By Abraham A. Fraenkel

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Z, I n the sequence mentioned in definition 111, the same set may, of course, appear several times. This does not alter the sum. For reasons of symmetry (and of symbolical simplicity) it would be 3, preferable to start, not from one of the sets, but from their sum. This will not change anything. 30 ELEMENTS. CONCEPT OF CARDINAL NUMBER [CH. I other - and in different ways - SO that to every number corresponds a single piece of fruit, and vice versa. The latter addition is essential. e. a single-valued or unique correspondence) even after dropping, for example, the banana from the first set : one may relate different apples t o each of the numbers 1, 2 , 3, 4, 5 , one orange to 6 and tho other orange to 7 as well as to S - a procedure that would not, however, make a single number correspond t o exery fruit, although a single fruit is related to every number.

V’e may thus formulate our results: THEOREM 3. The equivalence of sets (definition V) is a reflexive, symmetrical, and transitive relation; that is to say: S S ; S T implies T S ; X I’ and T W together imply S W . Hence in a totality of sets, such that every set is equivalent to a definite one, a n y two sets of the totality are equivalent to each other. Equivalence has different meanings in different branches of mathematics. However, all these meanings share the three properties expressed in theorem 3.

Not every sequence, however, is an enumerated set. For in a sequence l) a member may appear repeatedly, which has been excluded for sets. For the specialization efecticely denumerable, see 0 5, 4. CH. I, 0 31 45 DENUMERABLE SETS 2. Simplest Examples and Theorems. Let us consider a few instances of denumerable sets. As shown on p. 39, the set ( 2 , 3, 4, 5 , . . } is also denumerable. The same obviously holds for any set ill originating from the set of all natural numbers by dropping any finite number of elements.

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Abstract Set Theory by Abraham A. Fraenkel

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