By Yi Lin
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Proof: Let h : A → B be a mapping such that h establishes the similarity between A and h(A). If the theorem were not true, then we would have β < α and then B would be similar to a segment of h(A). 5. 11. The set W ( α) consisting of all ordinals less than α is wellordered by the relation ≤. Moreover, the type of W (α ) is α. Proof: Let A be a well-ordered set of type α. Associating the type of the segment O(a) with the element a ∈ A, we obtain a one-to-one mapping of A onto W ( α). It is easily seen that the following conditions are equivalent: (1) a 1 precedes a 2 or a 1 = a 2 .
A full line can be obtained similarly from a bent, open interval (Fig. 2). (c) Two finite sets are equipollent if, and only if, they contain equally many elements. (d) An infinite set can be equipollent to one of its proper subsets. This is shown by the equipollence of the sets {1,2,3, . . } and {2, 4, 6, . . }. 2. A full line obtained from a bent open interval. Naive Set Theory 25 thus exhibit in this respect an entirely different behavior from that of finite sets [see example (c)]. This property can therefore be used to distinguish finite and infinite sets independently of any enumeration.
Consequently, U is similar to the set of all rational numbers ordered naturally. 7, at least one element a of A and, because of (3), not more than one element of A. This, however, implies the assertion. For the set of all rational numbers can be mapped on U, whereby with every gap in the domain of rational numbers there is associated precisely one gap of U and, consequently, precisely one element of A. We have thus obtained a similarity mapping of the set of all real numbers on the set A. 5. Well-Ordered Sets An ordered set ( X, ≤ X ) is well-ordered if it is linearly ordered and every nonempty subset of X contains a first element with respect to the relation ≤ X .