The Schröder-Bernstein Theorem
The purpose of counting is to compare the size of one set with that of another; the most familiar method of counting the elements of a set is to arrange them in some appropriate order. The theory of ordinal numbers is an ingenious abstraction of the method, but it falls somewhat short of achieving the purpose. This is not to say that ordinal numbers are useless; it just turns out that their main use is elsewhere, in topology, for instance, as a source of illuminating examples and counterexamples. In what follows we shall continue to pay some attention to ordinal numbers, but they will cease to occupy the center of the stage. (It is of some importance to know that we could in fact dispense with them altogether. The theory of cardinal numbers can be constructed with the aid of ordinal numbers, or without it; both kinds of constructions have advantages.) With these prefatory remarks out of the way, we turn to the problem of comparing the sizes of sets.
KeywordsPartial Order Initial Segment Comparability Theorem Ordinal Number Cardinal Number
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