Reverse mathematicsReverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory.
Nicolas BourbakiNicolas Bourbaki (nikɔla buʁbaki) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the Éléments de mathématique (Elements of Mathematics), the group's central work.
David HilbertDavid Hilbert (ˈhɪlbərt; ˈdaːvɪt ˈhɪlbɐt; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).
Ordered pairIn mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors.
L. E. J. BrouwerLuitzen Egbertus Jan Brouwer (ˈbraʊ.ər; ˈlœy̯tsə(n) ɛɣˈbɛrtəs jɑn ˈbrʌu̯ər; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as the founder of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension.
Tarski's undefinability theoremTarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
UrelementIn set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. There are several different but essentially equivalent ways to treat urelements in a first-order theory. One way is to work in a first-order theory with two sorts, sets and urelements, with a ∈ b only defined when b is a set. In this case, if U is an urelement, it makes no sense to say , although is perfectly legitimate.
Primitive recursive arithmeticPrimitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , as a formalization of his finitistic conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitistic. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic.
Set-theoretic definition of natural numbersIn set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed in axiomatic set theory, and a system based on equinumerosity that was proposed by Gottlob Frege and by Bertrand Russell. Zermelo ordinals In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 = n ∪ {n} for each n. In this way n = {0, 1, ..., n − 1} for each natural number n.
Abraham RobinsonAbraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics. Nearly half of Robinson's papers were in applied mathematics rather than in pure mathematics. He was born to a Jewish family with strong Zionist beliefs, in Waldenburg, Germany, which is now Wałbrzych, in Poland.
