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In this paper we will trace the changing notion of infln1ty throughout Western thought. In the first part ideas from both mathematics and philosophy wl11 be brought together, for often in discussions on infinity the one 1s called upon to justify statements in the other. It would be impossible to ment1on all the uses of inf1nity as it is seen in almost every top1c in mathemat1cs, However, in the following parts we will study 1n detail Georg Cantor's theory of transfin1te numbers, the ar1thmetic of infinite sets. The Contlnuum Hypothesis, today's most perplexing mathematical problem concerning infinity will also be examined. We will conclude with a brief description of the present theorles of the infinite.