Or the Intensive and the Extensive Infinite

„In order to know the nature of the numbers according to which „geometrical distance” to be defined, for example, it would be necessary to know what happens both at indefinitely tiny and indefinitely large distance. Even today, these questions are without clearcut resolution.” (Roger Penrose, The Road to Reality)

Abstract

Today's mathematics is based on the continuum hypothesis (CH) about infinitesimals. Although Cantor formulated CH for extensive infinity, it can also be reformulated for intensive infinity. Thus, we have the foundation of a richer set theory, including classical mathematical analysis, in which the two-element numbers can now model the potential infinities corresponding to our experience, and the actual forms of infinities are captured not by their quantitative but by their special qualitative properties, and are modelled both "small" and "large".

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