By Redei L.

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Also BIRKHOFF (1948). § 15. Theorem of Well-ordering THEOREM 19 (theorem Of ZERMELO or the theorem of well-ordering). Any set can be well-ordered. Before the proof we note by way of introduction that: Given two semiordered sets A, B, we shall only then say that A is a subset of B (or B is an o verset of A), if, first of all, the set B contains the set A and any relation x < y (x, y E A) which is true in B, is also true in A. ) Naturally this also applies to the concept of the cuts of a semiordered set.

Consequently the assertion is proved. , n) are called finite, any other set is called infinite. , an) . THEOREM 8. A set is finite if, and only if, it is not equivalent to any of its proper subsets. First, we prove that it is impossible to map a finite set one-to-one onto any proper subset of itself. , n> . In the case n = l the statement is true. We assume its truth for any n and prove that it is true for n + 1. , that a one-to-one mapping x -+ ox of the set <1, . , n + 1> onto a proper subset S of it exists.

1), onto a proper subset of itself. This contradicts the induction assumption. 1) does not hold, then n + 1 E S. Then there certainly is an a (= 1, . , n) not belonging to S. 2) holds. 2)]. , n). 1), whereby we have proved the above statement. Later we shall prove the second part of Theorem 8. > by the one-to-one mapping (123... I` 234... > of all the natural numbers cannot be finite. Consequently it is infinite. ) of the natural numbers, we call countably infinite. The finite and the countably infinite sets are called countable sets.