transfinite

From German transfinit, coined by Georg Cantor, equivalent to trans- + finite.

(mathematics) Relating to transfinite numbers.


uncountable

The reasons for our failure were as uncountable as the grains of sand on a beach.

Mr. Chambre has since spent uncountable hours and even more uncountable dollars buying upward of 23,000 documents, and he figures to be just hitting speed.

1988 July 13, Douglas Martin, “About New York; Tasting History Through Papers That Age Well”, in The New York Times:

So many as to be incapable of being counted.

Cantor’s “diagonal proof” shows that the set of real numbers is uncountable.

(mathematics) Incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof.

ontelbaar

mathematics: incapable of being enumerated by natural numbers

Many languages do not distinguish countable nouns from uncountable nouns.

One meaning in law of the usually uncountable noun "information" is used in the plural and is countable.

(grammar, of a noun) That cannot be used freely with numbers or the indefinite article, and therefore usually takes no plural form. Example: information.

անհաշվելի

linguistics: about a noun which cannot be counted


uncountable

But inherent uncountables such as 'stuffs' can be conceptualized in two different ways, depending on whether they are viewed in terms of quantity or in terms of quality.

1988, Anna Wierzbicka, The Semantics of Grammar, page 440:


cardinal number

en-noun

cardinal number (plural cardinal numbers)

This cardinal number is the smallest of the infinite cardinal numbers; it is the one to which Cantor has appropriated the Hebrew Aleph with the suffix 0, to distinguish it from larger infinite cardinals. Thus the name of the smallest of infinite cardinals is ℵ₀.

1920, Bertrand Russell, Introduction to Mathematical Philosophy, page 83:

(mathematics) A generalized kind of number used to denote the size of a set, including infinite sets.


CH

CH (countable and uncountable, plural CHs)

continuum hypothesis
continuum hypothesis#English

(set theory) Initialism of continuum hypothesis.


ideal

en-adj

ideal (comparative more ideal, superlative most ideal)

more ideal (comparative)
most ideal (superlative)

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.

ideal point

An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

(mathematics) Not actually present, but considered as present when limits at infinity are included.


end

beginning

antonym(s) of “final point of something”

→ Japanese: エンド

From Middle English ende, from Old English ende, from Proto-West Germanic *andī, from Proto-Germanic *andijaz, from Proto-Indo-European *h₂entíos, from *h₂ent- (“front, forehead”).
See also Dutch einde, German Ende, Norwegian ende, Swedish ände; also Old Irish ét (“end, point”), Latin antiae (“forelock”), Albanian anë (“side”), Ancient Greek ἀντίος (antíos, “opposite”), Sanskrit अन्त्य (antya, “last”). More at and and anti-.
The verb is from Middle English enden, endien, from Old English endian (“to end, to make an end of, complete, finish, abolish, destroy, come to an end, die”), from Proto-Germanic *andijōną (“to finish, end”), denominative from *andijaz.

der

snitches get stitches and end up in ditches

(mathematics) An ideal point of a graph or other complex. See End (graph theory)


forever

/fəˈɹɛvə(ɹ)/

Univerbation of for ever, from Middle English for ever, for evere. By surface analysis, for + ever. First attested in the late 14c., and first attested in the late 17c. as one word. Noun first attested in 1858.

I shall love you forever.

Secondly, When a body is once in motion it will continue to move forever, unless something stops it. When a ball is struck on the surface of the earth, the friction of the earth and the resistance of the air soon stop its motion.

1839, Denison Olmsted, A Compendium of Astronomy, page 95:

If you want a picture of the future, imagine a boot stamping on a human face — for ever.

1949, George Orwell, Nineteen Eighty-Four:

(duration) For all time, for all eternity; for a lifetime; for an infinite amount of time.

ለዘለዓለም

for all time, for all eternity; for an infinite amount of time

για πάντα (gia pánta, literally “for always”)


cardinal

en-noun

cardinal (countable and uncountable, plural cardinals)

From Middle French cardinal, from Latin cardinālis (“pertaining to a hinge, hence applied to that on which something turns or depends, important, principal, chief”), from cardin-, cardō (“hinge”) + -ālis, adjectival suffix.

der

His uncle, a Cardinal, engages a Spanish youth of Moorish descent called Diego, an expert singer and player on the virginal, to unlock the secrets of the heart,[…]and cure him by the spell of his music.

1932, Maurice Baring, chapter 16, in Friday's Business:

Roman Catholic Church
Roman Catholic Church

(Roman Catholicism) One of the officials appointed by the pope in the Roman Catholic Church, ranking only below the pope and the patriarchs, constituting the special college which elects the pope.

Breezes blowing from beds of iris quickened her breath with their perfume; she saw the tufted lilacs sway in the wind, and the streamers of mauve-tinted wistaria swinging, all a-glisten with golden bees; she saw a crimson cardinal winging through the foliage, and amorous tanagers flashing like scarlet flames athwart the pines.

1907 August, Robert W[illiam] Chambers, “Afterglow”, in The Younger Set, New York, N.Y.: D. Appleton & Company, →OCLC, page 168:

Any of various species of New-World passerine songbird in the genus Cardinalis, or in the family Cardinalidae more generally, or of similar appearance and once considered to be related to the former; so called because of their red plumage. (See Wikipedia article for taxonomical information.)

кардинал

bird of the genus Cardinalis or family Cardinalidae

This cardinal number is the smallest of the infinite cardinal numbers; it is the one to which Cantor has appropriated the Hebrew aleph with the suffix 0, to distinguish it from larger infinite cardinals. Thus the name of the smallest of infinite cardinals is ₀א.

(mathematics) Short for cardinal number, a number indicating quantity, or the size of a set (e.g., 0, 1, 2, 3). (See Cardinal_number.)


infinity

en-noun

infinity (countable and uncountable, plural infinities)

From Middle English infinite, from Old French infinité, from Latin infinitas (“unlimitedness”), from negative prefix in- (“not”), + finis (“end”), + noun of state suffix -tas.


eternal

en-adj

eternal (comparative more eternal, superlative most eternal)

more eternal (comparative)
most eternal (superlative)

From Middle English eternal, from Old French eternal, from Late Latin aeternālis, from Latin aeternus (“eternal”), from aevum (“age”). Displaced native Old English ēċe.

hope springs eternal in the human breast

But here again it is another question, quite different from our having an idea of eternity, to know whether there were any real being, whose duration has been eternal.

1690, John Locke, An Essay Concerning Human Understanding:

Thy smoking altar shall be fat with food / Of incense and the grateful steam of blood; / Burnt-offerings morn and evening shall be thine, / And fires eternal in thy temple shine.

1700 [c. 1387–1400], John, transl. Dryden, “Palamon and Arcite”, in Fables, Ancient and Modern, translation of The Knight's Tale by Geoffrey Chaucer:

Organic life is nothing but a genetic mutation, an accident. Your lives are measured in years and decades. You wither and die.
We are eternal. The pinnacle of evolution and existence. Before us, you are nothing. Your extinction is inevitable. We are the end of everything.

2008, BioWare, Mass Effect (Science Fiction), Redwood City: Electronic Arts, →ISBN, →OCLC, PC, scene: Virmire:

In a bid to understand the eternal mystery that is woman, Bart goes to the least qualified possible source for advice and counsel: his father, who remarkably seems to have made it to his mid-30s without quite figuring out much of anything.

2012 May 27, Nathan Rabin, “TV: Review: THE SIMPSONS (CLASSIC): “New Kid On The Block” (season 4, episode 8; originally aired 11/12/1992)”, in The Onion AV Club:


ad infinitum

Unadapted borrowing from Latin ad infinitum (“to infinity”).

ubor

The idea of eternal return is a mysterious one, and Nietzsche has often perplexed other philosophers with it: to think that everything recurs as we once experienced it, and that the recurrence itself recurs ad infinitum!

1984, Milan Kundera, part I (Lightness and Weight), chapter I, in Michael Henry Heim, transl., The Unbearable Lightness of Being, London: Faber and Faber, translation of Nesnesitelná lehkost bytí, published 1985, →ISBN, page 3:


countable

[ˈkʰãʊ̯̃n.(ɾ)ə.bɫ̩]

From Middle English countable, equivalent to count (“to enumerate”) + -able.

可數 /可数

math: having a bijection with a subset of the natural numbers

en:either finite or countably infinite, in mathematics

(mathematics, of a set) Finite or countably infinite; having a one-to-one correspondence (bijection) with a subset of the natural numbers.

(mathematics, of a set) Countably infinite; having a bijection with the natural numbers.

numeroituva

math: having a bijection with the natural numbers


ocean

en-noun

ocean (countable and uncountable, plural oceans)

From Middle English *ocean, occean, occian, occyan, from Old French occean (later reborrowed or reinforced by Middle French ocean), from Latin Ōceanus, originally from Ancient Greek Ὠκεανός (Ōkeanós, “Oceanus”, a water deity). Displaced native Old English gārseċġ. Also commonly referred to as the ocean sea, the sea of ocean (compare Latin mare ōceanum; Old French mer oceane, occeanne mer). Compare Saterland Frisian Oceoan (“ocean”), West Frisian oseaan (“ocean”), Dutch oceaan (“ocean”), German Low German Ozeaan (“ocean”), German Ozean (“ocean”), Danish ocean (“ocean”), Swedish ocean (“ocean”), French océan (“ocean”), Italian oceano (“ocean”). Doublet of Oceanus/Okeanos.

der

it's not the size of the ship, it's the motion of the ocean

the boundless ocean of eternity

an ocean of difference

On the present occasion, Traddles was so hemmed in by the pagoda and the guitar-case, and Dora’s flower-painting, and my writing-table, that I had serious doubts of the possibility of his using his knife and fork; but he protested, with his own good-humour, ‘Oceans of room, Copperfield! I assure you, Oceans!’

1849 May – 1850 November, Charles Dickens, The Personal History of David Copperfield, London: Bradbury & Evans, […], published 1850, →OCLC:

Poison Ivy, Lord,'ll make you itch
You're gonna need an ocean
Of calamine lotion
You'll be scratchin' like a hound
The minute you start to mess around

1959, “Poison Ivy”, Jerry Leiber and Mike Stoller (lyrics), performed by The Coasters:

(figuratively) An immense expanse; any vast space or quantity without apparent limits.


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 Latin infīnītus, from in- (“not”) + fīnis (“end”) + the perfect passive participle ending -itus. Doublet of infinito. Displaced native Old English unġeendodlīċ.

Huxley's theory says that if you provide infinite monkeys with infinite typewriters, some monkey somewhere will eventually create a masterpiece – a play by Shakespeare, a Platonic dialogue, or an economic treatise by Adam Smith.

2012, Helen Donelan, Karen Kear, Magnus Ramage, Online Communication and Collaboration: A Reader:

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.


asymptote

circa 1650, from Ancient Greek ἀσύμπτωτη (asúmptōtē), the feminine of Apollonius Pergaeus' (circa 200 BC) Ancient Greek adjective ἀσύμπτωτος (asúmptōtos, “not falling together”), from ἀ- (a-, “not”) + συν- (sun-, “together”) + πτωτός (ptōtós, “fallen”).

af

The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F₁) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore.

1974, Stanisław Lem, translated by Michael Kandel, The Cyberiad:

mathematical analysis
mathematical analysis

(mathematical analysis) A straight line which a curve approaches arbitrarily closely as it goes to infinity. The limit of the curve; its tangent "at infinity".

asintota

a straight line which a curve approaches arbitrarily closely


asymptote

en-verb

asymptote (third-person singular simple present asymptotes, present participle asymptoting, simple past and past participle asymptoted)

asymptotes (present singular third-person)
asymptoting (participle present)
asymptoted (participle past)
asymptoted (past)

2006: Neil deGrasse Tyson, The Perimeter of Ignorance
As you become more scientific, yes, the religiosity drops off, but it asymptotes.

(mathematical analysis) To approach, but never quite touch, a straight line, as something goes to infinity.


eternity

en-noun

eternity (countable and uncountable, plural eternities)

sempiternity

antonym(s) of “existence outside of time”

From Middle English eternyte, from Old French eternité, from Latin aeternitās. Displaced native Old English ēcnes.

Eternity has generally been considered as divisible into two parts; which have been termed, eternity a parte ante, and eternity a parte post: that is, in plain English, that eternity which is past, and that eternity which is to come.

1829, John Wesley, Sermons on Several Occasions, 10th edition, volume 2, Sermon LVIII: On the Eternity of God, page 1:

This theory regards creation as an act of God in eternity past.

1886, Augustus Hopkins Strong, Systematic Theology: a Compendium and Commonplace-book Designed for the Use of Theological Students, page 190:

Those who like not the company of the saints on earth will get none of it in eternity; but, as godless company is their delight now, they will afterwards get enough of it, when they have eternity to pass in the roaring and blaspheming society of devils and reprobates in hell.

2000, Thomas Boston, Human Nature in Its Fourfold State, page 247:

(uncountable) Existence without end, infinite time.

We sometimes speak of eternity in contradistinction to time; and often say, "through time and into eternity;" and again "from eternity to eternity," which is simply another form of expressing the same idea, and "pass through time into eternity." in other words, time is a short period allotted to man in his probationary state—and we use the word time in contradistinction to the word eternity, merely for the accommodation of man in his finite sphere, that we may comprehend and learn to measure periods.

1879, Erastus Snow, “Rest Signifies Change, etc.”, in Brigham Young, editor, Journal of Discourses, volume 21, published 1881:

(uncountable, philosophy, theology) Existence outside of time.

Every niche was filled by a funeral urn, and by marble shapes that bent down in a pale eternity of sorrow.

1837, L[etitia] E[lizabeth] L[andon], “The Marriage”, in Ethel Churchill: Or, The Two Brides. […], volume I, London: Henry Colburn, […], →OCLC, page 220:

(countable) A period of time which extends infinitely far into the future.

أبَدِيَّة

time extending infinitely far into the future

de eeuwigheid

definite article required; often capitalized

It's been an eternity since we last saw each other.

I had to wait in the station for ten days - an eternity.

1899 February, Joseph Conrad, “The Heart of Darkness”, in Blackwood’s Edinburgh Magazine, volume CLXV, number M, New York, N.Y.: The Leonard Scott Publishing Company, […], →OCLC, part I, page 208:

(informal, hyperbolic) A comparatively long time.


boundless

en-adj

boundless (comparative more boundless, superlative most boundless)

more boundless (comparative)
most boundless (superlative)

'Tis the cruel gripe, / That lean hard-handed poverty inflicts, / The hope of better things, the chance to win, / The wiſh to ſhine, the thirſt to be amus'd, / That at the found of Winter's hoary wing, / Unpeople all our counties, of ſuch herds, / Of flutt'ring, loit'ring, cringing, begging, looſe, / And wanton vagrants, as make London, vaſt / And boundless as it is, a crowded coop.

1785, William Cowper, “The Garden”, in The Task, a Poem, in Six Books. By William Cowper […] To which are Added, by the Same Author, An Epistle to Joseph Hill, Esq. Tirocinium, or a Review of Schools, and The History of John Gilpin, London: Printed for J[oseph] Johnson, No. 72 St. Paul's Church-Yard, OCLC 221351486; republished as The Task. A Poem. In Six Books. To which is Added, Tirocinium: or, A Review of Schools, new edition, Philadelphia, Pa.: Printed for Thomas Dobson, bookseller, in Second-street, second door above Chestnut-street, 1787, OCLC 23630717, page 87


unlimited

The Scottish Region is issuing a Day Rail-Rover Ticket, available at 12 hours' notice, permitting unlimited travel in an area bounded by Edinburgh, St. Andrews, Dundee, Perth, Comrie, Callander, Stirling and Falkirk for 25s. (children, 12s. 6d.). . . .

1960 September, “Talking of Trains: News in Brief”, in Trains Illustrated, page 523:

limitless or without bounds; unrestricted


convergence

en-noun

convergence (countable and uncountable, plural convergences)

From converge + -ence (after Latin convergentia).

(mathematics) The process or property of approaching some limiting value; typically of an infinite series.

сходимост

the process of approaching some limiting value


lemniscate

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

From Latin lēmniscātus (“decorated with ribbons”).


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.


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.


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.


endless

From Middle English endeles, from Old English endelēas (“endless”), from Proto-Germanic *andijalausaz (“endless”), equivalent to end + -less.

endless time; endless praise

Trains from Lime Street to Edge Hill were hauled by an endless hempen rope worked by a stationary engine on the platform at the latter station.

1942 May-June, “Cable Operation at Liverpool and London”, in Railway Magazine, page 174:


cardinality

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.


universal set

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


vanishing point

vanishing point (plural vanishing points)

At 8h 35' 4", the least dent possible, quite black, appeared in the sun's limb. And at 8h 35' 6", the limb was restored to its perfect form, there having been a small trembling light, between the narrow watery border of Venus and the vanishing point of contact in the sun's limb for these two seconds of time.

1762, Samuel Dunn, “Some Observations of the Planet Venus, on the Disk of the Sun, June 6th, 1761,”, in Philosophical Transactions (1683-1775), volume 52, page 192:

It might cut the tuberculosis deathrate to the vanishing point.

1923 June 25, “Tuberculosis at Bay”, in Time:

It hit 100 m.p.h. with ease and sped toward the vanishing point down Highway 93A.

2003 August 25, Sally B. Donnelly, “The Need for Speed”, in Time:

The situation in which, place where, or point in time when some object or phenomenon is no longer observable or notable.

katoamispiste

point when an object or phenomenon is no longer observable


infinitely

en-adv

infinitely (comparative more infinitely, superlative most infinitely)

more infinitely (comparative)
most infinitely (superlative)

In an infinite manner; as of anything growing without bounds; endlessly.


limitless

en-adj

limitless (comparative more limitless, superlative most limitless)

more limitless (comparative)
most limitless (superlative)

Without limits in extent, size, or quantity; boundless.


everlasting

everlasting (comparative more everlasting, superlative most everlasting)

more everlasting (comparative)
most everlasting (superlative)

finite

antonym(s) of “existing or continuing without end”

/ˌɛvɚˈlæstɪŋ/

/ˌɛvəˈlɑːstɪŋ/

From Middle English, equivalent to ever + lasting.

(philosophy) Existing with infinite temporal duration (as opposed to existence outside of time).

sempiternal

existing with infinite temporal duration


continuum

en-noun

continuum (plural continuums or continua)

Borrowed from Latin continuum, neuter form of continuus, from contineō (“contain, enclose”).

(mathematics) The nondenumerable set of real numbers; more generally, any compact connected metric space.


celestial sphere

celestial sphere (plural celestial spheres)

(astronomy) The apparent outer limit of space; an abstract sphere of infinite radius which serves as the imaginary backdrop for celestial objects, and of which the visible sky is one hemisphere.


infiniteness

infiniteness (countable and uncountable, plural infinitenesses)

limitedness

antonym(s) of “quality of being immeasurably large”


boundlessness

boundlessness (usually uncountable, plural boundlessnesses)

The boundlessness of the night sky mocks our sense of scale.

The property of being boundless, of being without limits or ends.


infinite recursion

infinite recursion (countable and uncountable, plural infinite recursions)

(programming) Any recursion that continues without end.


undying

en-adj

undying (comparative more undying, superlative most undying)

more undying (comparative)
most undying (superlative)

My abandoned heart

Just doesn't understand

My undying love for you

Won't let me wait - Janet Jackson - Come Back to Me


infinite series

en-noun

infinite series (plural infinite series)

In 1734, Leonhard Euler solved the Basel problem by summing the infinite series #92;textstyle#92;frac#123;1#125;#123;1²#125;#43;#92;frac#123;1#125;#123;2²#125;#43;#92;frac#123;1#125;#123;3²#125;#43;#92;cdots#61;#92;sum#95;#123;n#61;1#125;#123;#92;infty#125;#92;frac#123;1#125;#123;n²#125;#61;#92;frac#123;#92;pi²#125;#123;6#125;.

(mathematical analysis) Any expression that represents the addition a countably infinite number of ordered summands, often explicitly indexed by the positive or nonnegative integers.


continued fraction

continued fraction (plural continued fractions)

Several results focus on the famous Rogers–Ramanujan continued fraction [47], [48, pp. 214-215], the only continued fraction appearing in Ramanujan's published papers.

1992, G. E. Andrews, B. C. Berndt, L. Jacobsen, R. L. Lamphere, “The Continued Fractions Found in the Unorganized Portions of Ramanujan's Notebooks”, in Memoirs of the American Mathematical Society, volume 99, number 477, page 1:

2000, Andrew Zardecki, 18: Continued Fractions in Time Series Forec[a]sting, Da Ruan (editor), Fuzzy Systems and Soft Computing in Nuclear Engineering, Physica-Verlag, Studies in Fuzziness and Soft Computing, page 397,
We achieve this by using well-known examples from the number theory pertaining to the continued fractions. Any sequence of natural numbers drawn from the probability distribution of the quotients of the continued fraction corresponding to an irrational number represents a typical sequence, in the sense that almost all sequences of quotients have this distribution.

We have seen that all rational numbers, and no other number, can be expressed as finite simple continued fractions.
The main reason of interest of continued fractions, however, is in their application to the representation of irrational numbers. To that end we shall need infinite simple continued fractions.

2009, M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo, translated by Daniele Gewurz, Elementary Number Theory, Cryptography and Codes, page 48:

(mathematics, number theory) A compound numerical expression consisting of an integer plus a fraction whose numerator is a positive integer and whose denominator is a continued fraction (an integer plus a fraction), and so on, with finite or infinite recursion.


convergent

en-adj

convergent (comparative more convergent, superlative most convergent)

more convergent (comparative)
most convergent (superlative)

(mathematical analysis, topology) Of a (typically infinite) sequence or series in a topological space, converging to a (finite, proper) limit.


convergent

(mathematics) The rational number obtained when a continued fraction is terminated after a finite number of terms.


endlessness

en-noun

endlessness (usually uncountable, plural endlessnesses)

From Middle English endelesnesse, from Old English endelēasnes; equivalent to endless + -ness.

The state or characteristic of being endless.


uncountable set

uncountable set (plural uncountable sets)

(mathematics) A set, containing infinite number of elements, whose elements can not be mapped one-to-one to the natural numbers. A set with a cardinality greater than that of the set of natural numbers.


real line

(mathematics) The real number system, viewed as a line.


countably

(mathematics) In a way that can be put into one-to-one correspondence with the natural numbers or any subset thereof.


infinitate

en-verb

infinitate (third-person singular simple present infinitates, present participle infinitating, simple past and past participle infinitated)

infinitates (present singular third-person)
infinitating (participle present)
infinitated (participle past)
infinitated (past)


uncountably

From uncountable + -ly or un- + countably.

The stars in the sky are uncountably many. Even a lifetime would not suffice to number them all.

And the dimensions of death that can result from such systems tripped in error, or through misperceptions of reality, are uncountably greater than those

1988 July 6, “Systems Easily Tripped in Error Bring Death in a Lake, Warning Us of...”, in Los Angeles Times:

a host of other producers fear that a vital link to New York's uncountably diverse populations is about to be cut.

1990 September 20, “Unplugging A Diverse Bit Of Cable TV”, in New York Times:

And the memory of man runneth not to a year when there was an episode of disproportion comparable to the planet-wide vapors occasioned by one of the year's uncountably numerous automobile accidents, this one in Paris.

1997 December 29, “Vapors And Serenity”, in Newsweek:

and the sheer quantity of material is uncountably greater

2006, Geoffrey Hawthorn, edited by David R. Olson and Michael Cole, Technology, Literacy and the Evolution of Society, Orality in Politics, page 185:

Stories of road travel in the United States have taken uncountably many forms, from John Steinbeck's "Travels With Charley" and Jack Kerouac's "On the Road" to "Little Miss Sunshine".

2007 January 17, “Hippie and redneck show is equal parts insane, inane”, in Boston Globe:

Too many to be counted (either by reason of being infinite or for practical constraints).


bottomless pit

A pit with no visible bottom and apparently infinite depth.


apeirogon

/əˈpiːɹɵɡɑn/

/əˈpeɪ̯ɹɵɡɑn/

(mathematics, geometry) A type of generalised polygon with a countably infinite number of sides and vertices;

A type of generalised polygon with a countably infinite number of sides and vertices;
(in the regular case) the limit case of an n-sided regular polygon as n increases to infinity and the edge length is fixed; typically imagined as a straight line partitioned into equal segments by an infinite number of equally-spaced points.


infinite regress

en-noun

infinite regress (plural infinite regresses)

(logic) A regress into an infinite sequence of propositions in an attempt to found the truth of the proposition Pᵢ on the truth of the proposition Pᵢ₊₁.


infinityfold

(rare) By a multiple of infinity; by infinity times as much or as many.


beginningless

Backwards, it spreads to the beginningless beginning. Forwards, it spreads to the endless end. We are always in the middle.

1980, Rajneesh, Osho, Yoga Vivek, Prem Chinmaya, Ananda Prem, The Supreme Doctrine: Discourses on the Kenopanishad:

Beginningless time is one of those concepts which, in our human state, we find difficult to comprehend.

2004, David Michie, Buddhism for Busy People:

Eternal; uncreated; having always existed.


Cantorian

en-adj

Cantorian (comparative more Cantorian, superlative most Cantorian)

more Cantorian (comparative)
most Cantorian (superlative)

Of or pertaining to Georg Cantor (1845-1918), German mathematician and creator of set theory.


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).


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


Peano curve

Named after its discoverer, Giuseppe Peano.

(mathematical analysis) A space-filling curve in the 2-dimensional plane.


space-filling curve

en-noun

space-filling curve (plural space-filling curves)

From space-filling curves it can be deduced that ℝ² and ℝ³ have the same cardinality as ℝ.

(mathematical analysis) a curve whose range contains the entire 2-dimensional unit square (or the 3-dimensional unit cube)


hyperreal number

hyperreal number (plural hyperreal numbers)

nonstandard analysis
nonstandard analysis

(mathematics) Any of a set of numbers which extends the set of real numbers by including infinitesimals and infinities, and is used in nonstandard analysis.


improper integral

improper integral (plural improper integrals)

(calculus) An integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity.


infinite loop

(programming) A loop which continues indefinitely.


semi-infinite

Limited at one end but extending to infinity in the other direction.


infinitary

(mathematics, logic) Of or pertaining to expressions of infinite length


superreal

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infinitism

(philosophy) The view that knowledge may be justified by an infinitely long chain of reasoning.


infinity pool

“Have you ever swum in an infinity pool?” Cattie had seen pictures. An infinity pool had one side that just … ended. Like the end of the world. Water went right up to the edge and fell away.

2012, Pete Hautman, All-in, Simon and Schuster, →ISBN, page 23:

infinity-edge pool
infinity-edge pool#English


infinity-edge pool

infinity-edge pool (plural infinity-edge pools)

[…] the American architect John Lautner is considered the father of infinity-edge pool design. His vanishing edge of water at the Silver Top residence in Silver Lake, California, received rave reviews in 1957 and fueled the fires of infinity-edge experimentation.

2012, Kim Inglis, Bali Home: Inspirational Design Ideas, Tuttle Publishing, →ISBN:

A swimming pool that produces a visual effect of water extending to the horizon or vanishing into a lake, bay, etc.


real number line

real number line (plural real number lines)

(mathematics) A geometric representation of the real number system.


extended real number system

extended real number system (plural real number systems)

(mathematics) The real number system adjoined with two extra symbols: ∞ and −∞, inheriting the ordering of the real number system, and defining −∞ to be less than any real number, and ∞ to be larger than any real number.


infinite monkey theorem

(informal) The proposition that a monkey hitting typewriter keys at random is bound to produce meaningful text (such as a work by William Shakespeare), given an infinite amount of time. Often extended to a ludicrous number of monkeys, such as a thousand or an infinite number.


Tipler cylinder

Tipler cylinder (plural Tipler cylinders)

Named after American physicist Frank J. Tipler, who recognized the possibility of such a device in 1974.

A hypothetical massive, infinitely long cylinder spinning along its longitudinal axis, enabling time travel.


hypertask

(philosophy) An uncountably infinite number of operations that occur sequentially within a finite interval of time.


supertask

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One might object that this scenario is truly a fiction, but the recent literature on supertasks shows that, although their reality is definitely still not established, at least its plausibility can be argued for.

2008 January 31, Bart Van Kerkhove, Jean Paul Van Bendegem, “Pi on Earth, or Mathematics in the Real World”, in Erkenntnis, volume 68, number 3, →DOI:

(philosophy) A quantifiably infinite number of operations that occur sequentially within a finite interval of time.


aleph number

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


infinity symbol

infinity symbol (plural infinity symbols)

A lemniscate; the figure ∞ that represents infinity.


projective plane

projective plane (plural projective planes)

Generally, a projective plane of "order" N has N²+N+1 points, and the same number of lines.

2004, David S. Dummit with Richard M. Foote, chapter 6, in Abstract Algebra, page 210

(geometry) A plane in which any pair of distinct points determine a line and any pair of distinct lines intersect at a point.


Ananta

From Sanskrit अनन्त (ananta, “endless”).

A male given name from Sanskrit, commonly used in India.


apeiro-

From Ancient Greek ἄπειρος (ápeiros, “infinite, boundless”).


apeirohedron

en-noun

apeirohedron (plural apeirohedrons or apeirohedra)

apeirohedrons (plural)
apeirohedra (plural)

/əˌpiːɹɵˈhiːdɹən/

/əˌpeɪ̯ɹɵˈhiːdɹən/

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There are exactly 12 regular apeirohedra that in some sense are reducible and have components that are regular figures of dimensions 1 and 2. These apeirohedra are blends of a planar regular apeirohedron, and a line segment { } or linear apeirogon {∞}. This explains why there are 12 = 6·2 blended (or non-pure) apeirohedra. For example, the blend of the standard square tessellation {4,4} and the infinite apeirogon {∞}, denoted {4,4}#{∞}, is an apeirohedron whose faces are helical apeirogons (over squares), rising above the squares of {4,4}, such that 4 meet at each vertex; the orthogonal projections of {4,4}#{∞} onto their component subspaces recover the original components, the square tessellation and the linear apeirogon.

2014, Daniel Pellicer with Egon Schulte, “Polygonal Complexes and Graphs for Crystallographic Groups”, in Robert Connelly, Asia Ivić Weiss, Walter Whiteley, editors, Rigidity and Symmetry, New York, N.Y.: Springer, →ISBN, page 331:

(mathematics, geometry) A polyhedron with an infinite number of faces.


surreal number

Coined by Donald Knuth in his 1974 novelette Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. The concept had been developed by British mathematician John Conway for his game theoretic research of the board game go. Conway had simply called them numbers, but subsequently adopted Knuth's term and used it in his 1976 book On Numbers and Games.

Conway's construction of surreal numbers relies on the use of transfinite induction.

Conway's approach was to build numbers from scratch using a construction inspired by his game theory research; the resulting class of surreal numbers proved much larger than the class of real numbers.

1986, Harry Gonshor, “An Introduction to the Theory of Surreal Numbers”, in Paperback, Cambridge University Press, published 1987, →ISBN:

• A surreal number X=(X_L,X_R) consists of two sets X_L and X_R of surreal numbers, such that no element from X_L is greater than any element from X_R.
• A surreal number Y=(Y_L,Y_R) is greater than another surreal number X=(X_L,X_R), X<Y, if and only if
− there is no x∈X_L such that Y<x, and
− there is no y∈Y_R such that y<X.

2012, Fredrik Nordvall Forsberg, Anton Setzer, A Finite Axiomatisation of Inductive-Inductive Definitions, Ulrich Berger, Hannes Diener, Peter Schuster, Monika Seisenberger (editors), Logic, Construction, Computation, Ontos Verlag, page 263, The class² of surreal numbers is defined inductively, together with an order relation on surreal numbers wich is also defined inductively

Here we shall follow Conway's exposition rather closely. Let L and R be two sets of numbers. Assume that no member of L is greater than or equal to any member of R. Then #92;#123;L#92;vertR#92;#125; is a surreal number. All surreal numbers are constructed in this fashion.

2018, Steven G. Krantz, Essentials of Mathematical Thinking, Taylor & Francis (Chapman & Hall/CRC Press), page 247:

(mathematics) Any element of a field equivalent to the real numbers augmented with infinite and infinitesimal numbers (respectively larger and smaller (in absolute value) than any positive real number).

Donald Knuth
John Horton Conway
On Numbers and Games
surreal number

nombre surréel

element of an extension of the real numbers that includes infinite numbers and infinitesimals


beginninglessness

Quality of having no beginning; eternalness.


Hilbert cube

Named after German mathematician David Hilbert (1862–1943)

named-after

topological space
topological space#English

(topology) A metrizable topological space which is the product of a countably infinite number of closed intervals.


Tychonoff cube

Named after Russian mathematician Андре́й Никола́евич Ти́хонов (Andréj Nikolájevič Tíxonov) (1906–1993).

lang

Андре́й Никола́евич Ти́хонов

Андре́й Никола́евич Ти́хонов

xlit

Андре́й Никола́евич Ти́хонов

Андре́й Никола́евич Ти́хонов

named-after

Андре́й Никола́евич Ти́хонов

A topological space which is a product of infinitely many closed intervals.


Cantor set

Named after German mathematician Georg Cantor (1845–1918).

The Lebesgue null sets include not only all countable sets but many sets having the cardinality of the continuum. We now present the standard example, the Cantor set, which is also of interest for other reasons.

1999, Gerald B. Folland, Real Analysis : Modern Techniques and Their Applications, 2nd edition, New York: John Wiley & Sons, Inc., →ISBN, →OCLC, §1.5, page 38:

connected component
connected component#English

(mathematical analysis, topology) A subset of an interval formed by recursively removing an interval in the middle of every connected component of the set.


infinitieth

Every number whose expression in decimals requires but a finite number of places of decimals is commensurable. Therefore, incommensurable numbers suppose an infinitieth place of decimals.

1892, Charles Sanders Peirce, The Law of Mind:

Watling's suggestion is of no help at all in explaining how to form an actual infinite by successive addition. To do so would necessitate adding an 'infinitieth' element, which is absurd.

1979, William Lane Craig, The Kalām Cosmological Argument, page 183:

I arrived at the foot of the tower with me hearty Master Builder crew, only to find the Kragle was all the way up on the infinitieth floor, guarded by a robot army.

2014, Phil Lord, Christopher Miller, The Lego Movie:


long line

(mathematics) A topological space, constructed by stringing together an uncountable number of intervals, which is "longer" than the real line.


Richard's paradox

First described by the French mathematician Jules Richard in 1905.

The paradox where, given the observation that certain English phrases unambiguously define real numbers while others do not, there is an infinitely long list of English phrases that unambiguously define real numbers, yet (using a similar technique to Cantor's diagonal argument) it is possible to generate another such phrase not in the list.


biinfinite

(mathematics) Infinite in both directions, having neither a first nor a last element.


omega point

To begin, let AB be a given segment, and l and m be sensed parallel lines with shared omega point Ω.

2018, Maureen T. Carroll, Elyn Rykken, Geometry: The Line and the Circle, page 319:

(geometry) A well-defined point outside the hyperbolic plane.


infinitist

This cannot be done, for any plausible account of the conditions under which P 0 is supported by P 1 will either require a justified belief (thus making infinitism circular) or will be incompatible with the infinitist commitment to Reasons are Supported.

2007 October 2, Andrew D. Cling, “The epistemic regress problem”, in Philosophical Studies, volume 140, number 3, →DOI:


Cantor dust

(mathematics) A multidimensional version of the Cantor set, formed by taking a finite Cartesian product of the Cantor set with itself.


point at infinity

en-noun

point at infinity (plural points at infinity)

Hyperbolic geometry provides each line with two points at infinity. Whether there is a piece of the line beyond the points at infinity, completing the piece lying in the finite to a closed curve, we cannot say, since we cannot move as far as the points at infinity, let alone beyond them.

1996, John Stillwell, Sources of Hyperbolic Geometry, page 82:

2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,
As we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.
This representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré, […] .

To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l’ and we call ω their common point at infinity.

2012, Norbert A'Campo, Athanase Papadopoulos, “Notes on non-Euclidean Geometry”, in Athanase Papadopoulos, editor, Strasbourg Master Class on Geometry, page 126:

(geometry, hyperbolic geometry) An ideal point.


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.


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.


pseudodimension

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Our algorithms rely on random sampling and their analysis leverages on Rademacher averages and pseudodimension, fundamental concepts from statistical learning theory.

2016, Matteo Riondato, Eli Upfal, “ABRA: Approximating Betweenness Centrality in Static and Dynamic Graphs with Rademacher Averages”, in arXiv:

(mathematics) The largest cardinality (or infinite) of a set of points (in a function class) that can be shattered.


teragon

(mathematics) A polygon that has fractal sides


Thomson's lamp

The idea was devised in 1954 by British philosopher James F. Thomson.

(philosophy) A hypothetical lamp in a philosophical paradox. The lamp is repeatedly turned on and off by means of a switch; at the end of this time period, regarded as the sum of an infinite series of intervals, it seems paradoxically that the lamp cannot be either on or off.


periodic continued fraction

periodic continued fraction (plural periodic continued fractions)

(mathematics) A continued fraction with a sequence of numerators and denominators that repeat indefinitely


infinite continued fraction

infinite continued fraction (plural infinite continued fractions)

(mathematics) A continued fraction that goes on forever


axiom of countable choice

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. One example is the axiom of countable choice, which states that if A#95;0,A#95;1,#92;dotsA#95;n#92;dots form a denumerable set of nonempty sets, their product is nonempty.[…]The axiom of countable choice is constantly used in analysis; it is often hidden so as not to sow confusion in the minds of the students (who are inclined to accept anything desired) or of the professors (who do not like to shake the foundations of the discipline).

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

we have no way to infer
∃R∀n[P0n→∀x(R_nx→Fx)∧n=Nx:R_nx)]
without an axiom of countable choice.

2012, Richard G. Heck, Jr., Reading Frege's Grundgesetze, Oxford University Press, page 271, But, once again, while we can easily prove ∀n[P0n→∃G(∀x(Gx→Fx)∧n=Nx:Gx)]

2013, Valentin Blot, Colin Riba, On Bar Recursion and Choice in a Classical Setting, Chung-chien Shan (editor), Programming Languages and Systems: 11th International Symposium, APLAS 2013, Proceedings, Springer, LNCS 8301, page 349,
We show how Modified Bar-Recursion, a variant of Spector's Bar-Recursion due to Berger and Oliva can be used to realize the Axiom of Countable Choice in Parigot's Lambda-Mu-calculus, a direct-style language for the representation and evaluation of classical proofs.

(set theory) A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function; equivalently, the statement that the direct product of a countable collection of nonempty sets is nonempty.


projective line

projective line (plural projective lines)

A projective line in P²( R ), consists of the projective points defined by coplanar directions in R ³.
In other words, just as the points in P²( R ) correspond to one-dimensional subspaces in R ³, the projective lines correspond to two-dimensional subspaces in R ³.

2007, Unnamed translator, Ana Irene Ramírez Galarza, José Seade, Introduction to Classical Geometries, [2002, Introducciòn a la Geometria Avanzada], Springer (Birkhäuser), page 97, In P2( R ), the projective lines are defined by two projective points, that is, by two linearly independent directions of R ³; if we take one vector for each direction, the two vectors generate a plane through the origin in R ³, that is, a subspace of dimension 2, and a projective line can be defined as follows

2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique] Springer, page 37,
Consider the projective line mathbf P¹, with homogeneous coordinates x and t and open sets U_0(x ne 0) and U_1(t ne 0).

The projective line is related to the Kronecker algebra,[…]In general one has to deal with the so-called weighted case which leads to the study of the canonical algebras and to the weighted projective lines (as Ringel pointed out in his survey [93]).

2009, Dirk Kussin, Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras, American Mathematical Society, page 13:

(projective geometry) A line that includes a point at infinity; a line in a projective space; a projective space of dimension 1.


axiom of infinity

(set theory) One of the axioms in axiomatic set theory that guarantees the existence of an infinite set.


epsilon number

From the Greek letter ε (“epsilon”), used to denote the numbers.

More generally, the epsilon numbers are the ordinals #92;varepsilon for which #92;varepsilon#61;#92;omega#92;varepsilon. The smallest epsilon number is #92;varepsilon#95;0. It is a countable ordinal, being the countable union of countable sets. By the Veblen fixed-point theorem, the class of epsilon numbers is unbounded.

1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240:

We show that the associated ordinals are the #92;varepsilon#95;0th epsilon number and the first #92;varepsilon#95;0-critical number, respectively.

1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17:

2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, [Revision of 1971 Addison-Wesley edition], page 203,
Thus there is at least one epsilon number, namely ε₀; we can easily show, in fact, that ε₀ is the least epsilon number.

(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.


transfinitely

We finish this section by briefly addressing how the reduction procedure for #92;mathsf#123;EMU#125; described so far can be formalized in the transfinitely iterated fixed point theory #92;widehat#123;#92;mathsf#123;ID

1999, Thomas Strahm, “First Steps into Metapredictivity in Explicit Mathematics”, in S. Barry Cooper, John K. Truss, editors, Sets and Proofs, Cambridge University Press, page 399:

After we have performed the deletion operation infinitely many times, we can look at what is left and perform the deletion operation on that, and thereby continue the process transfinitely.

2003, Thomas Forster, Logic, Induction and Sets, Cambridge University Press, page 147:

(manner) In a manner that requires a transfinite number of steps.


infinitely many

(mathematics) Having a quantity or cardinality that is infinite or unbounded.


Ford circle

Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.

Every Ford circle is tangent to the horizontal axis y#61;0, and any two Ford circles are either disjoint or meet at a tangent.

There is a unique Ford circle associated with every rational number. Additionally, the axis y#61;0 can be considered a Ford circle with infinite radius, corresponding to the case p#61;1,#92;q#61;0.

Thus the Ford circle [1], drawn tangent to the real axis at #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, and having radius #92;textstylea#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125;, must contain in its interior some points belonging to #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, such as #92;textstyle#92;theta#43;ia#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125; whose imaginary part lies between #92;textstyleq#95;n#123;-2#125; and #92;textstyleq#95;#123;n#43;1#125;#123;-2#125;.

1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:

A Farey fraction #92;textstyle#92;frackc defines a Ford circle #92;textstyle#92;mathcal#123;C#125;(k,c) in #92;textstyle#92;mathbbC. Its center is given by #92;textstylekc#43;i#92;frac 1#123;2c²#125; and its radius is #92;textstyle#92;frac1#123;2c²#125;. Two Ford circles #92;textstyle#92;mathcalC(a,b) and #92;textstyle#92;mathcalC(c,d) are tangent whenever #92;textstylead-bc#61;#92;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #92;textstyleF#95;N.

2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:

Ford circles are a collection of horocycles in #92;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#95;#123;a#47;b#125; and C#95;#123;c#47;d#125; are tangent if and only if #92;vertad-bc#92;vert#61;1.

2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:

(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers).

Ford-Kreis

any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle


Cantor's theorem

Named after German mathematician George Cantor (1845-1918) who stated and proved this theorem.

(set theory) A theorem in set theory which states for any set A, the set of all subsets of A (the power set of A, denoted by 𝒫(A)) has a strictly greater cardinality than A itself.


mucube

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

(geometry) A regular skew apeirohedron with six squares around each vertex, formed by an infinite number of cubic cells (specifically, cubes missing two opposite faces each, around cubic empty spaces with no faces) in Euclidean 3D space.


mutetrahedron

mutetrahedron (plural mutetrahedra or mutetrahedrons)

mutetrahedra (plural)
mutetrahedrons (plural)

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truncated tetrahedron
truncated tetrahedron

(geometry) A regular skew apeirohedron with six hexagons around each vertex, formed by an infinite number of truncated tetrahedron-like cells (specifically, truncated tetrahedra missing their triangle faces, with the resulting triangle-shaped holes joined to form empty spaces in the shape of faceless tetrahedra) in Euclidean 3D space.

六角六片三角孔ねじれ正多面体

ろっかくろくへんさんかくこうねじれせいためんたい


muoctahedron

muoctahedron (plural muoctahedra or muoctahedrons)

muoctahedra (plural)
muoctahedrons (plural)

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

truncated octahedra
truncated octahedron

(geometry) A regular skew apeirohedron with four hexagons around each vertex, formed by an infinite number of octahedron-like cells (specifically, truncated octahedra missing their square faces, and joined by the resulting square-shaped holes) in Euclidean 3D space.

六角四片四角孔ねじれ正多面体

ろっかくしへんしかくこうねじれせいためんたい


ideal triangle

In hyperbolic geometry, a triangle whose vertices are all ideal; i.e. points at infinity, rather than real.


limit cardinal

(set theory) A cardinal number that cannot be reached from another by repeated successor operations (a weak limit cardinal) or by repeated power set operations (a strong limit cardinal).


Zeno machine

(computing theory) A hypothetical computational model, related to Turing machines, that would be capable of carrying out computations involving a countably infinite number of algorithmic steps.


Grim Reaper paradox

A philosophical problem related to Benardete's paradox, but involving countably many Grim Reapers, each of which will kill the person in question at a certain time (if still alive), taking a certain period of time to do it.


diagonal argument

A proof, developed by Georg Cantor, to show that the set of real numbers is uncountably infinite.


omegath

(set theory, rare) In the position known as omega (a transfinite ordinal number referring to the next position after ordering a countably infinite set).


monotone convergence theorem

(mathematical analysis) A theorem which states that any sequence which is monotonic (either increasing or decreasing) and bounded is convergent.


ideal point

Synonym for omega point, in hyperbolic geometry. Also called: point at infinity


infinite hotel paradox

(set theory) A thought experiment illustrating a counterintuitive property of infinite sets, showing that a fully occupied hotel with infinitely many rooms may still accommodate any number of additional guests.


ideal polyhedron

ideal polyhedron (plural ideal polyhedrons or ideal polyhedra)

ideal polyhedrons (plural)
ideal polyhedra (plural)

In hyperbolic geometry, a polyhedron whose vertices are all ideal; i.e. points at infinity, rather than real. Its surface is topologically equivalent to a punctured sphere.