Usopeo

**Usopeo** (kwa Kiingereza: *Infinity*) ni hali ya kutokuwa na mipaka, kikomo wala mwisho. Huwakilishwa kwa alama ya **∞**, inayoitwa alama ya usopeo. Tangu zamani za Wagiriki wa kale, katika falsafa hali hiyo imekuwa mada ya mijadala mingi. 

Katika karne ya 17, kwa kuanzishwa kwa alama ya usopeo na mfumo wa kalkulasi, wataalamu wa hisabati walianza kufanyia kazi mfululizo usio na mwisho. Wakati wataalamu wa hisabati wakihangaika na misingi ya kalkulasi, haikubaki wazi ikiwa usopeo ungeweza kuchukuliwa kama namba au ukubwa. 

Mwishoni mwa karne ya 19, Georg Cantor alipanua somo la hisabati la usopeo kwa kusoma seti zisizo na mwisho, akionyesha kuwa zinaweza kuwa za saizi mbalimbali. Katika matumizi hayo, usopeo ni dhana ya kihisabati, na vitu vya kihisabati visivyo na mwisho vinaweza kusomwa na kutumiwa kama kitu kingine chochote cha kihisabati.

Dhana ya kihisabati ya usopeo inaboresha na kupanua dhana ya kale ya falsafa. Miongoni mwa kanuni za nadharia seti ya Zermelo–Fraenkel ni kanuni ya usopeo (axiom of infinity), ambayo inahakikisha uwepo wa seti zisizo na mwisho. Dhana ya kihisabati ya usopeo na utumiaji wa seti zisizo na mwisho hutumiwa sana katika hisabati, hata katika maeneo kama kumbu (combinatorics). Kwa mfano, uthibitisho wa Wiles wa nadharia ya mwisho ya Fermat unategemea uwepo wa ulimwengu wa Grothendieck, ambazo ni seti kubwa sana za usopeo.

## Tanbihi
## Marejeo
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## Vyanzo
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*D.P. Agrawal (2000). *Ancient Jaina Mathematics: an Introduction*, Infinity Foundation.
* Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
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* Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", *Indian Journal of History of Science*.
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* H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
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* O'Connor, John J. and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' , *MacTutor History of Mathematics archive*.
* O'Connor, John J. and Edmund F. Robertson (2000). 'Jaina mathematics' , *MacTutor History of Mathematics archive*.
* Pearce, Ian. (2002). 'Jainism', *MacTutor History of Mathematics archive*.
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## Viungo vya nje
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* *A Crash Course in the Mathematics of Infinite Sets *, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to *Infinite Reflections*, below. A concise introduction to Cantor's mathematics of infinite sets.
* *Infinite Reflections *, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
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* Hotel Infinity
* John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' , *MacTutor History of Mathematics archive*.
* John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics' , *MacTutor History of Mathematics archive*.
* Ian Pearce (2002). 'Jainism', *MacTutor History of Mathematics archive*.
* The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
* Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)

Jamii:Hisabati
Jamii:Falsafa