What remains is to show that

given two sets A and B, one is less than or equal to the other.

If one thinks of this problem for two "arbitrary" sets, one sees the

hopelessness of trying to actually define a map from one into the other.

I believe that almost anyone would have a feeling of unease about

this problem;

{{außer einigen Experten in dsm}}

namely that, since nothing is given about the sets, it

is impossible to begin to define a specific mapping. This intuition is,

of course, what lies behind the fact that it is unprovable in the usual

Zermelo-Frankel set theory. Cantor suggested a method of proving it.

It depended on the notion of a well-ordering, i.e., an ordering of a set

A in which every nonempty subset has a least element. If A and B

have well-orderings, it is not hard to show that either there is a unique

order preserving map of A onto an initial segment of B, or vice versa.

So if A and B are well-ordered, we can define a unique map which

shows that the cardinality of A is less than or equal to that of B,

or conversely. By an ordinal we simply mean an equivalence class of

well-orderings. It follows that the ordinals are themselves well-ordered.

Now if one assumes the well-ordering principle, that all sets have a

well-ordering, it follows that all the cardinal numbers have at least

one ordinal of that cardinality. Thus, we have in the "sequence" of

all ordinals, particular ordinals which we now call cardinals, which are

defined as ordinals whose cardinality is greater than that of any of its

predecessors.

[Paul J. Cohen: "The discovery of forcing", Rocky Mountain Journal of Mathematics 32, 4 (2002)]

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