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# Aleph-Null Bottles Of Beer On The Wall

Let’s quickly recap the most important (and weird) properties of the “natural numbers infinity” we call $\aleph_0$.

Welcome to another part of the “Maths For Normal People” series. You will find all the previous parts in this article.

So far, we have derived the following strange facts ($n$ stands for an arbitrary natural number, e.g. $1, 2, 3, 4,\ldots$).

$$\displaylines{ \aleph_0 + n = \aleph_0 \\ \aleph_0 - n = \aleph_0 \\ \aleph_0 \cdot n = \aleph_0 \\ \frac{\aleph_0}{n} = \aleph_0 \\ \aleph_0 - \aleph_0 \text{ makes no sense!} }$$

The apparent absurdity of infinity can be summed up by this truth:

$$\aleph_0 - 1 = \aleph_0$$

Sounds difficult? Let me put that in a song. Know 99 Bottles of Beer?

Aleph-null bottles of beer on the wall, Aleph-null bottles of beer,
Take one down, and pass it around, Aleph-null bottles of beer on the wall.

Aleph-null bottles of beer on the wall, Aleph-null bottles of beer,
Take one down, and pass it around, Aleph-null bottles of beer on the wall.

Aleph-null bottles of beer on the wall, Aleph-null bottles of beer,
Take one down, and pass it around, Aleph-null bottles of beer on the wall.
$$\vdots$$

You just keep singing as long as there are any bottles left ;-)

Want to read more? You will find the outline of all parts of the series in this article.

September 15, MMXI — Mathematics, Maths For Normal People, Infinity.