one

Bahamian Creole: wan

Belizean Creole: wan

Bislama: wan

Gullah: one

Jamaican Creole: wan

Nigerian Pidgin: wọ́n, wọn

Sranan Tongo: wan

Tok Pisin: wan

→ Atong (India): wan

→ Fanagalo: wan

PIE word
 *h₁óynos
From Middle English oon, on, oan, an, from Old English ān (“one”), from Proto-West Germanic *ain, from Proto-Germanic *ainaz (“one”), from Proto-Indo-European *h₁óynos (“single, one”).
Cognate with Scots ae, ane, wan, yin (“one”); North Frisian ån (“one”); Saterland Frisian aan (“one”); West Frisian ien (“one”); Dutch een, één (“one”); German Low German een; German ein, eins (“one”); Danish en (“one”); Swedish en (“one”); Norwegian Nynorsk ein (“one”), Icelandic einn (“one”); Latin ūnus (“one”) (Old Latin oinos); Russian оди́н (odín), Spanish uno. Doublet of a, an, and Uno.
The use as an indefinite personal pronoun may have been influenced by unrelated French on, although the Germanic languages widely use cognates for the same sense (usually in non-subject function, but also in subject function, e.g. Luxembourgish een).
Verb form from Middle English onen.

inh

one's damned if one does and one's damned if one doesn't

(set theory) The cardinality of the smallest nonempty set.


transitive

From Latin trānsitīvus, from trānsitus, from trāns (“across”) + itus, from eō (“to go”).

"Is an ancestor of" is a transitive relation: if Alice is an ancestor of Bob, and Bob is an ancestor of Carol, then Alice is an ancestor of Carol.

(set theory, of a relation on a set) Having the property that if an element a is related to b and b is related to c, then a is necessarily related to c.


singular

en-adj

singular (comparative more singular, superlative most singular)

more singular (comparative)
most singular (superlative)
sing. (alternative abbreviation)
sg. (alternative abbreviation)
sg (alternative abbreviation)

From Middle English singuler, from Old French, from Latin singulāris (“alone of its kind”), from Latin singulus (“single”).

(set theory, of a cardinal number) Not equal to its own cofinality.


mouse

en-noun

mouse (plural mice or (computing) mouses)

mice (plural)
mouses (plural)
mowse (alternative obsolete)

→ Bengali: মাউস (maus)

→ Brazilian Portuguese: mouse

→ French: souris (semantic loan)

→ Georgian: მაუსი (mausi)

→ Italian: mouse

→ Japanese: マウス (mausu)

→ Korean: 마우스 (mauseu)

→ Romanian: mouse

→ Spanish: mouse

desc

→ Thai: เมาส์ (máo), เม้าส์ (máo)

→ Thai: เมาส์ (máo), เม้าส์ (máo)

→ Vietnamese: chuột (semantic loan)

From Middle English mous, from Old English mūs, from Proto-West Germanic *mūs, from Proto-Germanic *mūs, from Proto-Indo-European *múHs.
Cognates
Germanic cognates include Old Frisian mūs, Old Saxon mūs (German Low German Muus), Dutch muis, Old High German mūs (German Maus), Old Norse mús (Swedish mus, Danish mus, Norwegian mus, Icelandic mús, Faroese mús).
Indo-European cognates include Ancient Greek μῦς (mûs), Latin mūs, Spanish mur, Armenian մուկ (muk), Old Church Slavonic мꙑшь (myšĭ) (Russian мышь (myšʹ)), Albanian mi, Persian موش (muš), Northern Kurdish mişk, Sanskrit मूष् (mūṣ).
The computing sense was coined by American engineer Bill English in 1965 and first used publicly in a publication titled "Computer-Aided Display Control", in reference to the similarity with the animal.

coin

the best laid plans of mice and men go oft astray

the best laid plans of mice and men often go awry

(set theory) A small model of (a fragment of) Zermelo-Fraenkel set theory with desirable properties (depending on the context).


ordinal number

ординал

generalized kind of number to denote the length of a well-order on a set

1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, Dover (Dover Phoenix), 2006, page 137,
For not only do the antinomies a) to e) disappear when we admit as elements of sets only such sets, ordinal numbers, and cardinal numbers as are bounded above by a fixed cardinal number, but we see also that paradoxes always arise if we collect into a set any sets, cardinal numbers, or ordinal numbers which are not bounded above by a fixed cardinal number.

1960 [D. Van Nostrand], Paul R. Halmos, Naive Set Theory, 2017, Dover, Republication, page 80,
Is there a set that consists exactly of all the ordinal numbers? It is easy to see that the answer must be no. If there were such a set, then we could form the supremum of all ordinal numbers. That supremum would be an ordinal number greater than or equal to every ordinal number. Since, however, for each ordinal number there exists a strictly greater one (for example, its successor), this is impossible; it makes no sense to speak of the "set" of all ordinals.

2009, Marek Kuczma, Attila Gilányi (editor), An Introduction to the Theory of Functional Equations and Inequalities, Springer (Birkhäuser), 2nd Edition, page 10,
If α is an ordinal number, then by definition any two well-ordered sets of type α are similar, i.e., there exists a one-to-one mapping from one set to the other. Consequently these sets have the same cardinality. Consequently to any ordinal number α we may assign a cardinal number, the common cardinality of all well-ordered sets of type α.

equivalence relation
equivalence relation

(set theory) Such a number generalised to correspond to any cardinal number (the size of some set); formally, the order type of some well-ordered set of some cardinality a, which represents an equivalence class of well-ordered sets (exactly those of cardinality a) under the equivalence relation "existence of an order-preserving bijection".


chain

From Middle English cheyne, chaine, from Old French chaine, chaene (“chain”), from Latin catēna (“chain”), from Proto-Indo-European *kat- (“to braid, twist; hut, shed”). Doublet of catena. Displaced native Middle English rakil and rakent (from Old English racente (“chain”)); see rackan.

der

a chain is only as strong as its weakest link

We first find an approximation of the chain partition, i.e. a small but not minimum size set of chains which cover all elements of the poset.

2003, Jeremy P. Spinrad, Efficient Graph Representations, American Mathematical Society, page 108:

(mathematics, set theory, order theory) A totally ordered set, especially a totally ordered subset of a poset.


AC

ACs (plural)
a.c. (alternative)
A/C (alternative)
a/c (alternative)
A.C. (alternative)
a.c. (alternative)
ac (alternative)
A.C. (alternative)
A.C. (alternative)
A. C. (alternative)
a.c. (alternative)

(set theory) Initialism of axiom of choice.


CH

en-noun

CH (countable and uncountable, plural CHs)

continuum hypothesis
continuum hypothesis#English

(set theory) Initialism of continuum hypothesis.


family

family (countable and uncountable, plural families)

Jamaican Creole: faambli, fambili

Tok Pisin: famili

Sranan Tongo: famiri

→ Chuukese: famini

→ Malay: famili

→ Maori: whāmere

From Late Middle English famylye, from Latin familia (“a household”). Displaced native Old English hīred. Doublet of familia.

عَائِلَة

group of people related by blood, marriage, law, or custom

the family that lays together stays together

the family that prays together stays together

the family that sleeps together keeps together

Let #92;mathcalF be a family of subsets over S.

Any group or aggregation of things classed together as kindred or related from possessing in common characteristics which distinguish them from other things of the same order.

(set theory, countable) A collection of sets, especially of subsets of a given set.


ideal

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.
In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.

Formally, an ideal I of a given set X is a nonempty subset of the powerset #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.

(set theory) A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.


set

From Middle English set, sete, sette (“that which is set, the act of setting, seat”), from Old English set (“setting, seat, a place where people remain, habitation, camp, entrenchment, a place where animals are kept, stall, fold”) and Old English seten (“a set, shoot, slip, branch; a nursery, plantation; that which is planted or set; a cultivated place; planting, cultivation; a setting, putting; a stopping; occupied land”), related to Old English settan (“to set”). Compare Middle Low German gesette (“a set, suite”), Old English gesetl (“assembly”). According to Skeat, in senses denoting a group of things or persons, representing an alteration of sept, from Old French sette (“a religious sect”), from Medieval Latin secta (“retinue”), from Latin secta (“a faction”). See sect. It is quite possible that the modern word is more of a merger between both, however.

cog

(set theory) A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects which may be contained within it.


power

en-noun

power (countable and uncountable, plural powers)

powers (plural)
powre (alternative obsolete)

→ German: Power

/ˈpʌʊ̯.(ə)ɹ/

From Middle English power, poer, from Old French poeir, from Vulgar Latin potēre, from Latin posse, whence English potent. Compare French pouvoir. Displaced the native Old English anweald.

հզորություն

physics: measure of the rate of doing work or transferring energy


domain

range

antonym(s) of “domain of definition of a function”

From Middle English demayne, demain (“rule”), from Old French demeine, demaine, demeigne, domaine (“power”), (French domaine), from Latin dominium (“property, right of ownership”), from dominus (“master, proprietor, owner”). See dame, and compare demain, danger, dungeon. Doublet of demesne, domino.

definisieversameling

(mathematics) the set on which a function is defined

(mathematics, set theory) The set of input (argument) values for which a function is defined.

(more generally, of a binary relation R between A and B) The set A; The subset of A consisting of elements a of A such that there exists an element b in B with (a,b) in R.


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

ядро

(mathematics, algebra) set of elements mapped to zero

If a function is continuous then its kernel is a closed set.

(mathematics, set theory, of a function f) The set of pairs of elements in the domain of f which are mapped to the same value; the equivalence relation x≡y⟺f(x)=f(y).


order

en-noun

order (countable and uncountable, plural orders)

orders (plural)
ordre (alternative obsolete)

→ Swahili: oda

From Middle English ordre, from Old French ordre, ordne, ordene (“order, rank”), from Latin ōrdinem, accusative of ōrdō (“row, rank, regular arrangement”, literally “row of threads in a loom”), from Proto-Italic *ordō (“to arrange”), probably ultimately from Proto-Indo-European *h₂or-d-, from *h₂er-.
Related to Latin ōrdior (“begin”, literally “begin to weave”). In sense “request for purchase”, compare bespoke. Doublet of ordo.

cog

1911 [Cambridge University Press], William Burnside, Theory of Groups of Finite Order, 2nd Edition, Reprint, Dover (Dover Phoenix), 2004, page 222,
In this case, the conjugate set contains n(n − 1)/x(x − 1) distinct sub-groups of order m, and H is therefore self-conjugate in a group K of order x(x − l)m.

For various reasons it turns out to be better to enlarge this set of invariants to include suitable normalizers of subgroups of odd prime order.

2000, Michael Aschbacher, Finite Group Theory, 2nd edition, Cambridge University Press, page 260:

(set theory, of a set or algebraic structure) The number of elements contained within (the given object); formally, the cardinality (of the given object).


class

en-noun

class (countable and uncountable, plural classes)

→ Assamese: ক্লাছ (klas)

desc

→ Bengali: ক্লাস (klaś), কেলাস (kelaś) — nonstandard

q

or from French <i class="Latn mention" lang="fr">classe</i>

→ Gulf Arabic: كلاس (klās) (or from French classe)

→ Hindustani:

Hindi: क्लास (klās)

Urdu: کلاس

→ Japanese: クラス (kurasu)

→ Korean: 클래스 (keullaeseu)

→ Nepali: क्लास (klās)

→ Scottish Gaelic: clas

→ Thai: คลาส (kláas)

→ Yoruba: kíláàsì

From Middle French classe, from Latin classis (“a class or division of the people, assembly of people, the whole body of citizens called to arms, the army, the fleet, later a class or division in general”), from Proto-Indo-European *kelh₁- (“to call, shout”). Doublet of clas and classis.

class action

class-action, class action lawsuit, class suit

The class of all sets is not a set.

Every set is a class, but classes are not generally sets. A class that is not a set is called a proper class.

In the present section we shall discuss the various systems of set theory which admit, beside sets, also classes. Classes are like sets, except that they can be very comprehensive; an extreme example of a class is the class which contains all sets.[…]The main point which will, in our opinion, emerge from this analysis is that set theory with classes and set theory with sets only are not two separate theories; they are, essentially, different formulations of the same underlying theory.

1973, Abraham Fraenkel, Yehoshua Bar-Hillel, Azriel Lévy, Foundations of Set Theory, 2nd edition, Elsevier, page 119:

(set theory) A collection of sets definable by a shared property, especially one which is not itself a set (in which case the class is called proper).

դաս

set theory: collection of sets definable by a shared property


atom

→ Swahili: atomi

From Middle English attome, from Middle French athome, from Latin atomus (“smallest particle”), from Ancient Greek ἄτομος (átomos, “indivisible”), from ἀ- (a-, “not”) + τέμνω (témnō, “I cut”). Atoms are so named because historically the notion was that they were indivisible, given that chemically they are indeed so; the splitting of atoms awaited a later era of science and technology, and nonchemical means. Doublet of atomus.

(mathematics, set theory) An element of a set that is not itself a set; an urelement.


omega

From Middle English, from Ancient Greek ὦ μέγα (ô méga), meaning “great ω” (omega is a long vowel in Ancient Greek).

(set theory) A transfinite ordinal number referring to the next position after ordering a countably infinite set.


homogeneous

From Medieval Latin homogeneus, from Ancient Greek ὁμογενής (homogenḗs, “of the same race, family or kind”), from ὁμός (homós, “same”) + γένος (génos, “kind”). Compare homo- (“same”) and -ous (adjectival suffix).

(mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).

(set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.


intersection

/ˈɪntəɹˌsɛkʃən/

/ˌɪntəɹˈsɛkʃən/

From Middle French intersection and its etymon Latin intersectiō. By surface analysis, intersect + -ion.

(set theory) The set containing all the elements that are common to two or more sets.


algebra

en-noun

algebra (countable and uncountable, plural algebras)

⇒ Chinese: 代數 /代数 (dàishù)

→ Japanese: 代数

→ Korean: 대수 (daesu)

→ Vietnamese: đại số

/ˈæl.d͡ʒɪ.bɹə/

/ˈæl.d͡ʒə.bɹə/

Borrowed from Medieval Latin algebra, from the Arabic word الْجَبْر (al-jabr, “reunion, resetting of broken parts”) in the title of al-Khwarizmi's influential work الْكِتَاب الْمُخْتَصَر فِي حِسَاب الْجَبْر وَالْمُقَابَلَة (al-kitāb al-muḵtaṣar fī ḥisāb al-jabr wa-l-muqābala, “The Compendious Book on Calculation by Completion and Balancing”).

mathematical analysis
mathematical analysis

(countable, set theory, mathematical analysis) A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).


union

union (countable and uncountable, plural unions)

→ Bengali: ইউনিয়ন (iuniẏon) (learned)

From Middle English unyoun, from Old French union, from Late Latin ūniō, ūniōnem (“oneness, unity”), from Latin ūnus (“one”). Doublet of unio.

der

(countable, set theory) The set containing all of the elements of two or more sets.


union

en-verb

union (third-person singular simple present unions, present participle unioning, simple past and past participle unioned)

unions (present singular third-person)
unioning (participle present)
unioned (participle past)
unioned (past)

(set theory) To combine sets using the union operation.


element

From Middle English element, from Old French element, from Latin elementum (“a first principle, element, rudiment”) (see further etymology there).
The verb is from Middle English elementen, from the noun.

element

simplest or essential part or principle of anything

One of the simplest or essential parts or principles of which anything consists, or upon which the constitution or fundamental powers of anything are based.

(set theory) One of the objects in a set.


amorphous

en-adj

amorphous (comparative more amorphous, superlative most amorphous)

more amorphous (comparative)
most amorphous (superlative)

From Ancient Greek ἄμορφος (ámorphos, “without form, shapeless, deformed”) (itself from ἀ- (a-, “without”) + μορφή (morphḗ, “form”) + -ous. By surface analysis, a- + -morphous.

der

(set theory, of a set) Infinite and not the disjoint union of two infinite subsets.


proper

en-adj

proper (comparative more proper, superlative most proper)

more proper (comparative)
most proper (superlative)
propa (alternative Geordie)
propre (alternative obsolete)

From Middle English propre, from Anglo-Norman proper, propre, Old French propre (French: propre), from Latin proprius.

luonnonvärinen

heraldry: having its natural or usual coloration

(set theory, of a class) Not being a set.

Accurate, strictly applied.
Not being a set.


choice

en-noun

choice (countable and uncountable, plural choices)

choices (plural)
choise (alternative obsolete)
choyse (alternative obsolete)

From Middle English chois, from Old French chois (“choice”), from choisir (“to choose, perceive”), possibly via assumed Vulgar Latin *causīre (“to choose”), from Gothic 𐌺𐌰𐌿𐍃𐌾𐌰𐌽 (kausjan, “to make a choice, taste, test, choose”), from Proto-Germanic *kauzijaną, from *keusaną (“to choose”), from Proto-Indo-European *ǵews- (“to choose”). Akin to Old High German kiosan (“to choose”), Old English ċēosan (“to choose”), Old Norse kjósa (“to choose”). More at choose.

der

you pays your money and you takes your choice

you pays your nickel and you takes your choice

5. ZF* is the theory obtained from the aforementioned axiomatics (without choice) by adding the Axiom of Inaccessible Cardinals to be explained in the next secion; similarly, we get ZFC*.

2016 July 15, Decio Krause, Jonas R.B. Arenhart, The Logical Foundations of Scientific Theories: Languages, Structures, and Models (Routledge Studies in the Philosophy of Mathematics and Physics), Routledge, →ISBN, page 51:

(set theory) Ellipsis of axiom of choice.


asymmetric

en-adj

asymmetric (comparative more asymmetric, superlative most asymmetric)

more asymmetric (comparative)
most asymmetric (superlative)

From a- + symmetric, as Ancient Greek ἀσυμμετρία (asummetría, “disproportion, deformity”).

der

(set theory) Of a relation R on a set S: having the property that for any two elements of S (not necessarily distinct), at least one is not related to the other via R.


symmetric

en-adj

symmetric (comparative more symmetric, superlative most symmetric)

more symmetric (comparative)
most symmetric (superlative)

From symmetry + -ic, from Latin symmetria, from Ancient Greek συμμετρία (summetría).

"Is a sibling of" is a symmetric relation.

(set theory) Of a relation R on a set S, such that xRy if and only if yRx for all members x and y of S (that is, if the relation holds between any element and a second, it also holds between the second and the first).

symétrique

of a relation R on a set S, such that xRy if and only if yRx for all members x and y of S


member

→ Japanese: メンバー (menbā)

→ Korean: 멤버 (membeo)

→ Malay: member

→ Punjabi: ਮੈਂਬਰ (maimbar)

From Middle English membre, from Old French membre, from Latin membrum (“limb, body part”), from Proto-Italic *memzrom, from Proto-Indo-European *mḗms, *mēms-rom (“flesh”). Akin to Gothic 𐌼𐌹𐌼𐌶 (mimz, “meat, flesh”), Crimean Gothic menus.
Coexists with native Middle English lim, limb (“member, limb, joint”) (from Old English lim (“limb, joint, main branch”)), and displaced Middle English lith (“limb, joint, member”) (from Old English liþ (“limb, member, join, tip”)).


infinite

en-adj

infinite (comparative more infinite, superlative most infinite)

more infinite (comparative)
most infinite (superlative)

From Middle English infinite, from Old French infinit and its etymon Latin īnfīnītus, from in- (“not”) + fīnis (“end”) + the perfect passive participle ending -ītus. By surface analysis, in- + finite. Doublet of infinito. Displaced native Old English unġeendodlīċ.

For any infinite set, there is a 1-1 correspondence between it and at least one of its proper subsets. For example, there is a 1-1 correspondence between the set of natural numbers and the set of squares of natural numbers, which is a proper subset of the set of natural numbers.

2009, Brandon C. Look, “Symbolic Logic II, Lecture 2: Set Theory”, in www.uky.edu/~look, archived from the original on 2018-06-19:

(set theory, of a set) Having infinitely many elements.


complement

en-noun

complement (countable and uncountable, plural complements)

From Middle English complement, from Latin complēmentum (“that which fills up or completes”), from compleō (“I fill up, I complete”) (English complete). Doublet of compliment. The verb is from the noun.

der

The complement of the odd numbers is the even numbers, relative to the natural numbers.

(set theory) Given two sets, the set containing one set's elements that are not members of the other set (whether a relative complement or an absolute complement).


bijection

PIE word
 *dwóh₁
From French bijection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

The present text has defined a set to be finite if and only if there exists a bijection onto a natural number, and infinite if and only if there does not exist any such bijection.

2002, Yves Nievergelt, Foundations of Logic and Mathematics, page 214:

Note in particular that a function is a bijection if and only if it's both an injection and a surjection.

2007, C. J. Date, Logic and Databases: The Roots of Relational Theory, page 167:

The basic idea is that two sets A and B have the same cardinality if there is a bijection from A to B. Since the domain and range of the bijection is not relevant here, we often refer to a bijection from A to B as a bijection between the sets, or a one-to-one correspondence between the elements of the sets.

2013, William F. Basener, Topology and Its Applications, unnumbered page:

(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.

Nicolas Bourbaki
bijection
Éléments de mathématique

биекция

function that is both a surjection and an injection


surjection

/sɜː(ɹ)ˈd͡ʒɛk.ʃən/

LL-Q1860 (eng)-Flame, not lame-surjection.wav

From French surjection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.
Ultimately borrowed from Latin superiectiō (“a throwing over or on; (fig.) an exaggeration, a hyperbole”).

bor

In some special cases, however, the number of surjections A#92;rightarrowB can be identified.

1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, Institute of Physics Publishing, page 220:

Let J#61;#92;cap#95;im#95;i be the (irredundant) primary decomposition of J. We associate to the pair (J,#92;omega) the element #92;textstyle#92;sum#95;i(m#95;i,#92;omega#95;i)#92;inG, where #92;omega#95;i is the equivalence class of surjections from L#47;m#95;iL#92;oplus(A#47;m#95;i)#123;n-1#125; to m#95;i#47;m#95;i² induced by #92;omega.

1999, M. Pavaman Murthy, “A survey of obstruction theory for projective modules of top rank”, in Tsit-Yuen Lam, Andy R. Magid, editors, Algebra, K-theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday, American Mathematical Society, page 168:

In Banach space theory, a mapping u#58;E#92;rightarrowF (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isometric isomorphism. Moreover, by the classical open mapping theorem, u is a surjection iff the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isomorphism.

2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43:

(set theory) A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function f:X→Y for which for every y∈Y, there is at least one x∈X such that f(x)=y.

Nicolas Bourbaki
Surjective function
Éléments de mathématique


belong

en-verb

belong (third-person singular simple present belongs, present participle belonging, simple past and past participle belonged)

belongs (present singular third-person)
belonging (participle present)
belonged (participle past)
belonged (past)

⇒ Chinese Pidgin English: belong

→ Jersey Dutch: belânge

→ Tok Pisin: bilong

From Middle English belongen, bilongen, from Middle English be- + longen (“to be fitting, be suitable”), from Old English langian (“to pertain to, suit”), equivalent to be- + long (“to belong”). Compare Saterland Frisian beloangje (“to attain, reach, meet”), Dutch belangen (“to concern”), German belangen (“to sue, concern”).

cog

Suppose x belongs to #92;mathbb#123;R#125;…

(intransitive, set theory) (followed by to) To be an element of (a set). The symbol ∈ means belongs to.


creative

en-adj

creative (comparative more creative, superlative most creative)

more creative (comparative)
most creative (superlative)

Borrowed from Late Latin creativus, from Latin creō. Equivalent to create + -ive. Displaced native Old English orþanclīċ.

a creative set

(set theory) A type of set of natural numbers, related to mathematical logic.


normal

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(set theory, of a function from the ordinals to the ordinals) Which is strictly monotonically increasing and continuous with respect to the order topology.


relation

en-noun

relation (countable and uncountable, plural relations)

From Middle English relacion, relacioun, from Anglo-Norman relacioun and Old French relacion (whence French relation), from Latin relātiō, noun of process form from perfect passive participle relātus (“related”), from verb referō (“I refer, I relate”), from prefix re- (“again”) + ferō (“I bear, I carry”). By surface analysis, relate + -ion. Doublet of relazione.

[…]Signs are, first of all, physical things: for example, chalk marks on a blackboard, pencil or ink marks on paper, sound waves produced in a human throat. According to Reichenbach, "What makes them signs is the intermediary position they occupy between an object and a sign user, i.e., a person." For a sign to be a sign, or to function as such, it is necessary that the person take account of the object it designates. Thus, anything in nature may or may not be a sign, depending on a person's attitude toward it. A physical thing is a sign when it appears as a substitute for, or representation of, the object for which it stands with respect to the sign user. The three-place relation between sign, object, and sign user is called the sign relation or relation of denotation.

1974, Thomas S. Szasz, M.D., chapter 7, in The Myth of Mental Illness, →ISBN, page 107:

Equality is a symmetric relation, while divisibility is not.

(set theory) Specifically, a set of ordered pairs; a binary relation.


filtration

filtration (countable and uncountable, plural filtrations)

(mathematics, set theory, probability theory) A totally ordered collection of subsets.


strict

en-adj

strict (comparative stricter or more strict, superlative strictest or most strict)

stricter (comparative)
more strict (comparative)
strictest (superlative)
most strict (superlative)

Borrowed from Latin strictus, past participle of stringere (“to draw tight, bind, contract”). Doublet of strait and stretto. See stringent, strain.

(set theory, order theory) Irreflexive; if the described object is defined to be reflexive, that condition is overridden and replaced with irreflexive.


collection

en-noun

collection (countable and uncountable, plural collections)

From Middle English colleccioun, collection, from Old French collection, from Latin collēctiō, collēctiōnem, from collēctus, from colligō (“collect together”), composed of con- + legō (“bring together, gather, collect”), ultimately from Proto-Indo-European *leǵ- (“to gather, collect”).

der

(set theory, topology, mathematical analysis) A set of sets; used because such a thing is in general too large to comply with the formal definition of a set.


correspondence

en-noun

correspondence (countable and uncountable, plural correspondences)

From Middle English correspondence, from Latin correspondentia. By surface analysis, correspond + -ence.

Congruity or similarity between different things, people, etc:


subset

proper subset

subset that is strictly less than the given other set

The set of integers is a subset of the set of real numbers.

The set #92;lbracea,b#92;rbrace is a both a subset and a proper subset of #92;lbracea,b,c#92;rbrace while the set #92;lbracea,b,c#92;rbrace is a subset of #92;lbracea,b,c#92;rbrace but not a proper subset of #92;lbracea,b,c#92;rbrace.

1963, David B. MacNeil, Modern Mathematics for the Practical Man, David Van Nostrand, Republished as 2013, David B. MacNeil, Fundamentals of Modern Mathematics: A Practical Review, Dover, page 3,
In the foregoing example, the set D of the first four letters of the alphabet, was a subset of the set A of all the letters of the alphabet, because A includes all the members of D.

Let A be a subset of the topological space X and take x#92;inX.

1997, Wolfgang Filter, K. Weber, Integration Theory, Chapman & Hall, page 5:

We say that a set S has a finite partition into subsets S#95;1,#92;dots,S#95;n, if S#61;S#95;i#92;cup#92;dots#92;cupS#95;n, where the subsets are pairwise disjoint, that is, S#95;i#92;capS#95;j#61;#92;empty, if i#92;nej. (We do not require that the subsets be nonempty.)

2007, Judith D. Sally, Paul J. Sally, Jr., Roots to Research: A Vertical Development of Mathematical Problems, American Mathematical Society, page 280:

(set theory, of a set S) A set A such that every element of A is also an element of S.

подмножество

set whose elements are within another given set


forcing

forcing (countable and uncountable, plural forcings)

(set theory) A technique used to prove the consistency of certain axioms in set theory. See forcing (mathematics).

כפייה

set theory: a technique used to prove the consistency of certain axioms


successor

successors (plural)
successour (alternative obsolete)

predecessor

person or thing that immediately follows another

From Anglo-Norman successour, from Latin successor.

A limit ordinal is not the successor of any ordinal.

(arithmetic, set theory) The integer, ordinal number or cardinal number immediately following another.

наследник

integer or cardinal immediately following another


aleph-null

From its mathematical symbol ℵ₀, from א, aleph, the first letter of the Hebrew alphabet.

(set theory) The first of the transfinite cardinal numbers; corresponds to the number of natural numbers.


computable

(mathematics, set theory) Of a countably infinite set, having a computable indicator function.


saturated

en-adj

saturated (comparative more saturated, superlative most saturated)

more saturated (comparative)
most saturated (superlative)

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(mathematics, set theory, of a set S, with respect to an equivalence relation ≡) Equal to a union of equivalence classes of ≡; such that if x∈S and x≡y then y∈S.

(mathematics, especially topology, with respect to a function f) Saturated with respect to the equivalence relation x≡y⟺f(x)=f(y) (the kernel, sense 10, of f).


transfinite number

en-noun

transfinite number (plural transfinite numbers)

It will be recalled that Cantor called the first transfinite number ℵ₀. He called the second transfinite number—the one describing the set of all real numbers— C. It has not been proved whether C is the next transfinite number after ℵ₀ or whether another number exists between them.

1961, Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, Courier Dover Publications, →ISBN, page 228:

Let R be a bicompact of dimensionality #92;operatorname#123;ind#125;(R)#92;le#92;alpha. If #92;alpha is an isolated transfinite number, than ^([sic]) at any point x#92;inR there exist arbitrarily small neighborhoods Vx with boundaries of dimensionality #92;operatorname#123;ind#125;#92;overline#123;Vx#125;#92;le#92;alpha-1.

1968, B. T. Levšenko, “Spaces of transfinite dimensionality”, in Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Soc., →ISBN, page 141:

After all, it was the ordinals that made precise definition of the transfinite cardinals possible. And until Cantor had introduced the order types of transfinite number classes, he could not define precisely any transfinite cardinal beyond the first power.

1990, Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, →ISBN, page 180:

For example, does there exist a transfinite number that is strictly bigger than ℵ₀ and strictly smaller than ℵ₁? In this case an instance of this in between number is too big to be put into one-to-one correspondence with the set of natural numbers, and too small to be put into one-to-one correspondence with the set of real numbers.

2009, John Tabak, Numbers: Computers, Philosophers, and the Search for Meaning, Infobase Publishing, →ISBN, page 153:

Having demonstrated the existence of a one-to-one correspondence, we can conclude that the class of the squares of all the natural numbers has the same transfinite number as the class of all the natural numbers! This result is not what might have been anticipated, seeing that the second class is a proper subset of the first.

2012, Benjamin Wardhaugh, A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing, Princeton University Press, →ISBN, page 136:

(set theory) Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc.


aleph-one

continuum hypothesis
continuum hypothesis

(set theory) The second of the transfinite cardinal numbers; according to the continuum hypothesis, it corresponds to the number of real numbers.


partition

partition (countable and uncountable, plural partitions)

/pɑɹˈtɪ.ʃən/

/pɑɹˈtɪ.ʃɪn/

Recorded c.1430, "division into shares, distinction," from Middle English particioun, from Old French particion (modern partition), from Latin partitio, partitionem (“division, portion”), from partitus, the past participle of partire (“to split (up), part(ition)”).

der

(set theory) A collection of non-empty, disjoint subsets of a set whose union is the set itself (i.e. all elements of the set are contained in exactly one of the subsets).


S

From the first letter of set and of stage, two fundamental concepts of the theory.

S has no axiom of extensionality and identity is absent from the other S axioms.

(set theory) An axiomatic set theory, developed by logician George Boolos, in which several of the axioms of ZF are derivable as theorems.


tuple

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From the ending of the words quintuple, sextuple; from Latin -plus.

(set theory) A finite sequence of terms.


ZF

Gödel's model is an example of a simple type of inner model that might be called a definable transitive inner model³), where the universe is replaced by a transitive subuniverse defined within ZF, the membership relation is just the original one restricted to the subuniverse, and where the axioms of ZF relativized to the subuniverse, are provable within ZF.

1967, Carol Karp, “A Proof of the Relative Consistency of the Continuum Hypothesis”, in John N. Crossley, editor, Sets, Models, and Recursion Theory, North-Holland, page 7:

1971, Ulrich Felgner, Models of ZF-Set Theory, Springer, Lecture Notes in Mathematics 223, page 21,
1. Corollary: ZF is not finitely axiomatizable.
2. Corollary: ZF is reflexive (i.e. the consistency of every finite subtheory of ZF can be proved within ZF).

1991 [Kluwer Academic], Fred Landman, Structures for Semantics, 1991, Springer, Softcover, page 56,
However, the problem with it, and the reason why it is not part of ZF strictly (apart from the fact that it implies the axiom of choice) is that it is rather arbitrary.

Zermelo-Fraenkel
w:Zermelo-Fraenkel set theory

(set theory) Initialism of Zermelo-Fraenkel (set theory): a particular axiomatic formulation of set theory without the axiom of choice.


superset

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The set of human beings is a superset of the set of human children.

The set of characters "LBPG" is a superset of the set of characters "PG".

(set theory) (symbol: ⊇) With respect to another set, a set such that each of the elements of the other set is also an element of the set.

nadmnožina

set containing all elements of another set


empty set

Start with the empty set. We define it as “the set of all objects not equal to themselves,” since there are no such objects. All empty sets have the same members—no members at all! Therefore, as sets they're identical, by definition of identity of sets. In other words, there's only one empty set. This unique empty set is our building block. Next comes the set whose only member is—the empty set. This set is not empty.

1997, Reuben Hersh, What Is Mathematics, Really?, Oxford University Press, page 257:

The existence of the empty set is entailed, in the presence of the axiom scheme of separation, by the existence of any set whatever.

2004, Michael Potter, Set Theory and its Philosophy, Oxford University Press, page 58:

A topological space may be very sparsely endowed with open sets. As we know, some spaces have only two, the empty set and the full space.

2007, A. K. Sharma, Measure Theory, Discovery Publishing House, page 1:

Their existence can be generated on the basis of the empty set, since one is simply the set of the empty set, two is simply that set plus the empty set and so forth.

2008, Sam Gillespie, The Mathematics of Novelty: Badiou's Minimalist Metaphysics, re.press, page 57:

(set theory) The unique set that contains no elements, denoted ∅ or {}.


cardinality

en-noun

cardinality (countable and uncountable, plural cardinalities)

The empty set has a cardinality of zero.

The cardinality of a set A is the least ordinal α such that there exists a bijection between A and α. We sometimes use the notation #92;alpha#61;#124;A#124; to indicate this.

2005, Johan de Jong, “Set Theory”, in The Stacks Project, retrieved 2018-02-26:

For fuzzy sets, the concept of set size or cardinality is both richer and more problematic than it is for crisp sets. It is richer because, as we shall see, we may use more than one kind of cardinality.

2006, Michael Smithson, Jay Verkuilen, Fuzzy Set Theory: Applications in the Social Sciences, SAGE Publications, page 37:

Clearly, in this example, the sensitivity to the cardinalities takes the weaker form F#91;M(A)#93;#61;#92;overset#123;#61;#125;#123;A#125; of a single-valued function from the measure to the cardinality rather than the stronger form M(A)#61;f(#92;overset#123;#61;#125;#123;A#125;) of a function from the cardinality to the measure.

2012, Adolf Grünbaum, Robert S. Cohen, Marx W. Wartofsky, Philosophical Problems of Space and Time, 2nd edition, Springer, page 487:

(set theory, of a set) The number of elements a given set contains.


power set

power sets (plural)
power-set (alternative)
powerset (alternative)

The power set of #92;#123;1,2#92;#125; is #92;left#92;#123;#92;empty,#92;#123;1#92;#125;,#92;#123;2#92;#125;,#92;#123;1,2#92;#125;#92;right#92;#125;.

Moreover, for notational convenience, we write the cardinality of a denumerable set as #92;aleph#95;0. Cardinality of the power set of a denumerable set is written as #92;aleph#95;1. We may thus extend this notation further by taking cardinality of the power set of the power set of a denumerable set as #92;aleph#95;2, etc. but we do not have the need for it right now.

2009, Arindama Singh, Elements of Computation Theory, Springer, page 16:

Theorem 4.1. A complete Boolean algebra B has a set of (complete and atomic) ca-free generators iff B is isomorphic to the power set of a power set.

2013, A. Carsetti, Epistemic Complexity and Knowledge Construction, Springer, page 94:

Exponentiation is essentially a move to the power set—the set of all subsets of a given set. This is one of the reasons why Bertrand Russell's paradox is indeed a paradox: We cannot find a universal set because no set can contain its own power set!

2015, Amir D. Aczel, Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers, Palgrave MacMillan, page 147:

(set theory, of a set S) The set whose elements comprise all the subsets of S (including the empty set and S itself).


null set

(set theory, sometimes proscribed) The empty set.


negligible set

(mathematics, set theory, measure theory) A set that is small enough that it can be ignored for some purpose.


universal set

(set theory) A set large enough to contain all sets under consideration in the current context.


proper subset

{2, 4, 5} is a proper subset of {1, 2, 3, 4, 5}, but {1, 2, 3, 4, 5} is not.

(set theory) a set that is a subset of the set, but not equal to it.

真子集

a set that is a subset of but not equal to another set


Venn diagram

Named after British mathematician and philosopher John Venn (1834–1923).

named-after

(set theory) A diagram representing some sets by contours of closed shapes, such as circles or ellipses (and sometimes also the universal set as a rectangle enclosing all of these shapes), and indicating the relationships between the sets: by overlapping the shapes to show that the corresponding sets have a non-empty intersection, and by possibly (but not necessarily) enclosing all of the sets (which are proper subsets of the universal set) within a universal set (represented typically by a rectangle); such that the total number of simply connected regions is 2ⁿ, where n is the number of depicted sets which are proper subsets of the universal set.


non-empty

(set theory) Of a set, containing at least one element; not the empty set.


disjoint

en-adj

disjoint (comparative more disjoint, superlative most disjoint)

more disjoint (comparative)
most disjoint (superlative)

From Middle English disjoynen, from Old French desjoindre (“disjoin”), from Latin disiungō, from dis- + iungō (“join”).

(set theory, not comparable) Of two or more sets, having no members in common; having an intersection equal to the empty set.


ordered pair

If an ordered pair were defined (in terms of sets) as (x,y):=a,a,b then the "first element" of an ordered pair S could be defined as CAR(S) where CAR(S) = x if and only if (∀y∈S.,x∈y). Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if (∃y∈S.,(∃z∈y.,x∈z)). If the two elements happened to be equal, then the ordered pair would still have cardinality two as would be naturally expected.

(set theory) An object containing exactly two elements in a fixed order, so that, when the elements are different, exchanging them gives a different object. Notation: (a, b) or ⟨a,b⟩.


equivalence relation

en-noun

equivalence relation (plural equivalence relations)

(set theory) A binary relation that is reflexive, symmetric and transitive.


binary relation

binary relation (plural binary relations)

A partially ordered set #92;langleA,#92;varrho#92;rangle consists of a nonvoid set A and a binary relation #92;varrho on A, such that #92;varrho satisfies properties (P1)-(P3).

1978, George Grätzer, General Lattice Theory, Academic Press, page 1:

1.30. Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.
It should be noted that a binary relation may be irreflexive and negatively transitive without being transitive; as an example, consider the standard inequality relation (≠).

1999, James C. Moore, Mathematical Methods for Economic Theory 1, Springer, page 24:

Definition If E is a non-empty set then by an order on E we mean a binary relation on E that is reflexive, anti-symmetric, and transitive.

2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1:

(set theory, order theory, "on" a set A) A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A).

(set theory, order theory, "on" or "between" sets A and B) A subset of the Cartesian product A×B.


setoid

(set theory) A set together with an equivalence relation.


equivalence class

equivalence class (plural equivalence classes)

(set theory) Any one of the subsets into which an equivalence relation partitions a set, each of these subsets containing all the elements of the set that are equivalent under the equivalence relation.

třída ekvivalence

Any one of the subsets into which an equivalence relation partition is set


productive

en-adj

productive (comparative more productive, superlative most productive)

more productive (comparative)
most productive (superlative)

Borrowed from Late Latin productivus, equivalent to product + -ive.

a productive set

(set theory) A type of set of natural numbers, related to mathematical logic; a set S is productive if there exists a total recursive function f such that ∀x∈ℕ,~~W_x⊆S⇒f(x)∈A⧵W_x, where W_x is a recursive function whose Gödel number is x.


saturation

en-noun

saturation (usually uncountable, plural saturations)

LL-Q1860 (eng)-Flame, not lame-saturation.wav

Etymology tree
Late Latin saturatiobor.
English saturation
Borrowed from Late Latin saturatio, saturationem.

etymon

(mathematics, set theory, topology, of a set S, with respect to an equivalence relation or function, countable) The smallest set containing S which is saturated with respect to the equivalence relation or function.


meager

en-adj

meager (comparative meagerer, superlative meagerest)

meagerer (comparative)
meagerest (superlative)
meagre (alternative Commonwealth)

Jamaican Creole: mawga

From Middle English megre, from Anglo-Norman megre, Old French maigre, from Latin macer, from Proto-Indo-European *mh₂ḱrós. Akin, through the Indo-European root, to Old English mæġer (“meager, lean”), West Frisian meager (“meager”), Dutch mager (“meager”), German mager, Icelandic magr whence the Icelandic magur,
Norwegian Bokmål mager and Danish mager. Doublet of maigre.

(set theory) Of a set: such that, considered as a subset of a (usually larger) topological space, it is in a precise sense small or negligible.


directed

directed (comparative more directed, superlative most directed)

more directed (comparative)
most directed (superlative)

(set theory) Having the properties of a directed set.


meagre

meagre (comparative meagrer, superlative meagrest)

meagrer (comparative)
meagrest (superlative)
meager (alternative)

Jamaican Creole: mawga

From Middle English megre, borrowed from Anglo-Norman megre, Old French maigre, from Latin macer, macrum, from Proto-Indo-European *mh₂ḱrós. Cognate with Old English mæġer (“meagre, lean”), Dutch mager (“lean”), German mager (“lean”), Icelandic magur (“lean”).


injection

en-noun

injection (countable and uncountable, plural injections)

/ɪnˈd͡ʒɛk.ʃən/

Nicolas Bourbaki
Éléments de mathématique

Borrowed from Middle French injection, from Latin iniectio. The mathematical sense is from French injection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

(set theory) A function that maps distinct x in the domain to distinct y in the codomain; formally, a f: X → Y such that f(a) = f(b) implies a = b for any a, b in the domain.


bounded

en-adj

bounded (comparative more bounded, superlative most bounded)

more bounded (comparative)
most bounded (superlative)

(set theory, order theory, of a poset X with partial order ≤) That contains a least element, a, and a greatest element, b, such that for all x ∈ X, a ≤ x ≤ b.

barita

that contains a least and a greatest elements


Cartesian product

en-noun

Cartesian product (plural Cartesian products)

Cartesian products (plural)
cartesian product (alternative)

From Cartesian + product, after French philosopher, mathematician, and scientist René Descartes (1596–1650), whose formulation of analytic geometry gave rise to the concept.

(set theory, of two sets X and Y) The set of all possible ordered pairs of elements, the being first from X, the second from Y, written X×Y. Formally, the set (x,y);|;x∈X;and;y∈Y.


supremum

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A sublattice of a lattice is a subset that is closed under pairwise infima and suprema.

2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 8:

The best way to describe the supremum of S is to say that it wants to be the greatest element of S. In fact, if S has a greatest element, then that element is the supremum.

2010, James S. Howland, Basic Real Analysis, Jones & Bartlett Publishers, page 9:

The key to an approach to vector optimization based on infimum and supremum is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions.

2011, Andreas Löhne, Vector Optimization with Infimum and Supremum, Springer, page vii:

partially ordered set
partially ordered set

(set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.

上确界

element of a set greater than or equal to all members of a given subset


set-builder notation

With this idea for describing a finite set of sets, it is easy to generalize the concept to a certain infinite family #92;mathcal#123;S#125;#95;2 of sets #92;mathcal#123;S#125;#95;2#61;#92;#123;A#95;i#92;verti#92;inN#92;#125;#61;#92;#123;A#95;1,A#95;2,A#95;3,#92;dots,A#95;n,#92;dots#92;#125;. Once again, the power of set builder notation triumphs. The sets #92;mathcal#123;S#125;#95;1 and #92;mathcal#123;S#125;#95;2 may be described more precisely with set builder notation than by enumeration.

2000, Kenneth E. Hummel, Introductory Concepts for Abstract Mathematics, CRC Press (Chapman & Hall/CRC), page 123:

Q=a/b|a∈Iandb∈I,b ne 0
This statement is read in English as "Q is the set of all numbers of the form a/b such that a and b are both integers, but b is not equal to zero."

2011, Tom Bassarear, Mathematics for Elementary School Teachers, Cengage Learning, 5th Edition, page 56, In this case, and in many other cases, we describe the set using set-builder notation

2012, Richard N. Aufmann, Joanne Lockwood, Intermediate Algebra, Cengage Learning, 8th Edition, page 6,
A second method of representing a set is set-builder notation. Set-builder notation can be used to describe almost any set, but it is especially useful when writing infinite sets. In set-builder notation, the set of integers > −3 is written
x|x>-3,x∈integers

(set theory) A mathematical notation for describing a set by specifying the properties that its members must satisfy.


poset

Abbreviation of partially ordered set. Coined by American mathematician Garrett Birkhoff in his Lattice Theory.

coinage

1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, →ISBN, page 76,
42. Definition. A poset (partially ordered set) (X, ≤) (usually written just X) is a set X together with a transitive, antisymmetric relation ≤ on X.
43. Definition. A linearly ordered set or chain is a poset (X, ≤), such that ∀a, b ∈ X, either a ≤ b or b ≤ a or a = b.

1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153,
We construct a complete set of reflection functors for the representations of posets and prove that they really have the usual properties. In particular, when the poset is of finite representation type, all of its indecomposable representations can be obtained from some "trivial" ones via relations. To define such reflection functors, a wider class of matrix problem is introduced, called "representations of bisected posets".

The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph.

1999, Manfred Stern, Semimodular Lattices: Theory and Applications, Cambridge University Press, page 189:

(set theory, order theory) A partially ordered set.


ZFC

Zermelo-Fraenkel set theory with Choice
w:Zermelo–Fraenkel set theory

(set theory) Initialism of Zermelo-Fraenkel set theory with Choice; the standard axiomatization of set theory, including the axiom of choice.


axiom of choice

en-noun

axiom of choice (countable and uncountable, plural axioms of choice)

A calque of German Axiom der Auswahl (now more commonly Auswahlaxiom), which first appeared in print with a description of the axiom in 1908, Ernst Zermelo, Untersuchungen über die Grundlagen der Mengenlehre I ["Investigations in the foundations of set theory I"], Mathematische Annalen, 65 (although the paper was dated 1907).

The axiom of choice is logically equivalent to the assertion that every vector space has a basis.

If V = L then the axioms of choice and the continuum hypothesis are both true, and the assertion that a measurable cardinal exists is false.

1993, Thomas Tymoczko, editor, Penelope Maddy: Does V Equal L?: New Directions in the Philosophy of Mathematics: An Anthology, page 357:

Throughout this work we adopt the Zermelo–Fraenkel (ZF) axioms of set theory with the Axiom of Choice, commonly abbreviated as ZFC. It follows from the work of Gödel and Cohen that if the ZF axioms are consistent, the Axiom of Choice can be neither proved nor disproved from the ZF axioms.

1993, Gary L. Wise, Eric B. Hall, Counterexamples in Probability and Real Analysis, page vii:

To clarify these ideas for the reader, let us show, without the axiom of choice, that a product of finitely many nonempty sets is nonempty: This is done by induction on the number n of sets.[…]The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable.

2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, page 169:

Perhaps what this debate about whether to accept the axiom of choice indicates is that the disjunction between regularity and randomness is as fundamental to our conception of the world as that between discreteness and continuity.

2004, Michael Potter, Set Theory and its Philosophy: A Critical Introduction, page 259:

There is a uniform status quo almost with just a few exceptions of a handful of mathematicians around the world. Most people accept the “infinite choice, infinite decimals” approach to real numbers, and the justification that has been created to substantiate this, and to overcome some of the difficulties that I have shown you is through an elaborate axiomatic framework. Mathematicians are not stupid! They realize that this theory is dubious, and so what they have done is they have created a rather elaborate axiomatic framework. And that axiomatic framework is something that we can study in set theory and logic; and its purported aim is to create a framework for mathematics; but its initial aims were very much directed towards solving the problems, overcoming the difficulties with real numbers as infinite decimals. Prominent amongst these axioms is this Axiom of Choice! The Axiom of Choice manifests itself in mathematics in many ways; but its primary role — OK — its primary role is exactly here at the level of infinite decimals and real numbers. It essentially asserts, as a matter of faith or belief, that it is possible to choose an infinite number of digits arbitrarily and independently, and that a legitimate mathematical object results. This is the key philosophical point. Are we, or are we not, in a position to be able to specify a first digit, a second digit, a third digit, a fourth digit, and so on to infinity? And are we allowed to call that specification a new mathematical object, a real number? The Axiom of Choice says that “Yes we are!”, so it is a statement of belief and its primary objective is to allow the building of this real number system.

2012, 47:31 – 49:50 from the start, in Difficulties with real numbers as infinite decimals (I) Real numbers + limits Math Foundations 91, episode 91, njwildberger (N. J. Wildberger), via YouTube:

(set theory) One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.

ընտրության աքսիոմ

axiom that any product of non-empty sets is non-empty


direct product

suora tulo

product of sets — see also Cartesian product

If A and B are sets, their direct product is the set of ordered pairs (a,b) with a in A and b in B.

(set theory) The set of all possible tuples whose elements are elements of given, separately specified, sets.


partial order

1986, Kenneth R. Goodearl, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, Softcover reprint 2010, page xxi,
A partial order on a set X is any reflexive, antisymmetric, transitive relation on X. In most cases, partial orders are denoted ≤.

1999, Paul A. S. Ward, An Online Algorithm for Dimension-Bound Analysis, Patrick Amestoy, P. Berger, M. Daydé, I. Duff, V. Frayssé, L. Giraud, D. Ruiz (editors), Euro-Par ’99 Parallel Processing: 5th International Euro-Par Conference, Proceedings, Springer, LNCS 1685, page 144,
The vector-clock size necessary to characterize causality in a distributed computation is bounded by the dimension of the partial order induced by that computation.

Consider an arbitrary finite set S. The family #92;mathcal#123;P#125; of all strict partial orders (asymmetric, transitive, cf. 1.8.3, p. 14) on S enjoys a remarkable property: any partial order P can be linked to any other partial order P’ by a sequence of steps each of which consists of changing the order either by adding one ordered pair of elements of S (imposing an ordering between two previously-incomparable elements) or by removing one ordered pair (causing two previously related elements to become incomparable), without ever leaving the family #92;mathcal#123;P#125;.

2008, David Eppstein, Jean-Claude Falmagne, Sergei Ovchinnikov, Media Theory: Interdisciplinary Applied Mathematics, Springer, page 7:

(set theory, order theory) (informal) An ordering of the elements of a collection that behaves like that of the natural numbers by size, except that some elements may not be comparable (if all elements are comparable, it is called a total order); (formal) a binary relation that is reflexive, antisymmetric, and transitive.


partially ordered set

partially ordered set (plural partially ordered sets)

半順序集合

はんじゅんじょしゅうごう

(set theory, order theory, loosely) A set that has a given, elsewhere specified partial order.

1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts, Real Analysis, 2005, Dover, page 28,
A partially ordered set means a pair (P,≻) consisting of a set P and a partial order ≻ in P. As usual, when the meaning is clear, we may suppress the notation of "≻" and speak of the partially ordered set P.
The ordered fields defined earlier are easily seen to be examples of partially ordered sets.

In sections 7-10 we shall consider random fields over some subsets T of the partially ordered set T_M.

1994, I. V. Evstigneev, P. E. Greenwood, Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting, American Mathematical Society, page 35:

The invention of a derivative of a finite partially ordered set by Nazarova and Roiter in the late 1960s or early 1970s was a seminal event in the subject of representations of finite partially ordered sets (see [Simson 92]).

2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 45:

(set theory, order theory, formally) The ordered pair comprising a set and its partial order.


antisymmetric

LL-Q1860 (eng)-Flame, not lame-antisymmetric.wav

1987, David C. Buchthal, Douglas E. Cameron, Modern Abstract Algebra, Prindle, Weber & Schmidt, page 479,
The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.

(i) The identity relation on a set A is an antisymmetric relation.
(ii) Let R be a relation on the set N of natural numbers defined by
 x R y #92;Leftrightarrow 'x divides y' for all x, y ∈ N.
This relation is an antisymmetric relation on N.

2006, S. C. Sharma, Metric Space, Discovery Publishing House, page 73:

(set theory, order theory, of a binary relation R on a set S) Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the property that, for any x, y ∈ S, if both xRy and yRx then x=y.

antisymetrický

(order theory; of a binary relation on a set)


partially ordered

This leads to the introduction of new kinds of abstract spaces—partially ordered linear spaces—and to their systematic use in functional analysis. The beginnings of a theory of partially ordered linear spaces are given in the works of L. V. Kantorovič in 1935-1937.

1963, L. V. Kantorovič, B. Z. Vulih, A. G. Pinsker, “Partially Ordered Groups and Partially Ordered Linear Spaces”, in A. L. Brudno, editor, American Mathematical Society Translations, Series 2, Volume 27: 18 papers on algebra, American Mathematical Society, page 51:

The chains in #92;mathcal#123;V#125; form a class #92;mathcal#123;C#125; which is partially ordered by inclusion.

1966, S. J. Taylor, Introduction to Measure and Integration, Cambridge University Press, page 22:

2008, Patrik Eklund, M. Ángeles Galán, Partially Ordered Monads and Rough Sets, James F. Peters, Andrzej Skowron (editors), Transactions on Rough Sets VIII, Volume 8, Springer, LNCS 5084, page 53,
In this paper we will show that partially ordered monads contain appropriate structure for modeling rough sets in a generalized relational setting.

(set theory, order theory, of a set) Equipped with a partial order; when the partial order is specified, often construed with by.


partial ordering relation

partial ordering relation (plural partial ordering relations)


independence number

independence number (plural independence numbers)

(set theory) The smallest cardinality of a maximal independent family of subsets of the natural numbers, usually denoted by lowercase Fraktur letter i.


total order

[…]we conclude §2.1 by showing how, given a triangulation #92;Delta (i.e., simplicial decomposition) of a closed oriented 3-manifilld M, and a total order '#92;le' on the set of vertices of #92;Delta, as well as a choice of a system #92;mathcal#123;B#125; of orthonormal bases for various Hilbert spaces that get specified in the process, we may obtain a complex number #92;langle(M,#92;Delta,#92;le,#92;mathcal#123;B#125;)#92;rangle.

2001, Vijay Kodiyalam, V. S. Sunder, Topological Quantum Field Theories from Subfactors, CRC Press (Chapman & Hall), page 2:

Example 6.2.2. Suppose A is a finite set and R is a partial order on A. Prove that R can be extended to a total order on A. In other words, prove that there is a total order T on A such that R ⊆ T.

2006, Daniel J. Velleman, How to Prove It: A Structured Approach, 2nd edition, Cambridge University Press, page 269:

2013, Nick Huggett, Tiziana Vistarini, Christian Wüthrich, 15: Time in Quantum Gravity, Adrian Bardon, Heather Dyke (editors), A Companion to the Philosophy of Time, Wiley, 2016, Paperback, page 245,
A binary relation R defines a total order on a set X just in case for all x, y, z ∈ X, the following four conditions obtain: (1) Rxx (reflexivity), (2) Rxy & Ryz → Rxz (transitivity), (3) Rxy & Ryx → x = y (weak antisymmetry), and (4) Rxy ∨ Ryx (comparability). Bearing in mind that the relata of the total order are not events in ℰ, but entire equivalence classes ℰ/S of simultaneous events, it is straightforward to ask ≤ to be a total order of ℰ/S.

(set theory, order theory) A partial order, ≤, (a binary relation that is reflexive, antisymmetric, and transitive) on some set S, such that any two elements of S are comparable (for any x, y ∈ S, either x ≤ y or y ≤ x).

lineární uspořádání

partial order that applies an order to any two elements

linear order

partial order which applies an order to any two elements


symmetric difference

symmetric difference (plural symmetric differences)

(set theory) The set that contains all elements belonging to either of given two sets but not to the both; the relative complement of the intersection in the union of given two sets.

فرق تماثلي

set of elements belonging only one of given two sets


set operation

(set theory) A mathematical operation that returns a set taking one or more sets as input.


relative complement

relative complement (plural relative complements)

(set theory, of set A "in" set B) The set containing exactly those elements belonging to B but not to A.

množinový rozdíl

set of elements contained in one set but not another


absolute complement

absolute complement (plural absolute complements)

(set theory) The set that contains exactly those elements belonging to the universal set but not to a given set.


set-theoretic difference

set-theoretic difference (plural set-theoretic differences)

(set theory, of two sets A and B) The set that contains exactly those elements belonging to A but not to B; the relative complement of B in A.


preorder

LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-preorder.wav

The relation of logical implication over sentences is an example of a preorder.

Example 436 For each set A, the relation #92;subseteq is a preorder on #92;mathcal#123;P#125;(A).

2002, Yves Nievergelt, Foundations of Logic and Mathematics, Springer (Birkhäuser), page 152:

Let us first compare S#95;1, S#95;2 and S#95;3 w.r.t. the partial preorder #92;succeq.

2010, S. Kaci, “Refined Preference-Based Argumentation Frameworks”, in Pietro Baroni, F. Cerutti, M. Giacomin, G. R. Simari, editors, Computational Models of Argument: Proceedings of COMMA 2010, IOS Press, page 306:

2000, Jean-Charles Pomerol, Sergio Barba-Romero, Multicriterion Decision in Management: Principles and Practice, Springer, Softcover, page 58,
It can easily be verified that the above relation is a preorder, i.e. that it is reflexive and transitive.

(set theory, order theory) A binary relation that is reflexive and transitive.


irreflexive

(set theory) Of a binary relation R on X: such that no element of X is R-related to itself.


multiset

Coined by Nicolaas Govert de Bruijn, multi- + set.

Two words are anagrams if they are made up of the same multiset of letters.

Membrane computing essentially deals with multisets, which are sets with multiplicities associated with their elements.

2002, Gheorghe Pǎun, Membrane Computing: An Introduction, Springer, page 49:

2006, Xin Chen, Lan Liu, Zheng Liu, Tao Jiang, On the Minimum Common Integer Partition Problem, Tiziana Calamoneri, Irene Finocchi, Giuseppe F. Italiano (editors), Algorithms and Complexity: 6th Italian Conference CIAC 2006, Proceedings, Springer, LNCS 3998, page 236,
A partition of a positive integer n is a multiset of positive integers that add up to exactly n, and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S.

2012, Edward A. Scheinerman, Mathematics: A Discrete Introduction Brooks/Cole (Cengage Learning), page 102,
Two multisets are the same provided they contain the same elements with the same multiplicities. For example, {1, 2, 3, 3} = {3, 1, 3, 2}, but {1, 2, 3, 3} ≠ {1, 2, 3, 3, 3}.

(set theory) A generalized type of set in which multiple occurrences of an element are permitted.

多重集

generalized type of set that may contain multiple occurrences of elements


countably infinite

1953 [Addison-Wesley], Bruce Elwyn Meserve, Fundamental Concepts of Algebra, 1982, Dover, page 36,
This one-to-one correspondence between the set of positive integers and the set of pairs of positive integers indicates that the set of pairs is countably infinite. Since the set of positive rational numbers is a subset of the set of all pairs of positive integers, the set of positive rational numbers is at most countably infinite. Then, since it is also at least countably infinite, the set of positive rational numbers is countably infinite.

Instead of saying that the aggregate of natural numbers is countably infinite (see Chapter 24), one can use Cantor's symbolism and state that its cardinal number is #92;aleph#95;0 (read aleph null).[…]Now the range of a random variable may be a finite set, a countably infinite set, or a continuum.

1970, Edna E. Kramer, “The Nature and Growth of Modern Mathematics”, in Paperback, Princeton University Press, published 1982, page 332:

The term “countably infinite” would seem to suggest that such a set is infinite. However, the definitions of “countably infinite” and “infinite” were made separately, and so we have to prove that countably infinite sets are indeed infinite (otherwise our notation would be rather misleading).

2000, Ethan D. Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics, Springer (Birkhäuser), page 211:

(set theory, of a set) That is both countable and infinite; having the same cardinality as the set of natural numbers; formally, such that a bijection exists from ℕ to the set.


totally ordered set

totally ordered set (plural totally ordered sets)

(set theory) A set having a specified total order.


well-order

well-orders (plural)
well order (alternative)

1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,
̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that
f≨_(XY)g⇒h•f≨_(XY)h•g
for every h:Y→Z.

2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71,
Some simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order.

2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378,
Definition 1226 (Von Neumann Well-Orders). A well-order X is said to be a von Neumann well-order if for every x∈X, we have x=y∈X|y<x (that is x is equal to the set Pred(x) consisting of its predecessors).
Clearly the examples listed by von Neumann above, namely
empty , empty, empty ,empty, empty ,empty,empty ,empty, …
are all von Neumann well-orders if ordered by the membership relation "∈," and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders.

(set theory, order theory) A total order of some set such that every nonempty subset contains a least element.


well-order

en-verb

well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)

well-orders (present singular third-person)
well-ordering (participle present)
well-ordered (participle past)
well-ordered (past)
well order (alternative)

The set of positive integers is well-ordered by the relation ≤.

1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111,
Starting from these special well-ordered subsets, it is then possible to well-order the entire set.

1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182,
To carry the analogy a bit further, the axiom of choice implies the ability to well order any set.

Then #92;le#95;C is a well defined order on C, and (C,#92;le#95;C) belongs to #92;mathcal#123;X#125; (that is, #92;le#95;C well orders C) and is an upper bound for #92;mathcal#123;C#125;.

2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 18:

(set theory, order theory, transitive) To impose a well-order on (a set).


antichain

First of all, Zorn's Lemma implies that every uncountable antichain in T is contained in a maximal uncountable antichain. So it suffices to make sure that T has no maximal uncountable antichains.

1997, Winfried Just, Martin Weese, Discovering Modern Set Theory II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 140:

2013, Vijay K. Garg, Maximal Antichain Lattice Algorithms for Distributed Computations, Proceedings, Davide Frey, Michel Raynal, Saswati Sarkar, Rudrapatna K. Shyamasundar, Prasun Sinha (editors), Distributed Computing and Networking: 14th International Conference, ICDCN, Springer, LNCS 7730, page 245,
We first define three different but isomorphic lattices: the lattice of maximal antichain ideals, the lattice of maximal antichains and the lattice of strict ideals.

2014, Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK, Springer, 5th Edition, page 199,
In 1928 Emanuel Sperner asked and answered the following question: Suppose we are given the set N=1,2,3,…,n. Call a family ℱ of subsets of N, which has partial order ⊆]) an antichain if no set of ℱ contains another set of the family ℱ. What is the size of a largest antichain? Clearly, the family ℱₖ of all k-sets satisfies the antichain property with |ℱₖ|=\binomnk. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size \binomn[n/2]= max ₖ\binomnk. Sperner's theorem now asserts that there are no larger ones.

(set theory, order theory, graph theory) A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤.


preordered

Most of the theory of this section on linear order spaces (and cones) can be generalized to preordered linear spaces (and wedges).

1980, Johann Schröder, Operator Inequalities, Academic Press, page 8:

It can be proved [see Herden (1989a,c, 1990, 1993a,b) and Herden and Mehta (1997a) that a continuous isotone (or strictly isotone) function defined on a closed subset K of a preordered topological space X has a continuous isotone (or strictly isotone) extension to a function g on X if and only if there is a separable system on X which satisfies certain "separation" conditions.

1998, Ghanshyam B. Mehta, “Chapter 1: Preference and Utility”, in Salvador Barbera, Peter Hammond, Christian Seidl, editors, Handbook of Utility Theory, Volume 1: Principles, Springer, page 30:

We remind the reader that the barycentric subdivision sd(P) of a preordered set P is the poset of chains in P, ordered by inclusion.

1999, Rafael Villarroel-Flores, Equivariant Homotopy Type of Categories and Preordered Sets, University of Minnesota, page 24:

(set theory, order theory, of a set) Equipped with a preorder.


Cantorian

en-adj

Cantorian (comparative more Cantorian, superlative most Cantorian)

more Cantorian (comparative)
most Cantorian (superlative)

(set theory, New Foundations) Smaller than its own power set; satisfying Cantor's theorem.


memberless

(set theory) Without members; empty or null.


Cantor-Bendixson theorem

Named after Georg Cantor (1845–1918), a German mathematician who invented set theory, and Ivar Otto Bendixson (1861–1935), a Swedish mathematician.

(set theory) A theorem which (in a simpler formulation) states that a closed uncountable set in Euclidean n-space is equal to the disjoint union of a perfect set and a countable set.

Georg Cantor
Ivar Otto Bendixson
Perfect set property


continuum hypothesis

In other words, it is consistent with the axioms^([of set theory]) to suppose that the continuum hypothesis is false. (It had been known for some time — see below — that it is consistent with the axioms of set theory to suppose that CH is true.)

2002, Steven G. Krantz, Handbook of Logic and Proof Techniques for Computer Science, Springer (Birkhäuser), page 65:

2013, Kenneth Reinhard, Introduction, Susan Spitzer (translator), Alain Badiou, The Incident at Antioch: A Tragedy in Three Acts / L’Incident d’Antioche: Tragédie en trois actes, Columbia University Press, page xxxiii,
After all, when Paul Cohen sent his proof of the undecidability of the continuum hypothesis (which Badiou considers an “event”) to Kurt Gödel in 1963, Gödel wrote back that “reading your proof had a similarly pleasant effect on me as seeing a good play."

In the end, it turned out that Cantor was unable to prove his continuum hypothesis precisely because it is unprovable.

2015, R. Bruce Elder, DADA, Surrealism, and the Cinematic Effect, Wilfrid Laurier University Press, page 582:

(set theory) The hypothesis which states that any infinite subset of ℝ must have the cardinality of either the set of natural numbers or of ℝ itself.

continuümhypothese

hypothesis that there is no cardinality between that of ℤ and that of ℝ


generalized continuum hypothesis

Generalization of the continuum hypothesis.

(set theory) The hypothesis that, for each ordinal α, there is no cardinal number strictly between א_α and 2^(א_α), i.e. 2^(א_α)=א_(α+1).


Burali-Forti paradox

the Burali-Forti paradox (canonical)
Burali-Forti's paradox (alternative)

Named after Cesare Burali-Forti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg Cantor.

1984, Michael Hallett, Cantorian Set Theory and Limitation of Size, Oxford University Press (Clarendon Press), 1986, Paperback, page 186,
Like them, Mirimanoff concentrates on the Burali-Forti paradox, and like Russell's analysis before, Mirimanoff shows how. in terms of size, the Burali-Forti paradox is basic and that if we solve this the other paradoxes will be solved too.

1994 [Routledge], Ivor Grattan-Guinness (editor), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Volume 1, 2003, Johns Hopkins University Press, Paperback, page 632,
In the first place, Berry rejected Russell's solution to the Burali-Forti paradox, claiming that it was easy to prove that the set of all ordinal numbers was a well-ordered set (and that Cantor had actually done it).

The Burali-Forti paradox was discovered by Cantor in 1895 and Burali-Forti in 1897, but was not regarded by them as a paradox.

2002, Marcus Giaquinto, The Search for Certainty: A Philosophical Account of Foundations of Mathematics, Oxford University Press (Clarendon Press), page 37:

(set theory) The paradox that supposing the existence of a set of all ordinal numbers leads to a contradiction; construed as meaning that it is not a properly defined set.

Burali-Forti paradox
Cesare Burali-Forti
Georg Cantor


NFU

(set theory) Initialism of New Foundations with urelements.


urelement

urelements (plural)
ur-element (alternative)

1996, Scientific Books staff (translators), Yuri L. Ershov, Definability and Computability, Scientific Books, page viii,
The introduction of urelements would seem to be a technical improvement; however, now we know that just such an extension of the notion of the admissible set led to the universal theory of computability based on the notion of definability by formulas with (in a broad sense) effective semantics.

There may be no urelements; as we shall see, we can still build plenty of sets in this case. At stage 0 we can always build the empty set. If there are no urelements, this is the only set we can build. If there is one urelement, a, we can build the sets #92;empty and #92;#123;a#92;#125;. If there are two urelements, a and b, the possible sets are #92;#123;a#92;#125;, #92;#123;b#92;#125;, and #92;#123;a,b#92;#125;; and so on if there are more urelements.
At stage 1, we can build any set containing urelements or sets built at stage 0; that is, any set whose members are already available at the beginning of stage 1. If there are no urelements, we can build #92;empty and #92;#123;#92;empty#92;#125;.

2012, Nicholas J. J. Smith, Logic: The Laws of Truth, Princeton University Press, page 448:

Let #92;mathcal#123;L#125;#92;alpha#95;#92;in be a version of the language of set-theory with urelements in which each occurrence of a quantifier is restricted by some V#95;#92;beta (#92;betalt;#92;alpha).

2013, Agustín Rayo, The Construction of Logical Space, Oxford University Press, page 95:

(set theory) A mathematical object which is not a set but which can be an element of a set.


Russell's paradox

Named after English mathematician, logician and philosopher Bertrand Russell.

named-after

One concludes that there must be something fishy about the Axiom of Comprehension, and, over time, the replacement of the Axiom of Comprehension by the Schema of Separation was seen to resolve Russell's paradox. Indeed, one cannot apply the Schema of Separation as in Russell's paradox unless one knows in advance that the collection of all sets is itself a set.

1999, R. C. Penner, Discrete Mathematics: Proof Techniques and Mathematical Structures, World Scientific, page 109:

The well-known theorem of Tarski that truth of sentences in any reasonably expressive language L cannot be defined in the language L itself is proved by a diagonalization argument similar to the argument involved in Russell's paradox.[…]It is usual to think that Russell's paradox excludes "large" sets like the universe, but this is actually not the case. An alternate solution to Russell's paradox (and other paradoxes) was proposed by Quine (1937) in his system "New Foundations" (NF): comprehension restricted to stratified formulae.

2001, M. Randall Holmes, “Tarski's Theorem and NFU”, in C. Anthony Anderson, Michael Zelëny, editors, Logic, Meaning and Computation: Essays in Memory of Alonzo Church, Springer (Kluwer Academic), page 469:

2013, Greg Frost-Arnold, Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science, Carus Publishing Company (Open Court), page 43,
Roughly, the idea is that Russell's paradox reveals that certain logics suffer serious problems, and therefore these logics should be avoided. […] Here again, Quine asserts that the real lesson of Russell's paradox is that we should give up quantifying over abstracta.

(set theory) The paradox that a set defined to contain all sets which do not contain themselves can neither consistently contain itself nor not contain itself.

Axiom schema of specification
Russell's paradox


Hasse diagram

Named after Helmut Hasse (1898–1979), though he was not the first to use them.

(set theory) A diagram which represents a finite poset, in which nodes are elements of the poset and arrows represent the order relation between elements. Transitivity of the order relation is tacit, in other words, if x<y and y<z then no arrow is drawn from x to z, but if there is no distinct z between x and y (such that x<z<y) then an arrow is draw from x to y.


pointclass

(mathematics, set theory) A collection of sets of points (ordinarily understood to be elements of some perfect Polish space), characterized by some sort of definability property.


antisymmetry

en-noun

antisymmetry (countable and uncountable, plural antisymmetries)

(mathematics, set theory) The condition of being antisymmetric.


connex

LL-Q1860 (eng)-Flame, not lame-connex.wav

(set theory) Such that, for all x and y in X, and for a binary relation R, either or both of xRy and yRx hold(s).


supercompact

en-adj

supercompact (comparative more supercompact, superlative most supercompact)

more supercompact (comparative)
most supercompact (superlative)

LL-Q1860 (eng)-Flame, not lame-supercompact.wav

(mathematics, set theory) Being a type of large cardinal with a variety of reflection properties.


Zorn's lemma

After Max August Zorn 1906-1993, German-born American mathematician. See German Zorn.

(set theory) A proposition of set theory stating that every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element.


real plane

(set theory) The Cartesian product of two real lines.


axiom of extensionality

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that two sets are equal if and only if they contain the same elements.


extensionality

en-noun

extensionality (usually uncountable, plural extensionalities)

ekstensjonalność

the principle of distinguishing only by externally observable properties

axiom of extensionality
axiom of extensionality

(mathematics, set theory) The principle, codified in the axiom of extensionality, that sets are equal if and only if they contain the same elements.


aleph number

(set theory) Any of a sequence of numbers used to represent the cardinality of infinite sets, denoted by the Hebrew letter aleph.


null sign

(set theory) The sign ∅, associated with empty sets.


Hartogs number

Hartogs numbers (plural)
Hartogs' number (alternative)

After German-Jewish mathematician Friedrich Hartogs (1874–1943).

1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160,
Let p be an infinite cardinal, |X|=p and let א=א(p) be the Hartogs number of p.

1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139,
If the Power Set Axiom is replaced by "א(x) is bound for every x" where
א(x)={a|∃f(f is one-to-one function from a into x)},
then the theory is denoted by ZFH (H stands for Hartogs' Number).

Since the proof of Hartogs' Theorem does not appeal to the Axiom of Choice, the Hartogs number of a set X exists whether or not X has a well-ordering.

2014, Barnaby Sheppard, The Logic of Infinity, Cambridge University Press, page 341:

(set theory) For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X.


semicreative

a semicreative set

(set theory) A type of set of natural numbers, related to mathematical logic.


trichotomous

From trichotomy + -ous. Modelled on dichotomous.

> is a trichotomous relation, while the relation "divides" over the natural numbers is not.

(set theory) Such that, for all x and y in X, and for a binary relation R, exactly one of xRy, yRx or x=y holds.


NBG

(set theory) von Neumann-Bernays-Gödel (axiomatic) set theory


Hintikka set

(set theory) A set with certain openness and completeness properties.


proper class

Thus, what is required is not that all proper classes be the size of the class of all sets, but that every proper class has at least as many members in #92;Omega#95;p as there are sets in #92;Omega#95;2, whatever is in #92;Omega#95;2.

1994, Shaughan Lavine, Understanding the Infinite, Harvard University Press, page 322:

2004, Hartry Field, The Consistency of the Naive Theory of Properties, Godehard Link, One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, page 308,
It is true that the absence of proper classes in ZF is sometimes awkward. It is also true that adding proper classes in the usual ways (either predicative classes as in Gödel—Bernays, or impredicative ones as in Morse-Kelley is conceptually unsettling: in each case (and especially in the more convenient Morse-Kelley case) they "look too much like just another level of sets", and the fact that there is no entity that captures the extension of predicates true of proper classes suggests the introduction of still further entities ("super-classes" that can have proper classes as members), and so on ad infinitum.

Any class is either a set or else a proper class. Every set is a class, but not all classes are sets. Specifically, the proper classes are not sets.

2009, Eric Steinhart, More Precisely: The Math You Need to Do Philosophy, Broadview Press, page 63:

(set theory) A class which is not a set.


Kurepa tree

Named after Yugoslav mathematician Đuro Kurepa.

named-after

(mathematics, set theory) A tree (T, <) of height ω₁, each of whose levels is at most countable, and has at least ℵ₂ branches.


well-orderable

(set theory, order theory) Capable of being well-ordered.


prewellordering

(mathematics, set theory) A binary relation that is transitive, total, and well-founded.


limit ordinal

(set theory) An ordinal number which is neither zero nor a successor of an ordinal.


premouse

(set theory) A generalization of a mouse that is not necessarily iterable.


choice function

en-noun

choice function (plural choice functions)

(set theory) A function whose domain is a family of nonempty sets, and which selects a member from each of those sets as its value.


dismantlable

(set theory) The property of an ordered set such that its elements can be listed in an order such that, for every element, the element is irreducible (has exactly one upper or lower cover) in the subset consisting of that element and all subsequent elements.


autoreducibility

Finally, we consider autoreducibility for ITRMs and show that randomness implies non-autoreducibility..

2015, Merlin Carl, “Randomness and Degree Theory for Infinite Time Register Machines”, in arXiv:

(mathematics, set theory) The state or quality of being autoreducible.


autoreducible

(mathematics, set theory) Of a set, that can be reduced to itself by a Turing machine that does not ask for its own input.


Scott's trick

(set theory) A method for giving a definition of equivalence classes for equivalence relations on a proper class, relying on the axiom of regularity but not on the axiom of choice.


comeager

(set theory) Alternative form of comeagre


comeagre

(set theory) Being the complement of a meagre set.


Galileo's paradox

Named after the scientist Galileo, though he did not originate the concept.

(set theory) A demonstration of a surprising property of infinite sets. Some positive integers are squares while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares; yet for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.


quasireflexive

(set theory) Of a relation ~ on a set S: such that every element that is related to some element is also related to itself, i.e. ∀x,y∈S: x~y ⇒ x~x ∧ y~y.


hypercontinuum

We show that negative iterates of a stable arc converges to this hypercontinuum in the considered metric.

2015, Alfonso Artigue, “Pseudo-Anosov maps and continuum theory”, in arXiv:

(set theory) The set of all subsets of a set of with the (infinite) cardinality of the continuum.


semilow

We show that in every non low \ce degree, there are sets with semilow#95;#123;1.5#125; complements without semilow complements as well as sets with semilow#95;2 complements and the outer splitting property that do not have semilow#95;#123;1.5#125; complements.

2015, Peter Cholak, Rachel Epstein, “Computably Enumerable Sets that are Automorphic to Low Sets”, in arXiv:

(set theory) Such that there exists a nonempty intersection (with another set) that is low.


surject

en-verb

surject (third-person singular simple present surjects, present participle surjecting, simple past and past participle surjected)

surjects (present singular third-person)
surjecting (participle present)
surjected (participle past)
surjected (past)

(mathematics, set theory) To form or to undergo surjection


directed set

(algebra, set theory) A set equipped with a preorder such that for any two elements a and b there is an element c such that a ≤ c and b ≤ c.


linear order

A set of numbers on a number line form a linear order.


pairwise disjoint

Proposition 4.5. Every permutation is a product of pairwise disjoint cycles, and this decomposition is unique up to the order of the terms.

2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 61:

2009, John M. Franks, A (Terse) Introduction to Lebesgue Integration, American Mathematical Society, page 27,
For example, if we had a collection of pairwise disjoint intervals of length 1/2,1/4,1/8,…1/2ⁿ,…,etc., then we would certainly like to be able to say that the measure of their union we is the sum ∑1/2ⁿ=1 which would not follow from finite additivity.

To show that all #92;Gamma#95;i-translates of F#95;i, are pairwise disjoint, it suffices to show that all #92;Gamma#95;#123;i,0#125;-translates of F#95;i are pairwise disjoint, since then the argument as above will show inductively that the #92;Gamma#95;#123;i,m#125;-translates of F#95;i are pairwise disjoint for all mgt;0.

2015, Su Gao, Stephen C Jackson, Brandon Seward, Group Colorings and Bernoulli Subflows, American Mathematical Society, page 158:

(mathematics, set theory, of a collection of two or more sets) Let A_λ_(λ∈Λ) be any collection of sets indexed by a set Λ. We call the indexed collection pairwise disjoint if for any two distinct indices, λ,μ∈Λ, the sets A_λ and A_μ are disjoint.

mútuament disjunts

such that any two distinct sets are disjoint


transitive set

(set theory) Any set X such that for any x ∈ X, if y ∈ x then y ∈ X; equivalently, such that if x ∈ X and x is not an urelement then x ⊆ X.


transitive closure

en-noun

transitive closure (plural transitive closures)

Transitive closure
Transitive set#Transitive closure

(set theory, order theory, of a binary relation R on a set X) The smallest binary relation on X that includes R and is transitive.

(set theory, of a set X) The smallest transitive set that contains X.


totally ordered

1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,
(A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above).

The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.

1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:

G is a transitive group of order automorphisms of a totally ordered set X,
θ is a convex G-congruence on the totally ordered set X,
ξ is the order type of the totally ordered set X,
◌̅η is the order type of some class Y=xη of the congruence η,

1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation

ζ is the order type of the totally ordered quotient set X/θ of the totally ordered set X by the congruence θ.

(set theory, order theory) That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset.