ABSTRACT
What has been discussed so far in this book indicates that the concepts of actual and potential infinite have been studied since antiquity and the question of whether or not they are the same has been debated since the very early part of the recorded history of Western civilization. Therefore, the history can be roughly divided into two segments: The early period that lasted more than two thousand years, and the later period that took place in the past one hundred plus years. During the former period, scholars mostly agreed that actual and potential infinite are different at least in temporal terms, while during the latter period, the boundary between these two kinds of infinite seemed to be blurred. However, the vase puzzle once again rekindled the light of askingwhether actual and potential infinite are the same, because the puzzle seems to suggest they are different not only in temporal terms but also in analytical terms. This indication of course is fundamentally different from the purely temporal arguments existing during the first segment of history. In this chapter, we will show that the answer to this age old question is both yes and no, constituting a new crisis in modern mathematics.
