Algebra Vol. I by Redei L.

By Redei L.

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Example text

5). This contradiction proves Theorem 4. This proof is a modification of the proof of LORENZEN (1959). Cf. also PICKERTG6RKE (1958) and KALMAR (1949-50). We apply Theorem 4 to the definition of the sum m + n and product in, n of two natural numbers m, n. ) and ml =m, mn'=mn+m are valid. ) NATURAL NUMBERS t3 It maybe proved that the well-known arithmetic rules for the natural numbers follow from these recursive definitions, which implies that axioms I and II in fact encompass all the essential properties of the natural numbers.

As these attempts have been unsuccessful, mathematicians have now accepted the axiom of choice. § 16. Transfinite Induction By means of (complete) induction we are able to prove only such propositions as can be suitably split up into countably many component propositions. The following is more general. THEOREM 20 (theorem of transfinite induction). Let a proposition A, correspond to each element v of a well-ordered set I. If the truth of A follows from the assumption of the truth of all propositions Ax (x < v) for each v, then all the propositions A, (v E I) are true.

Let a proposition A" be in unique correspondence with every natural number n. If for every n, the truth of A,, follows from the supposition of the truth of all propositions AX (x < n), then A is true for all it. If A were false for one n, then according to Theorem 5 there would be a least n of this nature. Since then all At (x < n) are true, the truth of A follows from the induction assumption. This contradiction provesTheorem6. A second kind of recursive definition is contained in the following.

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