B. Russell

From The Axiom of Infinity, 1904 by Bertrand Russell

Professor Keyser’s very interesting article ‘On the Axiom of Infinity’ contains a contention of capital importance for the theory of infinity. The view advocated by those who, like myself, believe all pure mathematics to be a mere prolongation of symbolic logic, is, that there are no new axioms at all in the later parts of mathematics, including among these both ordinary arithmetic and the arithmetic of infinite numbers. Professor Keyser maintains, on the contrary, that a special axiom is covertly invoked in all attempted demonstrations of the existence of the infinite. I believe that, in so thinking, he has been misled by the brevity, and perhaps obscurity, with which writers on this subject have usually stated their arguments. I am myself, as yet, obnoxious to the same charge ; for the strict and detailed proof, with all the apparatus of logical rigour, is too long to be given incidentally, (etc).

I presuppose, in setting forth the argument, the definition of number, and the proof that, with the suggested definition, every class has some perfectly well-defined number of terms. (Etc.)

* * *

. . . Professor Keyser . . . appears to think that, at this point, the advocates of infinity are content with a vague ‘and so on’ — a sort of etcetera which is intended to cover a multitude of things. But etceteras, common as they are in ordinary mathematics, where they are represented by rows of little dots, are not tolerated by the stricter symbolic logicians. (Etc.)

* * *

. . . proofs, such as the one from the fact that the idea of a thing is different from the thing, are not appropriate to pure mathematics, since they do, as Professor Keyser points out, assume premises not mathematically demonstrable. But such proofs are not on that account circular or otherwise fallacious. Accepting the five postulates enumerated by Professor Keyser as assumed by Dedekind, I deny wholly that any one of the five presupposes the actual infinite ; it is indeed their purpose to do so. But it is too common, in philosophizing, to confound implications with presuppositions. At this rate, all deduction would be circular. The contention advanced by Professor Keyser is essentially the following : If the conclusion (the existence of the infinite) were untrue, one of the premises would be untrue ; consequently the premises beg the conclusions , and the argument is circular. But in all correct deductions, if the conclusions is false, so is at least one of the premises. The falsehood of the premises presupposes the falsehood of the conclusion, but it by no means follows that the truth of the premises presupposes the truth of the conclusion. The root of the error seems to be that, where a deduction is very easily drawn, it comes to be viewed as actually part of the premises ; and thus very elementary arguments acquire the appearance, quite falsely, of petitiones principii.

Another point that calls for criticism is the psychological form of Professor Keyser’s statement of the axiom of infinity. He states this axiom (p. 551) as follows : ‘Conception and logical inference alike presuppose absolute certainty that an act which the mind finds itself capable of performing is intrinsically performable endless.’ This statement is rendered vague by the word intrinsically ; but I sincerely hope there is no such presupposition in inference, since it is a most certain empirical fact that the mind is not capable of endlessly repeating the same act. (Etc.)

First published in the Hibbert Journal, (2 July 1904), pp. 809-12.
per Essays in Analysis, edited by Douglas Lackey
New York : George Braziller 1973, pp. 256-259.

From Philosophy of Logic and Mathematics, 1973 by Douglas Lackey

The first essay ‘On the Axiom of Infinity’ dates from 1904. In 1903, the Columbia mathematician C. J. Keyser had argued that the current theory of transfinite numbers required the assumption, in an axiom, of the existence of a denumerably infinite number of entities.¹ In his reply, Russell agreed that the existence of an infinite number of entities was required for the theory, but denied that a separate assumption to this effect was needed, because the existence of an infinite number of entities could be proved from prior principles. The proof he provided, in which the infinite ‘entities’ all turn out to be classes, was criticized by Whitehead,² but [and?] Russell eventually decided that Keyser was right. It is difficult to establish just when the decision was made to incorporate an ‘axiom of infinity’ into the Principia system (etc).

    ¹ Keyser A 1903. See also Keyser A 1904. Keyser's axiom was not stated as such, but in a form logically equivalent to this.

    ² 'Does not the theory require the explicit recognition of at least an indefinite plurality, if not of the definite cardinal numbers'' Whitehead to Russell, 23 April 1905, Bertrand Russell Archives.

    ³ Principia *120.04, labelled the 'axiom of infinity' is equivalent to Keyser's assumption.

idem, pp. 253-4.

Selections from the Bibliography

A. HISTORICAL BACKGROUND

1903

Keyser, C. J.
Concerning the axiom of infinity and mathematical induction, Bulletin of the American Mathematical Society, 9, 424-34.

1904

Keyser, C. J.
On the axiom of infinity, Hibbert Journal 2, 532-52.

idem, p. 327.