In the first place, the definition of an infinite whole must not be held to deny that it has an assignable number of simple parts which do not reconstitute it. For example, the stretch of fractions from 0 to 1 has three simple parts, ⅓, ½, ⅔. But these do not reconstitute the whole, that is, the whole has other parts which are not parts of the assigned parts or of the sum of the assigned parts. Again, if we form a whole out of the number 1 and a line an inch long, this whole certainly has one simple part, namely 1. Such a case as this may be excluded by asking whether every part of our whole either is simple or contains simple parts. In this case, if our whole be formed by adding `n` simple terms to an infinite whole, the `n` simple terms can be taken away, and the question can be asked concerning the infinite whole which is left. But again, the meaning of our question seems hardly to be: Is our infinite whole an actual aggregate of innumerable simple parts? This is doubtless an important question, but it is subsequent to the question we are asking, which is: Are there always simple parts at all? We may observe that, if a finite number of simple parts be found, and taken away from the whole, the remainder is always infinite. For if not, it would have a finite number; and since the term of two finite numbers is finite, the original whole would then be finite. Hence if it can be shown that every infinite whole contains one simple part, it follows that it contains an infinite number of them. For, taking away the one simple part, the remainder is an infinite whole, and therefore has a new simple part, and so on. It follows that every part of the whole either is simple, or contains simple parts, provided that every infinite whole has at least one simple part. But it seems as hard to prove this as to prove that every infinite whole is an aggregate.(§ 143 ¶ 1)

If an infinite whole can be divided into a finite number of parts, one at least of these parts must be infinite. If this be again divided, one of its parts must be infinite, and so on. Thus no finite number of divisions will reduce all the parts to finitude. Successive divisions give an endless series of parts, and in such endless series there is (as we shall see in Parts IV and V) no manner of contradiction. Thus there is no method of proving by actual division that every infinite whole must be an aggregate. So far as this method can show, there is no more reason for simple constituents of infinite wholes than for a first moment in time or a last finite number.(§ 143 ¶ 2)

But perhaps a contradiction may emerge in the present case from the connection of whole and part with logical priority. It certainly seems a greater paradox to maintain that infinite wholes do not have indivisible parts than to maintain that there is no first moment in time or furthest limit to space. This might be explained by the fact that we know many simple terms, and some infinite wholes undoubtedly composed of simple terms, whereas we know of nothing suggesting a beginning of time or space. But it may perhaps have a more solid basis in logical priority. For the simpler is always implied in the more complex, and therefore there can be no truth about the more complex unless there is truth about the simpler. Thus in the analysis of our infinite whole, we are always dealing with entities which could not be at all unless their constituents were. This makes a real difference from the time-series, for example: a moment does not logically presuppose a previous moment, and if it did it would perhaps be self-contradictory to deny a first moment, as it has been held (for the same reason) self-contradictory to deny a First Cause. It seems to follow that infinite wholes would not have Being at all, unless there were innumerable simple Beings whose Being is presupposed in that of the infinite wholes. For where the presupposition is false, the consequence is false also. Thus there seems a special reason for completing the infinite regress in the case of infinite wholes, which does not exist where other asymmetrical transitive relations are concerned. This is another instance of the peculiarity of the relation of whole and part: a relation so important and fundamental that almost all our philosophy depends upon the theory we adopt in regard to it.(§ 143 ¶ 3)

The same argument may be otherwise stated by asking how our infinite wholes are to be defined. The definition must not be infinitely complex, since this would require an infinite unity. Now if there is any definition which is of finite complexity, this cannot be obtained from the parts, since these are either infinitely numerous (in the case of an aggregate), or themselves as complex as the whole (in the case of a whole which is not an aggregate). But any definition which is of finite complexity will necessarily be intensional, i.e. it will give some characteristic of a collection of terms. There seems to be no other known method of defining an infinite whole, or of obtaining such a whole in a way not involving any infinite unity.(§ 143 ¶ 4)

The above argument, it must be admitted, is less conclusive than could be wished, considering the great importance of the point at issue. It may, however, be urged in support of it that all the arguments on the other side depend upon the supposed difficulties of infinity, and are therefore wholly fallacious; also that the procedure of Geometry and Dynamics (as will be shown in Parts VI and VII) imperatively demands points and instants. In all applications, in short, the results of the doctrine here advocated are far simpler, less paradoxical, and more logically satisfactory, than those of the opposite view. I shall therefore assume, throughout the remainder of this work, that all the infinite wholes with which we shall have to deal are aggregates of terms.(§ 143 ¶ 5)

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.