Infinite iterative processes and transcendent objects: A model of construction of mathematical infinity from the APOS Theory

Authors: Diana Paola Villabona Millán, Solange Roa Fuentes

Abstract: The present study aims to examine the mental structures that a person
can develop to construct the mathematical concept of infinity in two particular contexts: “the paradox of Achilles and the tortoise” and the “Sierpin´ ski triangle”.
Based on the genetic generic decomposition of the infinite, proposed by
Roa-Fuentes and Oktaç (2014), this investigation focuses on the study of the
particular characteristics, mechanisms and structures produced by each context.
The analysis of data from work done by postgraduate students (in Mathematics
and Mathematics Education) shows how from the infinite iterative process
(potential infinity) advances towards to a transcend object (actual infinity).
Furthermore, the results reflect the importance of the coordination mechanism
in the construction of infinite iterative process.

Key words: APOS theory, paradoxes, Sierpin´ ski triangle, infinite iterative processes,
transcendent objects.


El infinito potencial y actual: descripción de caminos cognitivos para su construcción en un contexto de paradojas

Solange Roa Fuentes y Asuman Oktaç

Abstract:A generic genetic decomposition of infinite and two particular genetic decompositions is proposed in this paper: A paradox for tennis balls and one for the Hilbert Hotel paradox. These analyzes take as a basis the construction of infinite iterative processes and transcendent objects, related to the infinite potential and current, respectively. In addition an analysis of the characteristics
of the processes involved in each situation and the complexity of coordinating them with the set of natural numbers to construct infinite iterative processes is presented. The difficulty faced by an individual to coordinate processes of different nature is studied: converging and diverging to infinity as a build process.

Keywords: apos theory, infinite iterative processes, mechanisms and mental structures, transcendent object, paradoxes.