# Cardinality, The Cardinals do not Form a Set

## The Cardinals do not Form a Set

Let S be a set and let W be an ordinal that maps 1-1 into S.
Let T be the image of W.
Now W implements a well ordering on the members of T, and leaves the rest of S alone.
Conversely, if T is a well ordered subset of S,
an isomorphism maps T back to a unique ordinal W.
Let U be the set of all possible relations on S cross S.
(Note that U cannot be built without the power set axiom.)
Restrict U to those relations that implement a well ordering on a subset of S,
and have no ordered pairs from the rest of S.
Let D be the domain of a function f
that carries the well ordered subset to the corresponding ordinal.
Invoke the replacement axiom and the range of f becomes a set.
Yet the range is precisely the corresponding ordinal that fits inside S.
If every ordinal embeds in S then there is a set containing all the ordinals,
which is
impossible.
Therefore there is some ordinal that does not map into S.

Let W be the least ordinal that does not map into S.
If W can be mapped onto a lower ordinal,
compose the two functions to carry W into S.
Therefore W is a cardinal.
For any set S, there is a cardinal, i.e. a bigger set, that cannot embed in S.

If S is the set of all cardinals, there is some cardinal W that is not a subset of S.
Since both are ordinals, S is a proper subset of W.
Now S is transitive, and if it contains W it contains all the members of W,
which is impossible.
Hence there are cardinals other than those found in S.