The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. History Mathematical topics typically emerge and evolve through interactions among many researchers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation Some ontology
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Well-orderEvery non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers below for an example). If ≤ is a non-strict well-ordering, then < is a strict well-ordering. Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers). Ordinal numbers Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. Examples and counterexamples Natural numbers This is a well-ordered set of order type ω + ω. Integers Reals
Shape theory (mathematics)Shape theory is a branch of topology, generalizing the idea of homotopy theory to cases with unfavorable local properties. The overall goal of shape theory is to adapt the methods and results from homotopy theory to more general spaces, such as compact metric spaces or compact Hausdorff spaces. Shape theory was founded by the Polish mathematician Karol Borsuk in 1968. Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. It has homotopy groups isomorphic to those of a point, but is not homotopy equivalent to it; Whitehead's theorem does not apply because the Warsaw circle is not a CW complex. Borsuk's original shape theory has been replaced by a more systematic approach by inverse systems, pioneered by Sibe Mardešić, and independently, by Timothy Porter. For some purposes, like dynamical systems, more sophisticated invariants were developed under the name strong shape. Mardešić, Sibe (1997).
Whitehead's point-free geometryMotivation Point-free geometry was first formulated in Whitehead (1919, 1920), not as a theory of geometry or of spacetime, but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical. Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal first order theories described in this entry were devised by others in order to clarify and refine Whitehead's theories. The domain for both theories consists of "regions." Inclusion-based point-free geometry The axioms G1-G7 are, but for numbering, those of Def. 2.1 in Gerla and Miranda (2008). The fundamental primitive binary relation is Inclusion, denoted by infix "≤". The axioms are: Inclusion partially orders the domain. (reflexive) (transitive) WP4. (anti-symmetric) Given any two regions, there exists a region that includes both of them. Proper Part densely orders the domain. C is reflexive.
Zeroth-order logicOrdinal numberRepresentation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. He derived them by accident while working on a problem concerning trigonometric series. Two sets S and S' have the same cardinality if there is a bijection between them (i.e. there exists a function f that is both injective and surjective, that is it maps each element x of S to a unique element y = f(x) of S' and each element y of S' comes from exactly one such element x of S). . itself, there are uncountably many countably infinite ordinals, namely (next cardinal after ). Definitions
Ring theoryCommutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. History Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928). Commutative rings Algebraic geometry Noncommutative rings Some useful theorems General: Structure theorems: Structures and invariants of rings Dimension of a commutative ring . where .
MereotopologyHistory and motivation Preferred approach of Casati & Varzi Casati and Varzi (1999: chpt.4) set out a variety of mereotopological theories in a consistent notation. This section sets out several nested theories that culminate in their preferred theory GEMTC, and follows their exposition closely. The mereological part of GEMTC is the conventional theory GEM. We begin with some domain of discourse, whose elements are called individuals (a synonym for mereology is "the calculus of individuals"). We begin with a topological primitive, a binary relation called connection; the atomic formula Cxy denotes that "x is connected to y." (reflexive) (symmetric) Now posit the binary relation E, defined as: Exy is read as "y encloses x" and is also topological in nature. then E can be proved antisymmetric and thus becomes a partial order. Let parthood be the defining primitive binary relation of the underlying mereology, and let the atomic formula Pxy denote that "x is part of y". Notes
StructureThe structure of a DNA molecule is essential to its function. Load-bearing A traditional Sami food storage structure Gothic quadripartite cross-ribbed vaults of the Saint-Séverin church in Paris Buildings, aircraft, skeletons, anthills, beaver dams and salt domes are all examples of load-bearing structures. The structure elements are combined in structural systems. Load-bearing biological structures such as bones, teeth, shells, and tendons derive their strength from a multilevel hierarchy of structures employing biominerals and proteins, at the bottom of which are collagen fibrils. Biological Ribbon schematic of the 3D structure of the protein triosephosphate isomerase. Structural biology is concerned with the biomolecular structure of macromolecules, particularly proteins and nucleic acids. The function of these molecules is determined by their shape as well as their composition, and their structure has multiple levels. Chemical Mathematical Musical