Russell, Ch.35 of Principles of Mathematics, ‘Cantor’s First Definition of Continuity’, summary notes

 

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Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.35: Cantor’s First Definition of Continuity

Brief Summary

Cantor has an early definition of continuity [the later ‘ordinal’ definition is discussed in the next chapter.] For a series to be continuous, it must be perfect and cohesive. It is cohesive if it it is complete dense, having no finite gaps among its terms. It is perfect if “it consists of exactly the same terms as its first derivative; i.e. when all its points are limiting-points, and all its limiting-points belong to it.” [294]

 

Summary

 

§271

We will be talking about Cantor’s continuity. We previously considered a series continuous if it has a term between any two. “This definition usually satisfied Leibniz,† and would have been generally thought sufficient until the revolutionary discoveries of Cantor.” [290] [[Note, Russell obtains this reference from a line in one of Leibniz’ letters:

Eo ipso, dum puncta ita sita ponuntur, ut nulla duo sint, inter quae non detur medium, datur extensio continua. [‘Leibniz an des Bosses’ 26 Maji 1716’, p515 Philosophische Schriften]

When points are situated in such a way that there are no two points between which there is no midpoint, then, by that very fact, we have a continuous extension. [‘Leibniz to Des Bosses, 26 May 1716 [excerpts]’, p.201-202 Philosophical Essays]

Leibniz also invented infinitesimal calculus, which examines when points draw so close such that there are no points between but only an infinitesimal difference between them. That seems to contradict his claim here. However, note that in the above statement Leibniz says that the continuity of extension results from there being points between any two others. The infinitely small magnitude does not expand through extensive space. It is an intensity of variation.]] Previously Russell refered to the continuity of the rational numbers as compactness. From now on, he will never refer to this sort of continuity but rather only to the Cantorian sorts.

§272

”In order that a series should be continuous, it must have two characteristics: it must be perfect and cohesive.” [291]

(1) Cohesiveness

A series is cohesive when it contains no finite gaps. Quoting Cantor:

“We call T a cohesive collection of points, if for any two points t and t' of T, for a number ε given in advance and as small as we please, there are always, in several ways, a finite number of points t₁, t₂, . . . t_ν, belonging to T, such that the distances tt₁, t₁t₂, t₂t3, . . . t_νt' are all less than ε.”
[291]

[[So there are always points between others. This could also define the infinitesimal, but apparently it somehow does not.]]

The condition that distances in the series are to have no minimum is satisfied by real or rational numbers [...]. Hence every cohesive series must be compact, i.e. must have a term between any two. [292]

However, not every compact series is cohesive. [292]

§273

(2) Perfect series. “A series is perfect when it coincides with its first derivative.” [293] The points of a series can be of two kinds: isolated points and limiting points. “A finite series has only isolated points; an infinite series must define at least one limiting-point, though this need not belong to the series. A limiting-point of a series is defined by Cantor to be
a term such that, in any interval containing the term, there are an infinite number of terms of the series”. [293]  The limiting point does not need to be a part of the series. “The assemblage of all limiting-points is called the first derivative of the series. The first derivative of the first derivative is called the second derivative, and so on.”  [293] [It seems possibly that the Peano definition to follow is saying that the first derivative is a number very close to the lowest value of a class’s series of values. It seems it would be similar to the infinitesimal. It seems also to be the lower limit of the series.]

Peano gives the definition of the first derivative of a class of real numbers as follows: Let u be a class of real numbers, and let x be a real number (which may or may not be a u) such that the lower limit of the absolute values of the differences of | x from terms of u other than x is zero; then the class of terms x satisfying this condition is the first derivative of u.* This definition is virtually identical with that of Cantor, but it brings out more explicitly the connection of the derivative with limits. A series, then, is perfect, when it consists of exactly the same terms as its first derivative; i.e. when all its points are limiting-points, and all its limiting-points belong to it.
[293-294]

§274

 

But recall that in the series of rationals, there are sub-series with irrational limits, which means the series of rationals has limits falling outside it, and thus it does not contain all the terms of its first derivative, and hence it is not a perfect series. [294]

What we must say is, that a series is perfect when all its points are limiting-points, and when further, any series being chosen out of our first series, if this new series is of the sort which is usually regarded as defining a limit, then it actually has a limit belonging to our first series.
[294]

§275

 

In this section, Russell repeats “the arguments against assuming the existence of limits in the class of series to which the rational numbers belong.” [296-298. See these pages for the details.]

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

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

Otherwise:

Leibniz. Philosophical Essays. Ed. and Transl. Roger Ariew and Daniel Garber. Indianapolis/Cambridge: Hackett, 1989.

Leibniz. Die philosophischen Schriften von G. W. Leibniz, zweiter band. Ed. G.I. Gerhardt. Berlin. Weidmann, 1879. Available online at:
https://archive.org/details/diephilosophisc00leibgoog