The Infinite Challenge: Levels of Conceiving the Endlessness of Numbers

Abstract

To conceive the infinity of integers, one has to realize: (a) the unending possibility of increasing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greater than that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set is itself infinite (immeasurable gap). Three experiments probed these understandings via competitive games and choice tasks accompanied by in-depth interviews. Participants were children 6 to 15 years old and adults. The results suggest that roughly from about age 8 on, children grasp potential and actual infinity. However, for several additional years their conception of actual infinity is incomplete because the immeasurable gap between a finite and an infinite set is not entirely internalized. Even many adolescents and adults fail to appreciate this gap. Distinguishing between number concepts and their names facilitates conceiving aspects of infinity. Educational implications of these findings are discussed.

ACKNOWLEDGMENTS

The study was partly supported by the Sturman Center for Human Development, The Hebrew University of Jerusalem. Thanks are due to many students of the Hebrew University for their significant contributions to the experiments and their interpretations. I am grateful to Raymond Nickerson, the editor, David Tall, and another anonymous reviewer for helpful comments and suggestions. However, I alone am responsible for the views expressed in the article. Yael Oren took care of the figures. I received invaluable help from Raphael Falk throughout all the stages of working on this project.

Notes

Since many of my graduate students over the years helped in collecting data and discussing the issues, I often use the plural first-person pronoun in reporting the research.

^aBased on less than 10 participants.

^aBased on the infinity scores in Game 1 (Experiment 1, N = 345).

^bBased on positive scores in Comparison of Sets (Experiment 2, N = 80).

^cBased on verbal-understanding scores in Location of Sets (Experiment 3, N = 428). In the adult age group the percent was 52.

In his article on “The Relativity of Wrong” Asimov (1989) gives an example suggesting that in some sense considering an immense finite cardinal number almost infinite is not utterly (psychologically) wrong: “Newton's theories of motion and gravitation were very close to right, and they would have been absolutely right if only the speed of light were infinite. However, the speed of light is finite” (p. 42). He further explains that although the difference between infinite and finite is itself infinite, at the speed at which light actually travels, it takes it 0.0000000033 seconds to traverse a meter, and it would take 0 seconds if light traveled at infinite speed. Because the inverse of the huge finite speed is almost zero, that speed might reasonably seem at first to be almost infinite.

For years I have been embarrassed about confusing between trillions, billions, and their meaning in different countries. Now I realize that, though I should perhaps not be proud of it, being able to deal with symbolic representations of numbers is more important.