In mathematical logic , the Peano axioms , also known as the Dedekind—Peano axioms or the Peano postulates , are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann , who showed in the s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers.
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He attended the village school in Spinetta then he moved up to the school in Cuneo, making the 5km journey there and back on foot every day.
His parents bought a house in Cuneo but his father continued to work the fields at Tetto Galant with the help of a brother and sister of Giuseppe, while his mother stayed in Cuneo with Giuseppe and his older brother. Giuseppe's mother had a brother who was a priest and lawyer in Turin and, when he realised that Giuseppe was a very talented child, he took him to Turin in for his secondary schooling and to prepare him for university studies. Giuseppe took exams at Ginnasio Cavour in and then was a pupil at Liceo Cavour from where he graduated in and, in that year, he entered the University of Turin.
Among Peano's teachers in his first year at the University of Turin was D'Ovidio who taught him analytic geometry and algebra. In his second year he was taught calculus by Angelo Genocchi and descriptive geometry by Giuseppe Bruno. Peano continued to study pure mathematics in his third year and found that he was the only student to do so.
The others had continued their studies at the Engineering School which Peano himself had originally intended to do. Among his teachers in his final year were again D'Ovidio with a further geometry course and Francesco Siacci with a mechanics course. On 29 September Peano graduated as doctor of mathematics. Peano joined the staff at the University of Turin in , being appointed as assistant to D'Ovidio. He published his first mathematical paper in and a further three papers the following year.
Peano was appointed assistant to Genocchi for and it was in that Peano made a discovery that would be typical of his style for many years, he discovered an error in a standard definition. Genocchi was by this time quite old and in relatively poor health and Peano took over some of his teaching. Peano was about to teach the students about the area of a curved surface when he realised that the definition in Serret 's book, which was the standard text for the course, was incorrect.
Peano immediately told Genocchi of his discovery to be told that Genocchi already knew. Genocchi had been informed the year by Schwarz who seems to have been the first to find Serret 's error.
In there was published a text based on Genocchi 's lectures at Turin. This book Course in Infinitesimal Calculus although based on Genocchi 's lectures was edited by Peano and indeed it has much in it written by Peano himself. The book itself states on the title page that it is Genocchi seemed somewhat unhappy that the work came out under his name for he wrote So that nothing will be attributed to me which is not mine, I must declare that I have had no part in the compilation of the aforementioned book and that everything is due to that outstanding young man Dr Giuseppe Peano Peano received his qualification to be a university professor in December and he continued to teach further courses, some for Genocchi whose health had not recovered sufficiently to allow him to return to the University.
The existence of solutions with stronger hypothesis on f had been given earlier by Cauchy and then Lipschitz. In addition to his teaching at the University of Turin, Peano began lecturing at the Military Academy in Turin in The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations. However Emile Picard had independently discovered this method and had credited Schwarz with discovering the method first.
In Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic. A more significant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself. This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano.
In Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets. These were published in a pamphlet Arithmetices principia, nova methodo exposita which, according to  were The pamphlet was written in Latin and nobody has been able to give a good reason for this, other than  Genocchi died in and Peano expected to be appointed to fill his chair.
He wrote to Casorati , who he believed to be part of the appointing committee, for information only to discover that there was a delay due to the difficulty of finding enough members to act on the committee.
Casorati had been approached but his health was not up to the task. Before the appointment could be made Peano published another stunning result. He invented 'space-filling' curves in , these are continuous surjective mappings from [0,1] onto the unit square. Hilbert , in , described similar space-filling curves. It had been thought that such curves could not exist. Cantor had shown that there is a bijection between the interval [0,1] and the unit square but, shortly after, Netto had proved that such a bijection cannot be continuous.
Peano's continuous space-filling curves cannot be of course, otherwise Netto 's theorem would be contradicted. In December Peano's wait to be appointed to Genocchi 's chair was over when, after the usual competition, Peano was offered the post. In Peano founded Rivista di matematica , a journal devoted mainly to logic and the foundations of mathematics.
The first paper in the first part is a ten page Peano summarising his work on mathematical logic up to that time. Peano had a great skill in seeing that theorems were incorrect by spotting exceptions. Others were not so happy to have these errors pointed out and one such was his colleague Corrado Segre. When Corrado Segre submitted an article to Rivista di matematica Peano pointed out that some of the theorems in the article had exceptions. Segre was not prepared to just correct the theorems by adding conditions that ruled out the exceptions but defended his work saying that the moment of discovery was more important than a rigorous formulation.
Of course this was so against Peano's rigorous approach to mathematics that he argued strongly:- I believe it new in the history of mathematics that authors knowingly use in their research propositions for which exceptions are known, or for which they have no proof It was not only Corrado Segre who suffered from Peano's outstanding ability to spot lack of rigour. Of course it was the precision of his thinking, using the exactness of his mathematical logic, that gave Peano this clarity of thought.
Peano pointed out an error in a proof by Hermann Laurent in and, in the same year, reviewed a book by Veronese ending the review with the comment:- We could continue at length enumerating the absurdities that the author has piled up. But these errors, the lack of precision and rigour throughout the book take all value away from it.
From around , Peano embarked on a new and extremely ambitious project, namely the Formulario Mathematico. He explained in the March part of Rivista di matematica his thinking:- Of the greatest usefulness would be the publication of collections of all the theorems now known that refer to given branches of the mathematical sciences Such a collection, which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic In many ways this grand idea marks the end of Peano's extraordinary creative work.
It was a project that was greeted with enthusiasm by a few and with little interest by most. Peano began trying to convert all those around him to believe in the importance of this project and this had the effect of annoying them.
However Peano and his close associates, including his assistants, Vailati , Burali-Forti , Pieri and Fano soon became deeply involved with the work. When describing a new edition of the Formulario Mathematico in Peano writes:- Each professor will be able to adopt this Formulario as a textbook, for it ought to contain all theorems and all methods.
His teaching will be reduced to showing how to read the formulas, and to indicating to the students the theorems that he wishes to explain in his course. When the calculus volume of the Formulario was published Peano, as he had indicated, began to use it for his teaching. This was the disaster that one would expect. Peano, who was a good teacher when he began his lecturing career, became unacceptable to both his students and his colleagues by the style of his teaching.
One of his students, who was actually a great admirer of Peano, wrote:- But we students knew that this instruction was above our heads. We understood that such a subtle analysis of concepts, such a minute criticism of the definitions used by other authors, was not adapted for beginners, and especially was not useful for engineering students.
We disliked having to give time and effort to the "symbols" that in later years we might never use. The Military Academy ended his contract to teach there in and although many of his colleagues at the university would have also liked to stop his teaching there, nothing was possible under the way that the university was set up. The professor was a law unto himself in his own subject and Peano was not prepared to listen to his colleagues when they tried to encourage him to return to his old style of teaching.
The Formulario Mathematico project was completed in and one has to admire what Peano achieved but although the work contained a mine of information it was little used. However, perhaps Peano's greatest triumph came in In that year there were two congresses held in Paris. It was a triumph for Peano and Russell , who attended the Congress, wrote in his autobiography:- The Congress was the turning point of my intellectual life, because there I met Peano.
I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation.
In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument on which he embarked. As the days went by, I decided that this must be owing to his mathematical logic.
It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years Peano remained in Paris for this Congress and listened to Hilbert 's talk setting out ten of the 23 problems which appeared in his paper aimed at giving the agenda for the century.
Peano was particularly interested in the second problem which asked if the axioms of arithmetic could be proved consistent. Even before the Formulario Mathematico project was completed Peano was putting in place the major project of his life.
In Peano expressed interest in finding a universal, or international, language and proposed an artificial language "Latino sine flexione" based on Latin but stripped of all grammar.
He compiled the vocabulary by taking words from English, French, German and Latin. In fact the final edition of the Formulario Mathematico was written in Latino sine flexione which is another reason the work was so little used. Peano's career was therefore rather strangely divided into two periods. The period up to is one where he showed great originality and a remarkable feel for topics which would be important in the development of mathematics.
His achievements were outstanding and he had a modern style quite out of place in his own time. However this feel for what was important seemed to leave him and after he worked with great enthusiasm on two projects of great difficulty which were enormous undertakings but proved quite unimportant in the development of mathematics.
Of his personality Kennedy writes in  I am fascinated by his gentle personality, his ability to attract lifelong disciples, his tolerance of human weakness, his perennial optimism. Peano may not only be classified as a 19 th century mathematician and logician, but because of his originality and influence, must be judged one of the great scientists of that century.
He attended the village school in Spinetta then he moved up to the school in Cuneo, making the 5km journey there and back on foot every day. His parents bought a house in Cuneo but his father continued to work the fields at Tetto Galant with the help of a brother and sister of Giuseppe, while his mother stayed in Cuneo with Giuseppe and his older brother. Giuseppe's mother had a brother who was a priest and lawyer in Turin and, when he realised that Giuseppe was a very talented child, he took him to Turin in for his secondary schooling and to prepare him for university studies. Giuseppe took exams at Ginnasio Cavour in and then was a pupil at Liceo Cavour from where he graduated in and, in that year, he entered the University of Turin. Among Peano's teachers in his first year at the University of Turin was D'Ovidio who taught him analytic geometry and algebra. In his second year he was taught calculus by Angelo Genocchi and descriptive geometry by Giuseppe Bruno. Peano continued to study pure mathematics in his third year and found that he was the only student to do so.