: A subscript (or superscript) "0" is added in the latter case: This section uses the convention N = N 0 = N ∗ ∪ { 0 } {\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}} . Given the set N {\displaystyle \mathbb {N} } of natural numbers and the successor function S : N → N {\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to

1332-608: A subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether . Ideals generalize Ernst Eduard Kummer 's ideal numbers , devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem . (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind and Heinrich Martin Weber applied ideals to Riemann surfaces , giving an algebraic proof of

1406-530: A unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2). However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other. Independent studies on numbers also occurred at around the same time in India , China, and Mesoamerica . Nicolas Chuquet used the term progression naturelle (natural progression) in 1484. The earliest known use of "natural number" as

1480-452: A well-ordered set is Dedekind-infinite if and only if it is infinite. The term is named after the German mathematician Richard Dedekind , who first explicitly introduced the definition. It is notable that this definition was the first definition of "infinite" that did not rely on the definition of the natural numbers (unless one follows Poincaré and regards the notion of number as prior to even

1554-509: Is consistent (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong. The five Peano axioms are the following: These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic,

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1628-444: Is weakly Dedekind-infinite if it satisfies any, and then all, of the following equivalent (over ZF ) conditions: and it is infinite if: Then, ZF proves the following implications: Dedekind-infinite ⇒ dually Dedekind-infinite ⇒ weakly Dedekind-infinite ⇒ infinite. There exist models of ZF having an infinite Dedekind-finite set. Let A be such a set, and let B be the set of finite injective sequences from A . Since A

1702-409: Is Dedekind-finite is finite. There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice. A vaguely related notion is that of a Dedekind-finite ring . This definition of " infinite set " should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal , namely

1776-535: Is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infinite set is Dedekind-infinite. However, the equivalence of the two definitions is much weaker than the full strength of AC. In particular, there exists a model of ZF in which there exists an infinite set with no countably infinite subset. Hence, in this model, there exists an infinite, Dedekind-finite set. By

1850-446: Is Dedekind-infinite, e.g. the integers . Natural number In mathematics , the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers 0, 1, 2, 3, ... , while others start with 1, defining them as the positive integers 1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Sometimes,

1924-505: Is a free monoid on one generator. This commutative monoid satisfies the cancellation property , so it can be embedded in a group . The smallest group containing the natural numbers is the integers . If 1 is defined as S (0) , then b + 1 = b + S (0) = S ( b + 0) = S ( b ) . That is, b + 1 is simply the successor of b . Analogously, given that addition has been defined, a multiplication operator × {\displaystyle \times } can be defined via

1998-419: Is a subset of m . In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order . Richard Dedekind Julius Wilhelm Richard Dedekind ( German: [ˈdeːdəˌkɪnt] ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory , abstract algebra (particularly ring theory ), and

2072-552: Is based on set theory . It defines the natural numbers as specific sets . More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a one to one correspondence between the two sets n and S . The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem that can be stated and proved in Peano arithmetic can also be proved in set theory. However,

2146-575: Is based on an axiomatization of the properties of ordinal numbers : each natural number has a successor and every non-zero natural number has a unique predecessor. 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. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem . The set of all natural numbers

2220-610: Is commonly considered the founder of set theory . Likewise, his contributions to the foundations of mathematics anticipated later works by major proponents of logicism , such as Gottlob Frege and Bertrand Russell . Dedekind edited the collected works of Lejeune Dirichlet , Gauss , and Riemann . Dedekind's study of Lejeune Dirichlet's work led him to his later study of algebraic number fields and ideals . In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that: Although

2294-466: Is infinite can be easily proven in ZF: every finite set has by definition a bijection with some finite ordinal n , and one can prove by induction on n that this is not Dedekind-infinite. By using the axiom of countable choice (denotation: axiom CC) one can prove the converse, namely that every infinite set X is Dedekind-infinite, as follows: First, define a function over the natural numbers (that is, over

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2368-503: Is infinite, the function "drop the last element" from B to itself is surjective but not injective, so B is dually Dedekind-infinite. However, since A is Dedekind-finite, then so is B (if B had a countably infinite subset, then using the fact that the elements of B are injective sequences, one could exhibit a countably infinite subset of A ). When sets have additional structures, both kinds of infiniteness can sometimes be proved equivalent over ZF . For instance, ZF proves that

2442-410: Is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that N {\displaystyle \mathbb {N} } is not a ring ; instead it is a semiring (also known as a rig ). If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with

2516-429: Is standardly denoted N or N . {\displaystyle \mathbb {N} .} Older texts have occasionally employed J as the symbol for this set. Since natural numbers may contain 0 or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as: Alternatively, since

2590-422: Is used, such as algebra texts including 0, number theory and analysis texts excluding 0, logic and set theory texts including 0, dictionaries excluding 0, school books (through high-school level) excluding 0, and upper-division college-level books including 0. There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include division by zero and

2664-499: The Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context. A much later advance was the development of

2738-539: The Riemann–Roch theorem . In 1888, he published a short monograph titled Was sind und was sollen die Zahlen? ("What are numbers and what are they good for?" Ewald 1996: 790), which included his definition of an infinite set . He also proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function . The next year, Giuseppe Peano , citing Dedekind, formulated an equivalent but simpler set of axioms , now

2812-683: The University of Berlin , not Göttingen , was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries; they were both awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent , giving courses on probability and geometry . He studied for a while with Peter Gustav Lejeune Dirichlet , and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions . Yet he

2886-566: The University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern . Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals "). This thesis did not display the talent evident in Dedekind's subsequent publications. At that time,

2960-422: The axiom of choice (AC) (usually denoted " ZF "). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is strictly weaker than the axiom of countable choice (CC). (See the references below.) A set A is Dedekind-infinite if it satisfies any, and then all, of the following equivalent (over ZF ) conditions: it is dually Dedekind-infinite if: it

3034-518: The axiomatic foundations of arithmetic . His best known contribution is the definition of real numbers through the notion of Dedekind cut . He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as logicism . Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig . His mother

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3108-444: The foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century, Zermelo–Fraenkel set theory , today the most commonly used form of axiomatic set theory , was proposed as an axiomatic system to formulate a theory of sets free of paradoxes such as Russell's paradox . Using

3182-661: The square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers"); in modern terminology, Vollständigkeit , completeness . Dedekind defined two sets to be "similar" when there exists

3256-400: The whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the integers , including negative integers. The counting numbers are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are six coins on

3330-534: The Latin word for "none", was employed to denote a 0 value. The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes . Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all. Euclid , for example, defined a unit first and then a number as a multitude of units, thus by his definition,

3404-499: The above, such a set cannot be well-ordered in this model. If we assume the axiom of countable choice (i. e., AC ω ), then it follows that every infinite set is Dedekind-infinite. However, the equivalence of these two definitions is in fact strictly weaker than even the CC. Explicitly, there exists a model of ZF in which every infinite set is Dedekind-infinite, yet the CC fails (assuming consistency of ZF ). That every Dedekind-infinite set

3478-408: The axiom of countable choice we may choose one member from each of these sets, and this member is itself a finite subset of X . More precisely, according to the axiom of countable choice, a (countable) set exists, G = { g ( n ) | n ∈ N }, so that for every natural number n , g ( n ) is a member of f ( n ) and is therefore a finite subset of X of size n . Now, we define U as the union of

3552-445: The axioms of Zermelo–Fraenkel set theory with the originally highly controversial axiom of choice included ( ZFC ) one can show that a set is Dedekind-finite if and only if it is finite in the usual sense. However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice ( ZF ) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that

3626-538: The book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death. The 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory . (The word "Ring", introduced later by Hilbert , does not appear in Dedekind's work.) Dedekind defined an ideal as

3700-455: The finite ordinals) f  : N → Power(Power( X )) , so that for every natural number n , f ( n ) is the set of finite subsets of X of size n (i.e. that have a bijection with the finite ordinal n ). f ( n ) is never empty, or otherwise X would be finite (as can be proven by induction on n ). The image of f is the countable set { f ( n ) | n ∈ N }, whose members are themselves infinite (and possibly uncountable) sets. By using

3774-409: The first axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method ( Latin : Arithmetices principia, nova methodo exposita ). This approach is now called Peano arithmetic . It

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3848-461: The first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut (German: Schnitt ), now a standard definition of the real numbers. The idea of a cut is that an irrational number divides the rational numbers into two classes ( sets ), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example,

3922-428: The general acceptance of the axiom of choice among the mathematical community, these issues relating to infinite and Dedekind-infinite sets have become less central to most mathematicians. However, the study of Dedekind-infinite sets played an important role in the attempt to clarify the boundary between the finite and the infinite, and also an important role in the history of the AC. Since every infinite well-ordered set

3996-511: The idea that  0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BCE , but this usage did not spread beyond Mesoamerica . The use of

4070-425: The members of G . U is an infinite countable subset of X , and a bijection from the natural numbers to U , h  : N → U , can be easily defined. We may now define a bijection B  : X → X \ h (0) that takes every member not in U to itself, and takes h ( n ) for every natural number to h ( n + 1) . Hence, X is Dedekind-infinite, and we are done. Expressed in category-theoretical terms,

4144-446: The natural numbers are defined iteratively as follows: It can be checked that the natural numbers satisfy the Peano axioms . With this definition, given a natural number n , the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S ." This formalizes the operation of counting the elements of S . Also, n ≤ m if and only if n

4218-458: The natural numbers in the other number systems. Natural numbers are studied in different areas of math. Number theory looks at things like how numbers divide evenly ( divisibility ), or how prime numbers are spread out. Combinatorics studies counting and arranging numbered objects, such as partitions and enumerations . The most primitive method of representing a natural number is to use one's fingers, as in finger counting . Putting down

4292-403: The natural numbers naturally form a subset of the integers (often denoted Z {\displaystyle \mathbb {Z} } ), they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a superscript " ∗ {\displaystyle *} " or "+" is added in the former case, and

4366-435: The natural numbers, this is denoted as ω (omega). In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder or Euclidean division is available as a substitute: for any two natural numbers

4440-439: The next one, one can define addition of natural numbers recursively by setting a + 0 = a and a + S ( b ) = S ( a + b ) for all a , b . Thus, a + 1 = a + S(0) = S( a +0) = S( a ) , a + 2 = a + S(1) = S( a +1) = S(S( a )) , and so on. The algebraic structure ( N , + ) {\displaystyle (\mathbb {N} ,+)} is a commutative monoid with identity element  0. It

4514-449: The notion of set). Although such a definition was known to Bernard Bolzano , he was prevented from publishing his work in any but the most obscure journals by the terms of his political exile from the University of Prague in 1819. Moreover, Bolzano's definition was more accurately a relation that held between two infinite sets, rather than a definition of an infinite set per se . For a long time, many mathematicians did not even entertain

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4588-595: The ordinary natural numbers via the ultrapower construction . Other generalizations are discussed in Number § Extensions of the concept . Georges Reeb used to claim provocatively that "The naïve integers don't fill up N {\displaystyle \mathbb {N} } ". There are two standard methods for formally defining natural numbers. The first one, named for Giuseppe Peano , consists of an autonomous axiomatic theory called Peano arithmetic , based on few axioms called Peano axioms . The second definition

4662-578: The rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his sister Julia. Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the French Academy of Sciences (1900). He received honorary doctorates from the universities of Oslo , Zurich , and Braunschweig . While teaching calculus for

4736-471: The same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence . A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from

4810-399: The size of the empty set . Computer languages often start from zero when enumerating items like loop counters and string- or array-elements . Including 0 began to rise in popularity in the 1960s. The ISO 31-11 standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as ISO 80000-2 . In 19th century Europe, there

4884-547: The standard ones. Dedekind made other contributions to algebra . For instance, around 1900, he wrote the first papers on modular lattices . In 1872, while on holiday in Interlaken , Dedekind met Georg Cantor . Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes with Leopold Kronecker , who

4958-433: The successor of x {\displaystyle x} is x + 1 {\displaystyle x+1} . Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence ". This does not work in all set theories , as such an equivalence class would not be

5032-422: The table", in which case they are called cardinal numbers . They are also used to put things in order, like "this is the third largest city in the country", which are called ordinal numbers . Natural numbers are also used as labels, like jersey numbers on a sports team, where they serve as nominal numbers and do not have mathematical properties. The natural numbers form a set , commonly symbolized as

5106-419: The thought that there might be a distinction between the notions of infinite set and Dedekind-infinite set. In fact, the distinction was not really realised until after Ernst Zermelo formulated the AC explicitly. The existence of infinite, Dedekind-finite sets was studied by Bertrand Russell and Alfred North Whitehead in 1912; these sets were at first called mediate cardinals or Dedekind cardinals . With

5180-402: The two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is Fermat's Last Theorem . The definition of the integers as sets satisfying Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory

5254-423: The two uses of counting and ordering: cardinal numbers and ordinal numbers . The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0 ) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω . For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by

5328-701: Was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests at Braunschweig Main Cemetery . He first attended the Collegium Carolinum in 1848 before transferring to

5402-512: Was also the first at Göttingen to lecture concerning Galois theory . About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic . In 1858, he began teaching at the Polytechnic school in Zürich (now ETH Zürich). When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent

5476-430: Was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act. Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man". The constructivists saw

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