Equality of cardinal numbers

We do not introduce a different symbol for every individual set, because sometimes two different sets can have the same size. For instance, two finite sets have the same size if and only if one set can be obtained from the other by relabeling the elements; for instance, $\{1,2,3\}$ has the same cardinality as $\{a,b,c\}$. One of the fundamental properties of Cantor's cardinal numbers is that the same should hold for arbitrary sets, finite or not. This can be reworded as follows:

RULE 1: $    #S = #T &iff#iff;$ there exists a bijection $f: S &rarr#rightarrow;T$.

For instance, if $S= \{0,-1,-2,-3,\dots\}$, then $\char93 {\mathbb{N}}= \char93 S$ because there is a bijection ${\mathbb{N}}\rightarrow S$ sending each nonnegative integer $n$ to $-n$:

$$\longleftrightarrow 0$$ 0

$$1 \longleftrightarrow -1$$

$$2 \longleftrightarrow -2$$

$$3 \longleftrightarrow -3$$

$$\vdots$$