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Extra info for Mathematical, philosophical, religious and spontaneous students' explanations of the paradox of Achilles and the tortoise
The mathematical opefation of swnming the infinite series tell us where and M e n Achilles will catch the Tortoise I h e can catch the Tortoise at ail, that is a big '+ ( Salmon, 1970, p 27-28) Tme, the sum of the distances covered by Achilles are convergent but the paradox is trying to question whether it is at al1 possible for Achilles to ever catch the Tortoise, so assuming that h e p n catch up and then figuring out when this will ocwr will not bring us any closer to resolving out dilemma. Black argues that it is impossible to Say that Achilles has completed an infinite number of steps.
These sets were called denumerable. However, he was also able to prove that the set of real numbers contained more elements than the above mentioned sets and thus had a greater cardinality and was non denumerable. The cardinality of the cet of real nwnbero was called the continuum denoted by C. Cantor didn't stop there, he went on to fmd additional infinities with largw cardinality Vian the continuum. For example aie set of atl subsets of real numbers has cardinality 2c and 2c > C . The continuum is the cardinality of the number Iine.
Each pair was presented the paradox and allowed to work it out amongst themselves for about a half hour. Occasionally I interjeded in order to push forward the conversation or to gain further insight into the students' ideas by suggesting altemafe explanations. Explanatjon By Sfudenh with a Background in Jewish Studiès. I first presented the paradox to the pair of rabbinical students U and W. This O Group had no real fonnal secular educaüon past grade 8 or 9. They had a basic knowledge of arithmetic but had not studied anything but Jewish studies in at least the past 10 years.