By Abraham A. Fraenkel

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**Extra resources for Abstract Set Theory**

**Example text**

One should not overlook the fact that this definition explicitly uses the concept of natural number. Whoever endeavors to base the concept of number on the more general concepts of set-theory (cf. $10, 6), should either not use the concept finite previously - which would be difficult to carry out - or formulate the definition in a way which does not use natural number explicitly, as did Russell when introducing the term inductive I). I n this and the following sections we shall rely o n the theory of natural numbers; that is to say, we shall use arithmetic for the development of our theory.

I, 5 21 41 THE FUNDAMENTAL CONCEPTS. FINITE AND INFINITE survey will show us, however, that the two definitions have the same meaning I). We start with an assertion that is almost self-evident: THEOREM 5. A set which is equivalent t o a finite (infinite)set, is again finite (infinite). Proof. The proof based on definition \'I is left to the reader. We shall base our proof on definition V I I a,nd show that a set T which is equivalent t o a reflexive set X, is again reflexive. S being reflexive, a proper subset of it, S ' , exists such that X 8'.

But this is the case b), already found contradictory. The contradiction reached in each case shows that our supposition about S being equivalent to a proper subset was false. I n other words, the truth of our theorem for all sets of n elements implies its truth for all sets of n 1 elements. Since it is true for n = 1, the proof is completed. + The properties of infinite sets stand in sharp contrast to theorem 4. Let us for example take the set N of all natural numbers. A proper subset N' of N is created by dropping the element 1, viz.