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Infinite set not splittable into infinite sets
In set theory, an amorphous set is an infinite set that is not the disjoint union of two infinite subsets. Amorphous sets cannot exist if the axiom of
Amorphous_set
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
Topics referred to by the same term
An amorphous set in set theory Amorphous semiconductor (disambiguation) This disambiguation page lists articles associated with the title Amorphous. If
Amorphous_(disambiguation)
Set with an equinumerous proper subset
may be Dedekind-finite even if its underlying set is Dedekind-infinite, e.g. the integers. Amorphous set Moore, Gregory H. (2013) [unabridged republication
Dedekind-infinite_set
Finite collection of distinct objects
{\displaystyle S} into two sets, at least one of the two sets is I-finite. (A set with this property which is not I-finite is called an amorphous set.) II-finite. Every
Finite_set
Collection of mathematical objects
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects:
Set_(mathematics)
Use of braces for specifying sets
{Z} ,n=2k\}} — The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation
Set-builder_notation
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Set of the elements not in a given subset
In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the
Complement_(set_theory)
States of matter for water as a solid
are manufactured for nano scale uses due to their properties. In space, amorphous ice is the most common form as confirmed by observation. Thus, it is theorized
Phases_of_ice
Any collection of sets, or subsets of a set
"family of sets" because if one instead uses "set of sets" then the subsequent use of "set" can be confusing as to whether it is the containing set or one
Family_of_sets
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Finite ordered list of elements
n-tuple can be formally defined as the image of a function that has the set of the first n natural numbers as its domain (1, 2, ..., n). Tuples may be
Tuple
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations
Union_(set_theory)
British mathematician (born 1947)
1112/plms/s3-65.1.121. Truss, J. K. (June 1995). "The structure of amorphous sets". Annals of Pure and Applied Logic. 73 (2): 191–233. doi:10.1016/0168-0072(94)00024-W
John_Truss
Any one of the distinct objects that make up a set in set theory
mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four
Element_of_a_set
Informal set theories
Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined
Naive_set_theory
Mathematical set that can be enumerated
mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can
Universal_set
Mathematical set containing no elements
the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories
Empty_set
Sets whose elements have degrees of membership
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an
Fuzzy_set
Mathematical set of all subsets of a set
mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed
Power_set
Concept in axiomatic set theory
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom)
Axiom_schema_of_specification
Generalization of "n-th" to infinite cases
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite
Ordinal_number
Pair of logical equivalences
complement of the union of two sets is the same as the intersection of their complements The complement of the intersection of two sets is the same as the union
De_Morgan's_laws
Paradox in set theory
a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory
Russell's_paradox
Family of subsets representing "large" sets
In mathematics, a filter on a set is a collection of nonempty subsets which is closed under taking supersets and finite intersections. An example of filter
Filter_on_a_set
Soft, siliceous sedimentary rock
become filled with silica. Diatomite forms by the accumulation of the amorphous silica (opal, SiO2·nH2O) remains of dead diatoms (microscopic single-celled
Diatomaceous_earth
Elements in exactly one of two sets
symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection
Symmetric_difference
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
British electronic group
Gallagher's second solo album would be in collaboration with The Amorphous Androgynous, and was set for release in 2012. In August 2012, Gallagher mentioned in
The_Future_Sound_of_London
System of mathematical set theory
Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Mathematician (1845–1918)
January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established
Georg_Cantor
Solid material with highly ordered microscopic structure
third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and
Crystal
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
Class of mathematical set whose elements are all subsets
In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever
Transitive_set
Mathematical concept
elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle
Equivalence_class
Equalities for combinations of sets
read as: ( Left set ∖ Middle set ) ∖ Right set = ( Left set ∖ Right set ) ∖ ( Middle set ∖ Right set ) . {\displaystyle ({\text{Left set}}\,\setminus
List of set identities and relations
List_of_set_identities_and_relations
Alternative to the standard Zermelo–Fraenkel set theory
set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory
List of alternative set theories
List_of_alternative_set_theories
A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian
Nested_set_collection
English fashion designer and television presenter
Clients included Grace Jones, Julie Christie, and Cher. The company's lycra Amorphous dress is in the collection of the Victoria and Albert Museum. Young has
Esme_Young
Size of a set in mathematics
In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The
Cardinality
Porous form of silicon dioxide
Silica gel is an amorphous and porous form of silicon dioxide (silica), consisting of an irregular three-dimensional framework of alternating silicon
Silica_gel
System of mathematical set theory
of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of
Morse–Kelley_set_theory
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
American mathematician (1934–2007)
hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal. Cohen was born in Long Branch
Paul_Cohen
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Axiomatic set theories based on the principles of mathematical constructivism
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language
Constructive_set_theory
In mathematics, operation on sets
{\displaystyle A\sqcup B} of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So
Disjoint_union
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
Axiom of set theory
axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order
Axiom_of_regularity
Hydrated amorphous form of silica
amorphous form of silica (SiO2·nH2O); its water content may range from 3% to 21% by weight, but is usually between 6% and 10%. Due to the amorphous (chemical)
Opal
Concept in axiomatic set theory
power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a set P ( x
Axiom_of_power_set
Set with exactly one element
a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton
Singleton_(mathematics)
Maximal proper filter
In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words
Ultrafilter_on_a_set
German logician and mathematician (1871–1953)
mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929
Ernst_Zermelo
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be elements
Non-well-founded_set_theory
System of mathematical set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory
Zermelo_set_theory
System of mathematical set theory
The Kripke–Platek set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can
Kripke–Platek_set_theory
Axiomatic set theory devised by W.V.O. Quine
logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory
New_Foundations
Technique invented by Paul Cohen for proving consistency and independence results
In set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand
Forcing_(mathematics)
Resistance of a fluid to shear deformation
must be used. In the high and low temperature limits, viscous flow in amorphous materials (e.g. in glasses and melts) has the Arrhenius form: μ = A e
Viscosity
contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be
Paradoxes_of_set_theory
Property in general topology
of mathematics, a family A {\displaystyle {\mathcal {A}}} of subsets of a set X {\displaystyle X} is said to have the finite intersection property (FIP)
Finite_intersection_property
Axiom of Zermelo-Fraenkel set theory
axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory
Axiom_of_infinity
Infinite set that is not countable
mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related
Uncountable_set
Finite sets whose elements are all hereditarily finite sets
mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself
Hereditarily_finite_set
Thermoplastic polymer
mass of plastic can be produced. Unlike polyethylene, crystalline and amorphous regions differ only slightly in their density. However, the density of
Polypropylene
Transparent non-crystalline solid material
Glass is an amorphous (non-crystalline) solid. Because it is often transparent and chemically inert, glass has found widespread practical, technological
Glass
Amorphous poly alpha olefin (APAO; also known as atactic poly alpha olefin) is a commodity chemical used in multiple applications. In the mid-to-late-1950s
Amorphous_poly_alpha_olefin
British actress and film producer (born 2004)
movie stars and TV actors has become more porous, and it was into this amorphous environment that streaming was born – meaning that there's never really
Millie_Bobby_Brown
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every
Computable_set
Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere
List_of_types_of_sets
Process used in the textile industry
in the amorphous domains, later in the crystalline ones and at last in the polymers. As with wool, the tensions brought in by spinning are set free. During
Heatsetting
Polymer
and thermal history, polyethylene terephthalate may exist both as an amorphous (transparent) and as a semi-crystalline polymer. The semicrystalline material
Polyethylene_terephthalate
American actor (1967–2014)
Carolina referred to him as an "anti-star", whose real identity remained "amorphous and unmoored". Hoffman was acutely aware that he was often too unorthodox
Philip_Seymour_Hoffman
Substance composed of macromolecules with repeating structural units
including toughness, high elasticity, viscoelasticity, and a tendency to form amorphous and semicrystalline structures rather than crystals. Polymers are studied
Polymer
German mathematician (1831–1916)
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
Richard_Dedekind
American musician and songwriter (1958–2016)
showmanship. He came to be regarded as a sex symbol for his androgynous, amorphous persona, play with signifiers of gender, and defiance of racial stereotypes
Prince_(musician)
Brittle form of sugar that looks like glass
methamphetamine in the AMC TV series Breaking Bad. Actor Aaron Paul would eat it on set. Provost, Joseph J.; Colabroy, Keri L.; Kely, Brenda S.; Bodwin, Jeffrey;
Sugar_glass
Mathematical construction of a set with an equivalence relation
setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied
Setoid
System of mathematical set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
Mathematical logician and philosopher
Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier
Kurt_Gödel
Semiconducting material
efficiency, performance, and reliability. In its amorphous form, a-IGZO is a representative transparent amorphous oxide semiconductor (TAOS), a class of wide-band-gap
Indium_gallium_zinc_oxide
Proof in set theory
infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some
Cantor's_diagonal_argument
German-Israeli mathematician and Zionist (1891–1965)
contributions to axiomatic set theory, especially his additions to Ernst Zermelo's axioms, which resulted in the Zermelo–Fraenkel set theory. Abraham Adolf
Abraham_Fraenkel
Reversible transition in amorphous materials
transition, is the gradual and reversible transition in amorphous materials (or in amorphous regions within semicrystalline materials) from a hard and
Glass_transition
Term in set theory
set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set
Almost
Pokémon species
It also plays the role of the main character in Pokémon Pokopia. An amorphous species classified as a Normal-type Pokémon, Ditto appears as a short
Ditto_(Pokémon)
Concept in set theory
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any
Axiom_schema_of_replacement
Chemical compound
reveal that CrPO4 catalyzes the adsorption of divalent cations onto its amorphous surface through the cation exchange mechanism. The mechanism suggests
Chromium(III)_phosphate
Marvel Comics character
by Marvel Comics. The character is a sentient alien symbiote with an amorphous, liquid-like form, who survives by bonding with a host, usually human
Venom_(character)
Possible axiom of set theory
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation
Axiom_of_limitation_of_size
3-volume treatise on mathematics, 1910–1913
set is the starting set, and other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived
Principia_Mathematica
Material of moderate electrical conductivity
materials are crystalline solids, but amorphous and liquid semiconductors are also known. These include hydrogenated amorphous silicon and mixtures of arsenic
Semiconductor
12th episode of the 5th season of The Simpsons
family collides when running and lands into the couch as one big mass of amorphous glop. Commentary Matt Groening James L. Brooks David Mirkin Conan O'Brien
Bart_Gets_Famous
Concept in axiomatic set theory
{\displaystyle X} is a set of sets, then the union of all sets in X {\displaystyle X} is still a set. In more basic terms, for each set X {\displaystyle
Axiom_of_union
System in mathematical set theory
In mathematics, vague sets are an extension of fuzzy sets. In a fuzzy set, each object is assigned a single value in the interval [0,1] reflecting its
Vague_set
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET
AMORPHOUS SET