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...al number]]s. In fact, [[Cantor's theorem]] implies the existence of an "[[infinity]] of infinities". He defined the [[cardinal number|cardinal]] and [[ordinal ...tics]]) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of [[God]],<ref name = "nuozkv">Dauben, 1977, p. 86; Dauben, ...

...846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 189 ...hey accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref> ...

...'[[ad infinitum]]''. The theorem illustrates the perils of reasoning about infinity by imagining a vast but finite number, and vice versa. The [[probability]] ...the chance of not typing ''banana'' is roughly 0.17%). As ''n'' approaches infinity, the probability ''X''_''n'' [[limit (mathematics)|approaches]] z ...

...though two [[parallel lines]] can be considered as tangent at a [[point at infinity]] in [[inversive geometry]] (see [[#Inversive methods|below]]).<ref>{{cite ...inversion is said to send '''P''' to infinity. (In [[complex analysis]], "infinity" is defined in terms of the [[Riemann sphere]].) Inversion has the useful p ...

...e ring's particles to the area of the ring. It assumes values from zero to infinity. A light beam passing normally through a ring will be attenuated by the fa ...

{{bquote|For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and in ...66)</ref> In ''Pensées'', Pascal surveys several philosophical paradoxes: infinity and nothing, faith and reason, soul and matter, death and life, meaning and ...

...use ''x_k'' is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ...average energy in the higher-frequency modes goes to zero as ''ν'' goes to infinity; moreover, [[Planck's law of black body radiation]], which describes the ex ...

...isolated systems that do not require knowledge of spacetime properties at infinity, cf. {{Harvnb|Ashtekar|Krishnan|2004}}.</ref> ...DM and Bondi masses in asymptotically flat spacetimes|Bondi mass]] at null infinity.<ref>For a pedagogical introduction, see {{Harvnb|Wald|1984|loc=sec. 11.2}} ...

...ieves this by calculating the value of a [[geometric series]] that sums to infinity with the [[ratio]] {{frac|1|4}}. ...

...amp;data_type=PDF_HIGH&amp;type=PRINTER&amp;filetype=.pdf X-ray spectra at infinity from a relativistic accretion disk around a Kerr black hole], ''Astronomica ...

...uch an ''n'' may not exist, in which case the order of ''a'' is said to be infinity. The order of an element equals the order of the cyclic subgroup generated ...

...ond-order and higher-order perturbation calculations can give apparently [[infinity|infinite]] contributions to the sum. Such unphysical results are corrected ...