Cardinalities

But many Note that one cannot say "most" because the cardinalities of the two sets (functions that can and cannot be represented with a finite number of basis functions) are the same.

Hausdorff's general products and powers of cardinalities had led him to the concept of partially ordered set.

In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities.

Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.

Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization.

Some infinite cardinalities are greater than others.

The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities.

This is a quite different statement, and is absurd for large finite cardinalities.