Concept
Richard Dedekind
Related concepts (25)
Göttingen
Göttingen (ˈɡɜːtɪŋən, USalsoˈɡɛt-, ˈɡœtɪŋən; Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. The origins of Göttingen lay in a village called Gutingi, first mentioned in a document in 953 AD. The city was founded northwest of this village, between 1150 and 1200 AD, and adopted its name. In medieval times the city was a member of the Hanseatic League and hence a wealthy town.
Algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Leopold Kronecker
Leopold Kronecker (ˈkʁoːnɛkɐ; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man"). Kronecker was a student and lifelong friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia (now Legnica, Poland) in a wealthy Jewish family.
Successor function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin.
Ideal number
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field.
Dedekind-infinite set
In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.
Irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.
Logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.