Claim: eternal infinity and breathed infinity are the same size; or, \(|\mathbb{R}| = |(0,1)|.\)

Eternal infinity is every instant of time ever. We imagine that time never ends and goes on forever, and that the past also never ends, and stretches behind us forever. We think of all the moments of time, forever into the past and forever into the future.

To make this mathematically precise, we think of time as a continuous one-dimensional phenomenon. Contemplating all time is like contemplating the set of all points on a never-ending continuous line. In Basics of Set Theory, we called this continuous line of points the real number line, written \(\mathbb{R}\). Thus we can equate eternal infinity with the set \(\mathbb{R}\).

Breathed infinity, then, is a segment of this continuous line. It is all the time that passes from the beginning of an inhale to the end of an exhale. For simplicity, we can assume the breath takes one second, starting at \(t=0\) and ending at \(t=1\). For mathematical reasons, let's drop the two endpoints and only consider the points of time in between \(t=0\) and \(t=1\).

If we think of this subset of eternal infinity as a subset of the real line \(\mathbb{R}\), then we write it as \((0,1)\). This notation \((0,1)\) means all the real numbers in between zero and one (but not including zero or one). So we are going to equate breathed infinity with the set \((0,1)\).

The claim that eternal infinity is the same size as breathed infinity, then, is the claim that \(|\mathbb{R}| = |(0,1)|.\)