I'm reading the PCM (The Princteton Companion to Mathematics) article on set theory (IV.22) by Joan Bagaria. I'm trying to understand his comment on difference between cardinal and ordinal numbers.
In set theory, one likes to regard all mathematical objects as sets. For ordinals this can be done in a particularly simple way: we represent 0 by the empty set, and the ordinal number α is then identified with the set of all its predecessors. For instance, the natural number n is identified with the set {0, 1, . . . , n − 1} (which has cardinality n) and the ordinal ω + 3 is identified with the set {0, 1, 2, 3, . . . , ω,ω + 1,ω + 2}.
... cardinal numbers are used for measuring the sizes of sets, while ordinal numbers indicate the position in an ordered sequence. This distinction is much more apparent for infinite numbers than for finite ones, because then it is possible for two different ordinals to have the same size. For example, the ordinals ω and ω+1 are different but the corresponding sets {0, 1, 2, . . . } and {0, 1, 2, . . . , ω} have the same cardinality, as figure 1 shows. In fact, all sets that can be counted using the infinite ordinals we have described so far are countable. So in what sense are different ordinals different? The point is that although two sets such as {0, 1, 2, . . . } and {0, 1, 2, . . . , ω} have the same cardinality, they are not order isomorphic: that is, you cannot find a bijection φ from one set to the other such that φ(x) < φ(y) whenever x < y. Thus, they are the same “as sets” but not “as ordered sets.”
Informally, the cardinal numbers are the possible sizes of sets. A convenient formal definition of a cardinal number is that it is an ordinal number that is bigger than all its predecessors.
Could anyone explain with examples what the sentence in bold means? What could be an example that an ordinal number is equal to or smaller than some of its predecessors?