The et cetera in question is capable of justi fication without appealing, apparently at least, to the axiom of infinity, namely, by use of the so-called indirect method ofproof, the method known as reductio ad absurdum. Thus let it be supposed that the argument sought to be indefinitely extended by means of the phrase so on" does not admit of indefinite exten sion along the ordered sequence of integers. There will, then, be a first integer, say m +1, for which the property p fails. As, by hypoth esis, in + I is the first integer for which p fails, p belongs to the preceding integer m; but since p belongs to in, it also belongs, by (2), to in + 1. Hence the supposition that the argu ment does not admit of indefinite extension is false; and the conclusion is obvious. This procedure is convincing, but it is plainly less a natural completion than an uunindicatee forti fication of the process it supplements. It is, besides, not entirely clear that the axiom of infinity is not surreptitiously subsumed by it.
By far the most penetrating investigation of the nature of mathematical induction was made originally by Richard Dedekind. (See Bibliog
raphy below). His procedure and result are, in brief, as followg: Let S denote a system of elements (things of any kind) such that there is a scheme of law of depiction by which S may be depicted upon itself, that is, a scheme by which each element e of S may be thought as corresponding to one and but one element e' of S and so that no two elements of S shall be thought as corresponding to a same element of S. The correspondent e' of e is called the picture or image of e. Every part of S (including S itself as a special case) thus depicted upon itself is named chain under 9. Denote by A an arbitrary part of S and by A. the assemblage of all the elements common to all the chains (in S) that contain A. It is obvious that, S and # being given, there is one and but one A. for a given part A of S. Ae, which is easily seen to be itself a chain, is described as the chain of A under 9. Now let Z denote an assemblage of elements. Dede kind proves the following