Chapter: Can Cardinals be Ordered?
We have defined a comparison between cardinals using the sign "$\le$" for "lower or equal", like it is used in case of a usual order relation between sets. But is it really one? Well, not really, because cardinalities are not sets, they are classes of sets, by definition. However, it is possible to establish the main properties common to all order relations:
- Reflexivity: Obviously, for every set $A$ the identity function $f(a)=a$ for all $a\in A$ is injective. Therefore $|A|\le|A|.$
- Transitivity: If there are injective functions $f:A\to B$ and $g:B\to C,$ then we know that the
composition $g\circ f$ is also is injective. Therefore, $|A|\le|C|$.
- Antisymmetry: By definition of this property and transferred to the case of cardinals, this would mean that if there are injective functions $f:A\to B$ and $g:B\to A,$ then $|A|=|B|$, i.e. then there is a bijective function $h:A\to B.$
The last property is known as the Schröder-Bernstein theorem, which we will prove soon. Historically, there were many more or less complicated proofs of this theorem, some of them turned out to be wrong.
The first correct proof was provided by Richard Dedekind (1831 - 1916).
- Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001