No abstract

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity.KEY WORDS: actual and potential infinity, APOS Theory, classical paradoxes of the infinite, encapsulation, history of mathematics, human conceptions of the infinite, large finite sets

No abstract

This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical issues related to infinity.

No abstract