An unfortunate situation

If $S$ and $T$ are finite sets, and $S$ is a proper subset of $T$ (this means that $S \subseteq T$ but $S \not= T$), then $\char93 S < \char93 T$. Unfortunately, this is no longer true when we consider infinite sets!

For example, if $S=\{3,4,5,6,\dots\}$, then $S$ is a proper subset of ${\mathbb{N}}$, but according to the definition, $\char93 S = \char93 {\mathbb{N}}$, because there is a bijection from $S$ to ${\mathbb{N}}$:

$$3 \longleftrightarrow 0$$

$$4 \longleftrightarrow 1$$

$$5 \longleftrightarrow 2$$

$$6 \longleftrightarrow 3$$

$$\vdots$$

Moreover, this unfortunate situation is unavoidable if we want to keep Rules 1 and 2 (and we do). We just have to live with it.