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The simple infinite sequence. Development of arithmetic Let M be a Dedekind infinite set, f a one-to-one correspondence between M and a proper part Mf of M. Let 0 denote an element of M not in Mf. I denote generally by af the image f(a) of a, also by Pf, when PEM, the set of all pf = f(p) when p runs through P. Let N be the intersection of all subsets X of M possessing the two properties 1) OeX, 2) (x)(xeX-»x'eX). Then N is called a simple infinite sequence or the f-chain from 0. We may say that it is the natural number series.

Thus as often as yex, zeu, we have y u{z}ex. Since u is inductive finite, it follows that uex. Hence u H v is inductive finite, that is, v is inductive finite. It follows easily from this that each subset v of u, u inductive finite, must be an element of every set of subsets of the kind mentioned in the definition of inductive finiteness. Theorem 37. Ifu and v are inductive finite, so is u\j v. Proof. We consider the subset x of all subsets w of u such that w U v is inductive finite. Obviously Oex.

It suffices to assume meu. Let x be a set of subsets of u U {m} such that Oex and if yex and zeu U {m} then y U {z}ex. Further, let xf be the subset of x consisting of all elements of x which are cu. Then Oex f and as often as yex f , zeu, we have y u {z}ex and therefore also y U {z}ex f . Thus, u being inductive finite, uex f . But uex and meu U{m} yields u U{m}ex. Hence the theorem is correct. Theorem 36. Every subset of an inductive finite set u is inductive finite. Proof. Let v be £u. I consider the set x of subsets w of u such that w n v is inductive finite.