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rdfs:label: Нескінченність Infini 无穷 Άπειρο Nekonečno Бесконечность Infinity Infinito Senfineco Infinit Unendlichkeit Tak hingga Infinitu لانهاية 무한 Éigríoch Infinito Infinito (matematica) 無限 Oneindigheid Nieskończoność Oändlighet
rdfs:comment: Бесконе́чность — категория человеческого мышления, используемая для характеристики безграничных, беспредельных, неисчерпаемых предметов и явлений, для которых невозможно указание границ или количественной меры. Используется в противоположность конечному, исчисляемому, имеющему предел. Систематически исследуется в математике, логике и философии, также изучаются вопросы о восприятии, статусе и природе бесконечности в психологии, теологии, физике соответственно. Бесконечность обозначается символом . El concepto de infinito (símbolo: ∞) aparece en varias ramas de la matemática, la filosofía y la astronomía, en referencia a una cantidad sin límite o sin final, contrapuesto al concepto de finitud. En matemáticas el infinito aparece de diversas formas: en geometría, el punto al infinito en geometría proyectiva y el punto de fuga en geometría descriptiva; en análisis matemático, los límites infinitos; y en teoría de conjuntos como números transfinitos. Infinitu zenbaki errealen multzoko goi muga da. Kontzeptua matematikaren, filosofiaren eta astronomiaren hainbat adarretan agertzen da, mugarik gabeko edo amaierarik gabeko kopuru bati dagokionez, eta kopuru hori finitutasunaren kontzeptuaren aurkakoa da. Infinito (do latim infinitus, símbolo: ) é a qualidade daquilo que não tem fim. É um conceito usado em vários campos, como a matemática, filosofia e a teologia. É representado com o símbolo ∞, e na matemática é uma noção quase-numérica usada em proposições. Mas é na matemática que o conceito tem as suas raízes mais profundas, sendo a disciplina que mais contribuiu para a sua compreensão. Tak hingga adalah sesuatu yang tiada berbatas maupun berpenghujung, atau sesuatu yang lebih besar dari sebarang batas yang ditetapkan. Tak hingga sering dilambangkan dengan simbol ∞. Dalam percakapan sehari-hari orang dapat mengartikan tak hingga sebagai "sesuatu yang lebih besar dari segala yang mungkin". Sehingga kadang kata tak hingga digunakan untuk menerangkan benda hingga namun seakan berterusan tak henti-henti atau sukar untuk menghitungnya. Kadang pula orang bergurau tentang sesuatu yang lebih besar dari tak hingga, katakanlah tak hingga tambah satu. Tetapi dalam matematika bilangan seperti itu terdefinisi dalam sistem bilangan tertentu, seperti bilangan transfinit. 無窮或無限大，來自於拉丁文的「infinitas」，即「沒有邊界」的意思。其數學符號為∞。它在科學、神學、哲學、數學和日常生活中有著不同的概念。通常使用這個詞的時候並不涉及它的更加技術層面的定義。在神學方面，根據書面記載無窮這個符號最早被用於某些秘密宗教，通常代表人類中的神性，而書寫此符號時兩圓的不對等代表人神間的差距，例如神學家邓斯·司各脱（Duns Scotus）的著作中，上帝的無限能量是運用在無約束上，而不是運用在無限量上。在哲學方面，無窮可以歸因於空間和時間。在神學和哲學兩方面，無窮又作為無限，很多文章都探討過無限、絕對、上帝和芝諾悖論等的問題。在數學方面，無窮與下述的主題或概念相關：數學的極限、阿列夫數、集合論中的類、、羅素悖論、超實數、射影幾何、擴展的實數軸以及絕對無限。在一些主題或概念中，無窮被認為是一個超越邊界而增加的概念，而不是一個數。 Le mot « infini » (-e, -s) est un adjectif servant à qualifier quelque chose qui n'a pas de limite en nombre ou en taille. Il vient du latin infīnītus, dérivé de fīnītus « limité » (avec in-, préfixe négatif), issu lui-même du verbe fīnĭo, fīnīre (« délimiter », mais aussi : « préciser », « déterminer », et intransitivement « finir »), et du nom fīnis (souvent au pluriel, fīnes : « bornes, limites d'un champ », « frontières d'un pays ») ; il signifie donc, littéralement « qui est sans borne », mais aussi « indéterminé » et « indéfini ».Lorsqu'il est substantivé, l'infini désigne fondamentalement une notion mathématique, ainsi qu'un concept philosophique, métaphysique ou théologique, dont les paradoxes ont nourri depuis longtemps et nourrissent encore l'histoire de la pensée dans le monde e 無限（むげん、infinity、∞）とは、限りの無いことである。「限界を持たない」というだけの単純に理解できそうな概念である一方で、有限な世界しか知りえないと思われる人間にとって、無限というものが一体どういうことであるのかを厳密に理解することは非常に難しい問題を含んでいる。このことから、しばしば哲学、数学、論理学や自然科学などの一部の分野において考察の対象として無限という概念が取り上げられ、そして深い考察が得られている。本項では、数学などの学問分野において、無限がどのように捉えられ、どのように扱われるのかを記述する。 El concepte d'infinit apareix en diverses branques de la filosofia, la matemàtica i l'astronomia, en referència a una quantitat sense límit o final, contraposat al concepte de finitud. En matemàtiques l'infinit, representat amb el símbol ∞, és la fita superior del conjunt del nombres reals. Tanmateix, no es tracta d'un nombre en si, sinó d'un concepte al qual hom només s'hi pot aproximar mitjançant límits. Per exemple, en la funció , quan x tendeix a 0 (és a dir, s'aproxima cada cop més a 0), tendeix a l'infinit (es fa cada cop més gran), però no es diu que arriba al valor "infinit". Oändlighet är ett begrepp för obegränsning och obundenhet i storlek, antal eller utsträckning. Dess motsats är . Man skiljer mellan potentiell oändlighet och faktisk oändlighet. Den matematiska symbolen är en lemniskata (∞). Der Begriff Unendlichkeit bezeichnet die Negation bzw. Aufhebung von Endlichkeit, weniger präzise auch deren Gegenteil. Sein mathematisches Symbol ist das Unendlichzeichen . Theoretisch beschreibt der Begriff „unendlich“ ein Objekt (z. B. eine Kugel) oder einen Vorgang ohne Ende bzw. Schluss, aber möglicherweise mit Anfang oder Beginn. In der Geometrie würde also ein Strahl oder eine Kreisbahn als unendlich beschrieben werden. Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor e In matematica il concetto di infinito (simbolo , talvolta detto lemniscata) ha molti significati, in correlazione con la nozione di limite, sia in analisi classica sia in analisi non standard. Nozioni di infinito sono usate in teoria degli insiemi e in geometria proiettiva. Nekonečno (∞) je abstraktní pojem, který označuje kvantitu (množství) něčeho, co je tak veliké, že nemá konec (od slova konec je odvozeno slovo konečný), typicky se nedá spočítat, změřit, a pokud ano, tak je větší než každé konečné číslo. Přesto se řadí mezi čísla. Objekt, který je tak veliký, že má atributy nekonečna, se někdy nazývá přídavným jménem nekonečný. Nekonečno nemá hranice, ale není totéž co neohraničenost. Nepřítomnost hranic je podmínkou nutnou, nikoli však postačující. Nekonečno lze ztotožnit s neohraničenosti pouze v Eukleidově geometrii, obecně je nutné rozlišovat ne/konečnost topologickou a metrickou. Např. kulová plocha (povrch koule) je (metricky) konečná, ale neohraničená. Stejně jako nekonečno. Sa mhatamaitic, uimhir níos mó ná aon uimhir eile. Ba é John Wallis (1616-1703), matamaiticeoir Sasanach, an chéad duine a d'úsáid an tsiombail ∞ di. Thaispeáin Cantor is matamaiticeoirí eile sa 19ú-20ú céad castacht choincheap na héigríche, agus iad ag obair ar thacartheoiric. Níos déanaí tugadh uimhreacha atá éigríochta go mór nó éigríochta go beag isteach sa chalcalas. Нескінче́нність (символ: ∞) — категорія людського мислення, яка використовується для характеристики безмежних, невичерпних предметів і явищ, для яких є неможливим вказання меж або кількісної міри. Використовується на противагу скінченному, обчислюваному, такому, що має межу. Термін нескінче́нність може описувати декілька різних понять, залежно від області застосування, будь це математика, фізика, філософія, теологія чи повсякденне життя. Nieskończoność (symbol: ∞) – byt nieograniczony (w sensie wielkości bądź ilości), który przyjęło się oznaczać za pomocą znaku podobnego do „przewróconej ósemki” (lemniskata). En matematiko kaj filozofio senfineco aŭ nefinio aŭ infinito havas plurajn signifojn, kiuj ĉiuj esprimas la ideon, ke la priparolata objekto aŭ procezo en ia senco ne havas finon. Ekde antikvaj tempoj filozofoj kaj matematikistoj klopodis kompreni kaj analizi nefinion, sed ofte temis pri misgvidaj argumentoj, ĝis Georg Cantor pli formale kaj rigore analizis la koncepton fine de la 19-a jarcento, interalie enkondukante klaran manieron distingi grandecojn de nefinio. Matematiko uzas la simbolon ∞ (unikode U+221E) por signi nefiniecon, atribuita al angla matematikisto John Wallis. كلمة لانهاية (بالإنجليزية: infinity)‏ تدل على «ما لا حدود له» أو «اللامنتهي» أو «غير المحدود» تستخدم بعدة مفاهيم مختلفة لكن يجمع بينها جميعا فكرة واحدة هي «عدم وجود نهاية». من هذا المنطلق فهي ترتبط بالفلسفة والرياضيات والإلهيات والحياة اليومية أيضا. وأوّل من استعمل الرمز المعروف الآن (∞) لهذا التعبير، كان جون واليس سنة 1655 في مؤلَّفيه: الأوّل De Sectionibus Conicis وبعدها في Arithmetica Infinitorum. في الرياضيات، اللانهاية تستخدم كعدد تقاس به كمية غير محدود، ويُرمز لها بالحرف (∞). وهو كيان مختلف عن أي كيان عددي آخر في خاصياته وسلوكه. Το άπειρο (σύμβολο: ) αποτελεί αφηρημένη έννοια που περιγράφει κάτι χωρίς κανένα όριο και έχει σημασία σε μια σειρά από επιστήμες, κυρίως τα μαθηματικά και τη φυσική. Η λέξη άπειρο προέρχεται από το στερητικό πρόθεμα "α-" και τη λέξη "πέρας" που σημαίνει τέλος. Αναφέρεται σε διαφορετικές έννοιες (που συνήθως συνδέονται με την έννοια του "χωρίς τέλος") που προκύπτουν στη φιλοσοφία, τα μαθηματικά και τη θεολογία. Oneindigheid staat in de betekenis van niet-eindig tegenover het begrip eindig. Het is een begrip in de filosofie en de natuurwetenschappen (zie daarvoor ook universum). In de wis- en natuurkunde heeft oneindig een min of meer kwantitatieve betekenis en wordt als symbool voor oneindig een lemniscaat (∞) gebruikt (ongeveer een liggende acht, en daarom ook wel zo genoemd).
owl:differentFrom: dbr:Infiniti_(disambiguation)
rdfs:seeAlso: dbr:Hilbert's_paradox_of_the_Grand_Hotel
foaf:depiction: n34:Números_hiperreales.png n34:Peanocurve.svg n34:Riemann_sphere1.svg n34:Reflections_1090029.jpg n34:Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif
dcterms:subject: dbc:Philosophy_of_mathematics dbc:Mathematical_objects dbc:Infinity dbc:Concepts_in_logic
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dbo:abstract: Нескінче́нність (символ: ∞) — категорія людського мислення, яка використовується для характеристики безмежних, невичерпних предметів і явищ, для яких є неможливим вказання меж або кількісної міри. Використовується на противагу скінченному, обчислюваному, такому, що має межу. Термін нескінче́нність може описувати декілька різних понять, залежно від області застосування, будь це математика, фізика, філософія, теологія чи повсякденне життя. Infinitu zenbaki errealen multzoko goi muga da. Kontzeptua matematikaren, filosofiaren eta astronomiaren hainbat adarretan agertzen da, mugarik gabeko edo amaierarik gabeko kopuru bati dagokionez, eta kopuru hori finitutasunaren kontzeptuaren aurkakoa da. Le mot « infini » (-e, -s) est un adjectif servant à qualifier quelque chose qui n'a pas de limite en nombre ou en taille. Il vient du latin infīnītus, dérivé de fīnītus « limité » (avec in-, préfixe négatif), issu lui-même du verbe fīnĭo, fīnīre (« délimiter », mais aussi : « préciser », « déterminer », et intransitivement « finir »), et du nom fīnis (souvent au pluriel, fīnes : « bornes, limites d'un champ », « frontières d'un pays ») ; il signifie donc, littéralement « qui est sans borne », mais aussi « indéterminé » et « indéfini ».Lorsqu'il est substantivé, l'infini désigne fondamentalement une notion mathématique, ainsi qu'un concept philosophique, métaphysique ou théologique, dont les paradoxes ont nourri depuis longtemps et nourrissent encore l'histoire de la pensée dans le monde entier. Le symbole de l'infini, en mathématiques et au-delà des mathématiques, est « ∞ », inventé par le mathématicien John Wallis au XVIIe siècle, signe dont l'origine est controversée et dont la forme peut évoquer un « 8 » horizontal (mais ce n'est pas en référence au chiffre 8 que ce signe fut choisi) ; cette forme a été rapprochée de celles de la lemniscate de Bernoulli ou du ruban de Möbius. Celles-ci sont d'ailleurs aussi des figurations possibles de l'infini ou de l'inaccessible mouvement perpétuel, mais figurations « à la limite » justement, pour un concept qui est par définition ou par nature impossible à figurer. La notion d'infini a fortement marqué l'histoire de la philosophie occidentale au moins depuis l'antiquité grecque, notamment dans la tentative de concilier la théologie chrétienne (dans ses réflexions sur le caractère à la fois infini et réel ou "actuel" de la nature divine) avec la pensée néoplatonicienne et l'aristotélisme, et singulièrement avec la distinction conceptuelle d'Aristote entre « "l'infini en acte", effectif et concret, qui ne peut se réaliser dans la nature, et "l'infini en puissance", celui que peuvent imaginer les hommes, le seul à exister », mais seulement en tant que concept abstrait et potentialité. Les grandes étapes de cette réflexion ont probablement été, depuis le mystérieux Livre des XXIV philosophes, du IVe au XVe siècle, les œuvres de : Marius Victorinus, Boèce, Jean Damascène, Thomas d'Aquin, Raymond Lulle, Jean Duns Scot, Nicole Oresme, et surtout Nicolas de Cues (voir la section « Une avancée vers l'Infini » de l'article consacré à ce dernier). Sans oublier que c'est entre autres sa vision d'un univers infini constitué d'une infinité de mondes finis chantant tous la gloire de Dieu présent partout (au risque, pour l'Inquisition de confondre le Créateur avec sa création) qui a valu le bûcher à Giordano Bruno, à l'orée du XVIIe siècle. Toutefois, cette notion d'infini a connu depuis le XVIIe siècle dans la pensée occidentale, ainsi que dans l'histoire des mathématiques et des sciences, un rebond spectaculaire, notamment à partir du développement du calcul infinitésimal par Fermat, Leibniz et Newton (et déjà dans les œuvres de Descartes, de Pascal et de Spinoza). Ces innovations en mathématiques (mais aussi en physique) ont induit un changement radical de perspectives : ainsi, Alexandre Koyré affirme que « la substitution d'un univers infini et homogène au cosmos fini et hiérarchiquement ordonné de la pensée antique et médiévale implique et nécessite la refonte des principes premiers de la raison philosophique et scientifique ». In matematica il concetto di infinito (simbolo , talvolta detto lemniscata) ha molti significati, in correlazione con la nozione di limite, sia in analisi classica sia in analisi non standard. Nozioni di infinito sono usate in teoria degli insiemi e in geometria proiettiva. Nieskończoność (symbol: ∞) – byt nieograniczony (w sensie wielkości bądź ilości), który przyjęło się oznaczać za pomocą znaku podobnego do „przewróconej ósemki” (lemniskata). كلمة لانهاية (بالإنجليزية: infinity)‏ تدل على «ما لا حدود له» أو «اللامنتهي» أو «غير المحدود» تستخدم بعدة مفاهيم مختلفة لكن يجمع بينها جميعا فكرة واحدة هي «عدم وجود نهاية». من هذا المنطلق فهي ترتبط بالفلسفة والرياضيات والإلهيات والحياة اليومية أيضا. وأوّل من استعمل الرمز المعروف الآن (∞) لهذا التعبير، كان جون واليس سنة 1655 في مؤلَّفيه: الأوّل De Sectionibus Conicis وبعدها في Arithmetica Infinitorum. في الثقافة الشعبية، اللانهاية عادة هي شيء يمكن تشبيهه «بأكبر عدد ممكن» أو أبعد مسافة ممكنة، ففي ذهن الكثير يبقى التساؤل: ما هو بعد اللانهاية، لكن الكثير أصبح يعتبر سؤال ما بعد اللانهاية أمرا سخيفا لأن اللانهاية تمثل رمز لما لا يمكن تخيل ما هو أكبر منه. في الرياضيات، اللانهاية تستخدم كعدد تقاس به كمية غير محدود، ويُرمز لها بالحرف (∞). وهو كيان مختلف عن أي كيان عددي آخر في خاصياته وسلوكه. El concepto de infinito (símbolo: ∞) aparece en varias ramas de la matemática, la filosofía y la astronomía, en referencia a una cantidad sin límite o sin final, contrapuesto al concepto de finitud. En matemáticas el infinito aparece de diversas formas: en geometría, el punto al infinito en geometría proyectiva y el punto de fuga en geometría descriptiva; en análisis matemático, los límites infinitos; y en teoría de conjuntos como números transfinitos. Tak hingga adalah sesuatu yang tiada berbatas maupun berpenghujung, atau sesuatu yang lebih besar dari sebarang batas yang ditetapkan. Tak hingga sering dilambangkan dengan simbol ∞. Dalam percakapan sehari-hari orang dapat mengartikan tak hingga sebagai "sesuatu yang lebih besar dari segala yang mungkin". Sehingga kadang kata tak hingga digunakan untuk menerangkan benda hingga namun seakan berterusan tak henti-henti atau sukar untuk menghitungnya. Kadang pula orang bergurau tentang sesuatu yang lebih besar dari tak hingga, katakanlah tak hingga tambah satu. Tetapi dalam matematika bilangan seperti itu terdefinisi dalam sistem bilangan tertentu, seperti bilangan transfinit. Ada juga definisi lain dalam bidang teori himpunan yang mengatakan bahwa tak hingga bukan benar-benar bilangan, tapi hanya merujuk kepada kardinalitas, yaitu besarnya sejenis himpunan. Karena ia tidak berlaku seperti bilangan yang biasa kita pakai dalam aritmetika, ia dapat digunakan untuk menjelaskan sifat-sifat beberapa objek matematika. Contohnya, berapa digit yang ada dalam representasi desimal untuk bilangan π. Atau seperti yang mengatakan bahwa limit untuk adalah tak hingga yang positif ketika menuju kepada 0 dari sisi positif. 無限（むげん、infinity、∞）とは、限りの無いことである。「限界を持たない」というだけの単純に理解できそうな概念である一方で、有限な世界しか知りえないと思われる人間にとって、無限というものが一体どういうことであるのかを厳密に理解することは非常に難しい問題を含んでいる。このことから、しばしば哲学、数学、論理学や自然科学などの一部の分野において考察の対象として無限という概念が取り上げられ、そして深い考察が得られている。本項では、数学などの学問分野において、無限がどのように捉えられ、どのように扱われるのかを記述する。 Der Begriff Unendlichkeit bezeichnet die Negation bzw. Aufhebung von Endlichkeit, weniger präzise auch deren Gegenteil. Sein mathematisches Symbol ist das Unendlichzeichen . Theoretisch beschreibt der Begriff „unendlich“ ein Objekt (z. B. eine Kugel) oder einen Vorgang ohne Ende bzw. Schluss, aber möglicherweise mit Anfang oder Beginn. In der Geometrie würde also ein Strahl oder eine Kreisbahn als unendlich beschrieben werden. Präzisierung fand der „Unendlichkeitsbegriff“ vor allem in der Mathematik, wesentlich initiiert durch das Werk Bernard Bolzanos, Georg Cantors und Richard Dedekinds, welches in die Mengenlehre und insbesondere in die Theorie der unendlichen Mengen und der transfiniten Kardinalzahlen mündete. Insbesondere erfuhr er dabei auch erstmals eine Erweiterung um Eigenschaften, die sich nicht aus der obigen negativen Definition ableiten lassen. Oändlighet är ett begrepp för obegränsning och obundenhet i storlek, antal eller utsträckning. Dess motsats är . Man skiljer mellan potentiell oändlighet och faktisk oändlighet. Den matematiska symbolen är en lemniskata (∞). Sa mhatamaitic, uimhir níos mó ná aon uimhir eile. Ba é John Wallis (1616-1703), matamaiticeoir Sasanach, an chéad duine a d'úsáid an tsiombail ∞ di. Thaispeáin Cantor is matamaiticeoirí eile sa 19ú-20ú céad castacht choincheap na héigríche, agus iad ag obair ar thacartheoiric. Níos déanaí tugadh uimhreacha atá éigríochta go mór nó éigríochta go beag isteach sa chalcalas. Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic. In physics and cosmology, whether the Universe is spatially infinite is an open question. Infinito (do latim infinitus, símbolo: ) é a qualidade daquilo que não tem fim. É um conceito usado em vários campos, como a matemática, filosofia e a teologia. É representado com o símbolo ∞, e na matemática é uma noção quase-numérica usada em proposições. Mas é na matemática que o conceito tem as suas raízes mais profundas, sendo a disciplina que mais contribuiu para a sua compreensão. El concepte d'infinit apareix en diverses branques de la filosofia, la matemàtica i l'astronomia, en referència a una quantitat sense límit o final, contraposat al concepte de finitud. En matemàtiques l'infinit, representat amb el símbol ∞, és la fita superior del conjunt del nombres reals. Tanmateix, no es tracta d'un nombre en si, sinó d'un concepte al qual hom només s'hi pot aproximar mitjançant límits. Per exemple, en la funció , quan x tendeix a 0 (és a dir, s'aproxima cada cop més a 0), tendeix a l'infinit (es fa cada cop més gran), però no es diu que arriba al valor "infinit". Nekonečno (∞) je abstraktní pojem, který označuje kvantitu (množství) něčeho, co je tak veliké, že nemá konec (od slova konec je odvozeno slovo konečný), typicky se nedá spočítat, změřit, a pokud ano, tak je větší než každé konečné číslo. Přesto se řadí mezi čísla. Objekt, který je tak veliký, že má atributy nekonečna, se někdy nazývá přídavným jménem nekonečný. Nekonečno nemá hranice, ale není totéž co neohraničenost. Nepřítomnost hranic je podmínkou nutnou, nikoli však postačující. Nekonečno lze ztotožnit s neohraničenosti pouze v Eukleidově geometrii, obecně je nutné rozlišovat ne/konečnost topologickou a metrickou. Např. kulová plocha (povrch koule) je (metricky) konečná, ale neohraničená. Stejně jako nekonečno. Nekonečno má důležité místo v matematice (zvláště v geometrii a teorii množin), v historii matematiky, k jeho studiu přispěli mimo jiné čeští vědci Bernard Bolzano a Petr Vopěnka. Nekonečno vyprovokovalo mnohé úvahy i ve filosofii a teologii. Symbol ∞ pro nekonečno zavedl anglický matematik John Wallis v 17. století. 無窮或無限大，來自於拉丁文的「infinitas」，即「沒有邊界」的意思。其數學符號為∞。它在科學、神學、哲學、數學和日常生活中有著不同的概念。通常使用這個詞的時候並不涉及它的更加技術層面的定義。在神學方面，根據書面記載無窮這個符號最早被用於某些秘密宗教，通常代表人類中的神性，而書寫此符號時兩圓的不對等代表人神間的差距，例如神學家邓斯·司各脱（Duns Scotus）的著作中，上帝的無限能量是運用在無約束上，而不是運用在無限量上。在哲學方面，無窮可以歸因於空間和時間。在神學和哲學兩方面，無窮又作為無限，很多文章都探討過無限、絕對、上帝和芝諾悖論等的問題。在數學方面，無窮與下述的主題或概念相關：數學的極限、阿列夫數、集合論中的類、、羅素悖論、超實數、射影幾何、擴展的實數軸以及絕對無限。在一些主題或概念中，無窮被認為是一個超越邊界而增加的概念，而不是一個數。 Oneindigheid staat in de betekenis van niet-eindig tegenover het begrip eindig. Het is een begrip in de filosofie en de natuurwetenschappen (zie daarvoor ook universum). In de wis- en natuurkunde heeft oneindig een min of meer kwantitatieve betekenis en wordt als symbool voor oneindig een lemniscaat (∞) gebruikt (ongeveer een liggende acht, en daarom ook wel zo genoemd). En matematiko kaj filozofio senfineco aŭ nefinio aŭ infinito havas plurajn signifojn, kiuj ĉiuj esprimas la ideon, ke la priparolata objekto aŭ procezo en ia senco ne havas finon. Ekde antikvaj tempoj filozofoj kaj matematikistoj klopodis kompreni kaj analizi nefinion, sed ofte temis pri misgvidaj argumentoj, ĝis Georg Cantor pli formale kaj rigore analizis la koncepton fine de la 19-a jarcento, interalie enkondukante klaran manieron distingi grandecojn de nefinio. Matematiko uzas la simbolon ∞ (unikode U+221E) por signi nefiniecon, atribuita al angla matematikisto John Wallis. Estas grave distingi diversajn uzojn de la koncepto "senfineco" en diversaj kampoj de matematiko. Ekz-e, en analitiko, "nefinia" estas koncepto uzata por paroli pri limo de vico, kiu ne havas reelnombran aŭ kompleksnombran limon; en tia kunteksto havas sencon distingi inter pozitiva kaj negativa nefinioj. En aroteorio aliflanke, ne havas sencon paroli pri negativa senfineco, sed tamen ekzistas pluraj malsamaj nefinioj: Ekz-e oni povas diri, ke la aro de reelaj nombroj estas pli granda ol la aro de naturaj nombroj, kvankam ambaŭ estas nefiniaj. La kialo estas, ke la naturaj nombroj ne sufiĉas por numeri la reelajn. Το άπειρο (σύμβολο: ) αποτελεί αφηρημένη έννοια που περιγράφει κάτι χωρίς κανένα όριο και έχει σημασία σε μια σειρά από επιστήμες, κυρίως τα μαθηματικά και τη φυσική. Η λέξη άπειρο προέρχεται από το στερητικό πρόθεμα "α-" και τη λέξη "πέρας" που σημαίνει τέλος. Αναφέρεται σε διαφορετικές έννοιες (που συνήθως συνδέονται με την έννοια του "χωρίς τέλος") που προκύπτουν στη φιλοσοφία, τα μαθηματικά και τη θεολογία. Στα μαθηματικά, το "άπειρο" χρησιμοποιείται συνήθως σε περιπτώσεις όπου αντιμετωπίζεται σαν να ήταν αριθμός (δηλαδή για τη σειρά ή το μέγεθος κάποιου πράγματος, π.χ. "άπειρος αριθμός στοιχείων") αλλά είναι διαφορετικό είδος αριθμού από τους πραγματικούς αριθμούς. Το άπειρο βρίσκεται στα όρια, στους , στις της θεωρίας συνόλων, στα σύνολα, στο παράδοξο του Ράσελ, στη , στους , στην , στο και στο του Κάντορ. Ο Γκέοργκ Κάντορ επισημοποίησε πολλές ιδέες που σχετίζονται με το άπειρο και τις άπειρες σειρές κατά τα τέλη του 19ου και στις αρχές του 20ού αιώνα. Στη θεωρία που ανέπτυξε, υπάρχουν άπειρες σειρές διαφόρων μεγεθών (πληθικότητα). Για παράδειγμα, το σύνολο των ακεραίων είναι αριθμήσιμο, ενώ το άπειρο σύνολο των πραγματικών αριθμών είναι υπεραριθμήσιμο. Бесконе́чность — категория человеческого мышления, используемая для характеристики безграничных, беспредельных, неисчерпаемых предметов и явлений, для которых невозможно указание границ или количественной меры. Используется в противоположность конечному, исчисляемому, имеющему предел. Систематически исследуется в математике, логике и философии, также изучаются вопросы о восприятии, статусе и природе бесконечности в психологии, теологии, физике соответственно. Бесконечность обозначается символом . Исторически первые проблемы бесконечности — вопросы конечности пространства и времени, количества вещей в мире, более сложные проблемы — возможность бесконечного деления континуума, возможность оперирования с бесконечными объектами (проблема актуальной бесконечности), природа и поведение бесконечно малых величин — инфинитезималей, наличие различных типов бесконечности и соотношение между ними. Наиболее глубокое исследование бесконечности предпринято в математической теории множеств, в которой построено несколько систем измерений различных видов бесконечных объектов, однако без дополнительных искусственных ограничений такие построения вызывают многочисленные парадоксы, пути их преодоления, статус теоретико-множественных построений, их обобщений и альтернатив являются основным направлением исследований бесконечности у философов современности.
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