2 May 2014

Russell, Ch.36 of Principles of Mathematics, ‘Ordinal Continuity’, summary notes


by Corry Shores
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Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.36: Ordinal Continuity

Brief Summary

Cantor has an early definition of continuity. We now address his later ordinal one, which might [in simplified terms] define a continuum as a dense series of terms whose values are all definable by means of limits contained with in it.






Previously we examined Cantor’s first definition of continuity. Since “in at least two points, some reference to either numbers or numerically measurable magnitudes”, this definition is not purely ordinal [it does not define continuity merely on the ordering of its terms but also refers to some more absolutely designated value, which may have been limits in that case.] [299] But since continuity seems like an ordinal notion, Cantor tried as well to define it without elements extraneous to order.


So we will examine Cantor’s definition of the continuum in his later works.

(1) We begin with the “the type of series presented by the rational numbers greater than 0 and less than 1, in their order of magnitude.” [299] We call this type of series η. And we define it thus: (1a)  “It is denumerable, that is, by taking its terms in a suitable order (which, however, must be different from that in which they are given), we can give them a oneone correspondence with the finite integers”; and (1b) “The series has no first or last | or last term” [299-300], and (1c) “There is a term between any two, i.e. the series is compact” [300a].

(2) We then prove that these 3 above characteristics define the rationals. And “Thus all series which are denumerable, endless,* and compact, are ordinally similar.” [300]

(3) We now consider a fundamental series contained in a one-dimensional series M. (We also note that a limit comes at the end of a progression).

(4) We prove that “no fundamental series can have more than one limit, and that, if a fundamental series has a limit, this is also the limit of all coherent series; also that two fundamental series, of which one is part of the other, are coherent”. [300]

(5) We give the name ‘principal term of M’ to any term of M which is the limit of a fundamental series in M. M is ‘condensed in itself’ if all its terms are principal terms. M is ‘closed’ if all its fundamental series have a limit in M. And it is ‘perfect’ if it is both closed and condensed in itself. And “All these properties, if they belong to M, belong to any series which is ordinally similar to M.” [300]

(6) We now define continuum. [Put in simpler terms, a continuum is a dense series of terms whose values are all definable by means of limits contained with in it.]

Let θ be the type of the series to which belong the real numbers from 0 to 1, both inclusive. Then θ, as we know, is a perfect type. But this alone does not characterize θ. It has further the property of containing within itself a series of the type η, to which the rationals belong, in such a way that between any two terms of the θ-series there are terms of the η-series. Hence the following definition of the continuum:

A one-dimensional continuum M is a series which (1) is perfect, (2) contains within itself a denumerable series S of which there are terms between any two terms of M.



Perhaps this series denumerability is questionable. “to be a denumerable collection is to be a collection whose terms are all the terms of some progression.” [301] Russell then gives a purely ordinal proof that no denumerable series can be perfect, concluding: “Hence the ordinal definition of the continuum is complete” [302]


Russell then explains how the continuity here can be illustrated by the integers.

Any completed segment of the series, however, is a continuous series, as the reader can easily see for himself. The denumerable compact series contained in it is composed of those infinite classes which contain all numbers greater than some number, i.e. those containing all but a finite number of numbers. Thus classes of finite integers alone suffice to generate continuous series.


The definition that he gives for this depends upon progressions [see text for above for more details]. What is interesting here is that a progression is a series of descrete terms, but we are using it to define continuity. Russell now inquires “what can be done by compact series without progressions.” [302]

There is no à priori reason why, in any series, the limit of any class should always be also the limit of a fundamental series; this seems, in fact, to be a prerogative of series of the types to which rationals and real numbers respectively belong. In our present case, at least, though our series is, in the above general sense, condensed in itself, there seems no reason for supposing its terms to be all of them limits of fundamental series, and in this special sense the series may not be condensed in itself.


Previously Russell confined the terms of a class to just segments that are definable by fundamental series. He now examines what results from this. [After some analysis] he concludes:

If it is the fact—as it seems to be—that, starting only from a compact series, so many of the usual theorems are indemonstrable, we see how fundamental is the dependence of Cantor’s ordinal theory upon the condition | that the compact series from which we start is to be denumerable.



Before continuing to the philosophical questions raised by the continuum, we will examine in the next chapter Cantor’s theorems by looking at his transfinite cardinal and transfinite ordinal numbers.

it is now time to consider what mathematics has to say concerning infinity. Only when this has been accomplished, shall we be in a position adequately to discuss the closely allied philosophical problems of infinity and continuity. [306]

[Note, it would be interesting to see if we would follow Russell to his conclusions in light of non-standard analysis, a more recent development in the mathematics of infinity. See for example Katz and Sherry’s discussions.]


Sources [unless otherwise notes, all bracket page citations are from]:

Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].



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