Infinity Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ===Zeno: Achilles and the tortoise=== {{Main|Zeno's paradoxes#Achilles and the tortoise}} [[Zeno of Elea]] ({{c.}} 495 – {{c.}} 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,<ref name="Zeno's paradoxes">{{cite web|url=https://plato.stanford.edu/entries/paradox-zeno/ |title=Zeno's Paradoxes |date=October 15, 2010 |website=Stanford University |access-date=April 3, 2017}}</ref> especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by [[Bertrand Russell]] as "immeasurably subtle and profound".<ref>{{harvnb|Russell|1996|p=347}}</ref> [[Achilles]] races a tortoise, giving the latter a head start. *Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward. *Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further. *Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further. *Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further. Etc. Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him. Zeno was not attempting to make a point about infinity. As a member of the [[Eleatic]]s school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument. Finally, in 1821, [[Augustin-Louis Cauchy]] provided both a satisfactory definition of a limit and a proof that, for {{math|0 < ''x'' < 1}},<ref>{{cite book|last=Cauchy|first=Augustin-Louis|author-link=Augustin-Louis Cauchy|access-date=October 12, 2019|title=Cours d'Analyse de l'École Royale Polytechnique|year=1821|publisher=Libraires du Roi & de la Bibliothèque du Roi|url=https://books.google.com/books?id=UrT0KsbDmDwC&pg=PA1|page=124}}</ref> <math display="block">a+ax+ax^2+ax^3+ax^4+ax^5+\cdots=\frac{a}{1-x}.</math> Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with {{math|1=''a'' = 10 seconds}} and {{math|1=''x'' = 0.01}}. Achilles does overtake the tortoise; it takes him <math display="block">10+0.1+0.001+0.00001+\cdots=\frac {10}{1-0.01}= \frac {10}{0.99}=10.10101\ldots\text{ seconds}.</math> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page