Confused intuition, everywhere and nowhere.

Breathed infinity and eternal infinity are the same size. This presents several counterintuitive facts.

In essence, we've shown that the real line \(\mathbb{R}\) (i.e. all of time) is the same size as a proper subset of itself. We can look at the real line, and see that the interval from \(0\) to \(1\) is only a small piece of it -- and yet the real line is the same size as that interval. There are the same number of points (i.e. instants) on the whole line as in the small interval. And a similar argument works with arbitrarily small intervals; we could take the interval from \(0\) to \(0.000001\), rather than \(0\) to \(1\).

Rather than consider the time it takes to breathe, we could consider the time it takes light to travel between one person’s smile and another person’s eyes; this contains the same number of instants as all of eternity.

This bizarre property is so fundamental to infinite sets, that it in fact characterizes them. In other words, every infinite set exhibits this property, and conversely if you have a set that is the same size as a proper subset of itself, that set must be infinite.

Our intuition gets confused again if we combine the results of this section with some in the others.

Breathed infinity and eternal infinity are the same size; here we are equating eternal infinity with the number of points on the real line. In the second module, it is shown that breathed infinity is larger than countable infinity. As explained in the In-Depth explanation of the second module, the collection of fractions (i.e. rational numbers, those of the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers but \(b\) is not zero) is the same size as countable infinity.

Putting this together, the infinite set of rational numbers is smaller than the infinite set of real numbers.

But where are the rational numbers, on the real line? They're everywhere. Given any point on the real line, and some arbitrarily small interval near that point, there will be infinitely many rational numbers in that interval. Mathematicians say that the rational numbers are dense on the real line.

For example, \(\pi\) is an irrational number (i.e. a real number that isn't rational), approximately equal to \(3.1415926\). So we have fractions

\[\frac{3}{1}, \frac{31}{10}, \frac{314}{100}, \frac{3141}{1000}, \frac{31415}{10000}, \frac{314159}{100000}, ...\]

that get us as close to \(\pi\) as we care to go.

The rational numbers are dense on the real line. But, as argued above, there are infinitely more real numbers than rational numbers. So if you throw a dart at the real line, the chances of hitting a rational number are zero.

They are everywhere, in one sense, but in another sense they are nowhere.