Actual and Potential Infinity

New Zeno and Actual Infinity

In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveller from reaching his destination. In this paper it is argued, contra Priest and Yablo, that the paradox must be resolved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical form as Yablo’s Paradox. It is suggested that constructivis...

An objective justification for actual infinity?

This document is a record of my contribution to a panel discussion which took place on July 27, 2011 as part of the Infinity and Truth Workshop held at the Institute for Mathematical Sciences, National University of Singapore, July 25–29, 2011. In preparation for the panel discussion, Professor Woodin asked each panelist to formulate a yes/no question to be asked of a benevolent, omniscient mat...

Infinity at stake Selected arguments on the actual infinity Against dogmatism and intolerance

Rethinking the Numbers: Quadrature and Trisection in Actual Infinity

The problems of squaring the circle or “quadrature” and trisection of an acute angle are supposed to be impossible to solve because the geometric constructibility, i.e. compass-and-straightedge construction, of irrational numbers like � is involved, and such numbers are not constructible. So, if these two problems were actually solved, it would imply that irrational numbers are geometrically co...

Conceptual Metaphor and the Cognitive Foundations of Mathematics: Actual Infinity and Human Imagination*

The Infinite is one of the most intriguing ideas in which the human mind has ever engaged. Full of paradoxes and controversies, it has raised fundamental issues in domains as diverse and profound as theology, physics, and philosophy. The infinite, an elusive and counterintuitive idea, has even played a central role in defining mathematics, a fundamental field of human intellectual inquiry chara...