Claim: breathed infinity is larger than digital infinity; or, \(|(0,1)| > |\mathbb{N}|\).

Breathed infinity is the infinity of instants in a single breath. Every moment of lived consciousness that passes from the beginning of an inhale to the end of an exhale. It is an uninterrupted flow of points, a brief segment of a continuum. Let's denote this set by \(\mathscr{B}\).

To make \(\mathscr{B}\) more mathematically precise, we imagine it as a piece of the continuous real line that represents time. Or imagine looking at an analog stopwatch run as you breathe in and then breathe out; the movement of the watch hand carves out a line segment, capturing every instant along the way.

For the sake of simplicity, assume your breath takes one second. Ignore \(t=0\) and \(t=1\), but take every instant between \(0\) and \(1\). This set, thought of as a subset of the real line, is written \((0,1)\).

Figure: Real line with interval between zero and one marked.

So we will assume \(\mathscr{B}\) is the same size as \((0,1)\).

And in the first module, we showed that digital infinity is the same size as \(\mathbb{N}\).

Thus we will justify the claim that breathed infinity is larger than digital infinity, by proving the mathematical statement \(|(0,1)| > |\mathbb{N}|\).