Part 1 of Cantor’s
Über unendliche lineare Punktmannig-faltigkeiten
(On infinite linear point-sets)
• English Translation •

This is a new (2021) English translation of Part 1 of Cantor’s work Über unendliche lineare Punktmannig-faltigkeiten (On infinite linear point-sets). Also available online are English translations of Part 2, Part 3 and Part 5 (Grundlagen). (Footnote: Translator’s note: As published by Teubner, Leipzig 1883. This is the first part of a sequence of papers by Cantor coming under the overall title of Über unendliche lineare Punktmannig-faltigkeiten (On infinite linear point-sets), which were published in the Mathematische Annalen between 1879 and 1884:
Part 1. Mathematische Annalen 15, 1879, pp. 1-7
Part 2. Mathematische Annalen 17, 1880, pp. 355-358
Part 3. Mathematische Annalen 20, 1882, pp. 113-121
Part 4. Mathematische Annalen 21, 1883, pp. 51-58
Part 5. (Grundlagen) Mathematische Annalen 21, 1883, pp. 545-591
Part 6. Mathematische Annalen 23, 1884, pp. 453-488
This translation uses a 1984 Springer reprint of the above collection, e-ISBN-13:978-3-7091-9516.1, DOl: 10.1007/978-3-7091-9516-1.)

English translation by James R Meyer, copyright 2021 www.jamesrmeyer.com

Notes: This translation uses current terminology if possible where the older terminology might cause confusion for modern readers. The footnotes are Cantor’s, unless indicated by green text and “Translator’s note:”.

On infinite linear point-sets: Part 1 of 6

In a previous article (Footnote: Ein Beitrag zur Mannigfaltigkeitslehre (A development in the theory of sets), Crelle’s Journal, Vol. 84, pp. 242-258, 1878.
Translator’s note, see an online English translation of that article at A Contribution to the Theory of Sets.) I have demonstrated that a very wide range of geometrical and arithmetical sets, both continuous and discontinuous, can be unambiguously and completely assigned to a straight line or a discontinuous component of it. We call these sets of points linear point manifolds, or for brevity, linear point-sets. Accordingly, these sets either form a continuous straight line, finite or infinite, or all their points are included in such a line as elements of the set, and as such are of special interest. It should therefore not be an unworthy endeavor if we devote a series of considerations to them; in the following I want to first investigate their classification. Various considerations and the associated classification principles lead us to classify the linear point-sets into certain groups.

To begin with one of these considerations, I recall the concept of the derivative of a given point-set P, which I presented in a work on trigonometric sequences. (Footnote: Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen (On the generalization of a theorem in the theory of trigonometric sequence), Mathematische Annalen, Vol. 5, pp. 123-132, 1872. ) In a recently published book by Dini (Footnote: Ulisse Dini, Fondamenti per la teorica della funzioni di variabili reali (Fundamentals of the theory of functions of real variables), T. Nistri e C., Pisa 1878. ) we see this term developed even further, taking it as the starting point for a number of remarkable generalizations of well-known analytical propositions. (Footnote: See also Giulio Ascoli, of Real Academia dei Lincei, in Nuove ricerche sulla serie di Fourier (New researches in Fourier sequence), Coi tipi del Salviucci, 1878.) Incidentally, the concept of the derivative of a given set is not restricted to linear sets, but also applies in the same way to planar, spatial and n‑dimensional continuous and discontinuous sets. As we will show later, the simplest and at the same time most complete explanation or determination of a continuum is based on this concept.

The derivative P′ of a linear point-set P is the set of all those points that are limit points of P, where it is immaterial whether the limit point is also a point of P or not. Since, by this definition, the derivative of any point-set P is a specific point-set P′ , then the derivative of this can also be sought, which is then called the second derivative of P and denoted by P′′; by continuing this procedure one obtains the v^th derivative of P, which is denoted by P^(v).

It can happen that the progression of the derivatives P′, P′′ leads to a derivative P⁽ⁿ⁾, which consists of points that only occur in finite numbers in each finite region, so that P⁽ⁿ⁾ has no limit points and consequently has no derivative; in this case we say of the point-set P that it is of the first number-class and of the n^th type. But if the sequence of derivatives of P, the sequence P′, P′′, P′′′, …, P^(v), … does not terminate, we say that the point-set P is of the second number-class.

It is easy to see from this that if P is of the first number-class and n^th type, then P′, P′′, P′′′, … also belong to the first number-class and are respectively of the n  -  1^st, n - 2^nd, n - 3^rd … type. In addition, if P belongs to the second number-class, the same also applies to all of its derivatives P′, P′′. It is also worth noting that all points of P′′, P′′′, … are always points of P′, while a point belonging to P′ is not necessarily also a point of P.

Furthermore, important characteristics of a point-set P emerge if its behavior is considered in relation to a given, continuous closed interval [α, β ] (whose end points are included in the interval). It can be the case that some or all points of this interval are also points of P, or it can be the case that no point of the interval [α, β ] is a point of P, in which case we say that P lies completely outside of the interval [α, β ]. If P lies partially or completely in the interval [α, β ], the remarkable case can arise that every interval [ γ, δ ] that is contained in [α, β ], no matter how small, contains points of P, and in such a case we say that P is everywhere-dense in the interval [α, β ]. Examples of such point-sets that are everywhere-dense in the interval [α, β ] are:

1. The point-set where all points of the interval [α, β ] are its elements.
2. The point-set, which consists of all points in the interval [α, β ] that are rational numbers.
3. The point-set which consists of all points in the interval [α, β ] that are rational numbers of the form ^2n + 1⁄_2m (where n and m are integers).

From this explanation of the expression “everywhere-dense in a given interval” it follows that if a point-set in an interval [α, β ] is not everywhere-dense, then there must exist an interval [ γ, δ ] in that interval [α, β ] in which there is not a single point of P. Furthermore, it can be shown that if P is everywhere-dense in the interval [α, β ], then the same not only applies for P′ , but it is also the case that P′ has all the points of the interval [α, β ] as its elements. This property of P′ can also be taken as the starting point for the definition of being everywhere-dense in an interval, in that one can say: a point-set P is called everywhere-dense in an interval [α, β ] if its derivative P′ contains all points of [α, β ] as its elements.

If P is everywhere-dense in an interval [α, β ], then P is also everywhere-dense in every other interval [α′, β′ ] which is contained in that interval. A point-set P that is everywhere-dense in an interval [α, β ] is necessarily of the second number-class. Hence P′ and also P′, P′′, … are everywhere-dense in the interval [α, β ] and this progression of the derivatives of P is unlimited, that is, P belongs to the second number-class. From this we can draw the conclusion that a point-set P of the first number-class in any given interval [α, β ] is certainly not everywhere-dense, and consequently an interval [ γ, δ ] can always be found within [α, β ] which does not contain a single point of P.

Conversely, the question arises as to whether every point-set of the second number-class is such that it can constitute an interval [α, β ] in which it is everywhere-dense; this question will be dealt with later. (Footnote: Translator’s note: See this paragraph in Part 2 of this work.)

We now come to a completely different, but no less significant, reason for classifying linear point-sets, namely, their cardinality. In the aforementioned article (Footnote: Translator’s note: Ein Beitrag zur Mannigfaltigkeitslehre (A development in the theory of sets), Crelle’s Journal, Vol. 84, pp. 242-258, 1878.
Translator’s note, see an online English translation of that article at A Contribution to the Theory of Sets.) we generally say of any two sets M and N, whether they are geometrical, arithmetical, or given by some other similar well-defined concept, that they have the same cardinal-number if one is able to set them to correspond one to the other according to some specific rule, in such a way that for every element of M there is an element of N and vice versa, for every element of M there is an element of N.

Depending on whether two sets have equal or different cardinal-numbers, they can either be assigned to the one and the same number-class, or to two different number-classes. These general rules just mentioned can now be specifically applied to linear point-sets, and hence such sets can be categorized into specific number-classes. All point-sets of the same number-class all have the same cardinal-number, while point-sets which are assigned to different classes have different cardinal-numbers. Hence any particular point-set can be viewed as representative of the number-class to which it belongs.

First of all, we consider the number-class of countable infinite point-sets, that is, those point-sets which have the same cardinal-number as the sequence of natural numbers 1, 2, 3, …, v, … and are therefore in the form of a simply infinite sequence with a general term dependent on v. For example, all point-sets of the first type belong to this number-class. But many point-sets of the second type also fall into this number-class, such as:

1. The point-set which consists of all points of an interval which are rational numbers. (Footnote: Translator’s note: Ein Beitrag zur Mannigfaltigkeitslehre (A development in the theory of sets), Crelle’s Journal, Vol. 84, p. 250, 1878.
   Translator’s note, see an online English translation of that article at A Contribution to the Theory of Sets.)
2. The point-set which consists of all points of an interval which are algebraic numbers. (Footnote: Translator’s note, see: Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen (On a property of the set of all real algebraic numbers), Crelle’s Journal, Vol. 77, p. 258, 1874.
   Translator’s note, see an online English translation of that article at On a Property of the Set of all Real Algebraic Numbers.)

Secondly, we consider linear point-sets that are representative of an arbitrary continuous interval, e.g. the set of all points which are ≥ 0 and ≤ 1. This class includes, for example:

1. Every continuous interval [α, β ].
2. Any point-set consisting of several separate, continuous intervals [α, β ], [α′, β′ ], [α′′, β′′ ], … , either finitely many or infinitely many.
3. Any point-set which emerges from a continuous interval by removing from it either a finite or a countably infinite point-set ω₁, ω₂, …, ω_v, … . (Footnote: Translator’s note, see: Ein Beitrag zur Mannigfaltigkeitslehre (A development in the theory of sets), Crelle’s Journal, Vol. 84, p. 254, 1878.
   Translator’s note, see an online English translation of that article at A Contribution to the Theory of Sets. )

Whether these two number-classes are the only ones into which linear point-sets fall will not be investigated at this point, but on the other hand we want to show that they really are distinct number-classes. In order to prove this it is necessary to show that any two representatives of these two number-classes (one of each) cannot be made to completely and unambiguously correspond one-to-one to each other.

We choose again the continuous interval [0, 1] as a representative of the second class. If this set also belonged to the first class, then there would exist a simply infinite sequence:

ω₁, ω₂, …, ω_v, …

consisting of all real numbers ≥ 0 and ≤ 1, so that every such real number ζ would be present at a specific position in that sequence. But a very general theorem contradicts this, which I have already rigorously proved, namely: (Footnote: Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen (On a property of the set of all real algebraic numbers), Crelle’s Journal, Vol. 77, pp. 258-262, 1874.
Translator’s note, see an online English translation of that article at On a Property of the Set of all Real Algebraic Numbers.)

“If one has a simply infinite sequence

ω₁, ω₂, …, ω_v, …

of real, unequal numbers, which progresses according to some rule, then in every given interval [α, β ] there is a number η (and consequently there are infinitely many) which is not in that sequence (as any member of the sequence).”

In view of the great interest which is attached to this theorem, not only in the present discussion, but also in many other arithmetical and analytic relationships, it is not unjustified to develop that argument more clearly by the use of some simplifying modifications.

On the basis of the sequence:

ω₁, ω₂, …, ω_v, …

which we will denote by the term (ω), and an arbitrary interval [α, β ], where α < β, we shall now show that a real number η can be found in this interval, which does not appear in (ω):

I: First we notice that if our set (ω), in the interval [α, β ], is not everywhere-dense, there must be another interval [ γ, δ ] within this interval, the numbers of which do not belong to (ω). One can then choose any number of the interval [ γ, δ ] to be η and it must be in the interval [α, β ] but it certainly does not appear in our sequence (ω). This case therefore does not present any particular difficulties, and we can move on to a more difficult case.

II: Now assume that the set (ω) is everywhere-dense in the interval [α, β ]. In this case every interval [ γ, δ ] no matter how small, which is in [α, β ], contains numbers in our sequence (ω). In order to show that nonetheless numbers η exist in the interval [α, β ] which do not appear in (ω), we make the following consideration:

Since there are certainly numbers in our sequence:

ω₁, ω₂, …, ω_v, …

that are in the interval [α, β ], then one of these numbers must have the smallest index, let it be ω_x1 and another has the next largest index, we call it ω_x2.

Let the smaller of the two numbers ω_x1 and ω_x2 be denoted by α′ and the larger by β′ (Their equality is excluded because of the assumption that our sequence consists only of unequal numbers). Then according to the definition:

α < α′ < β′ < β

and furthermore by:

x₁ < x₂

it can be observed that all of the numbers ω_μ of our sequence, where μ ≤ x₂, cannot lie in the interior of the interval [α′, β′ ], as is immediately evident from the definition of the numbers ω_x1, ω_x2 .

Similarly, if ω_x3, ω_x4 are the two numbers in our sequence with the smallest indices, which fall within the interior of the interval [α′, β′ ], then let the smaller of the numbers ω_x3, ω_x4 be α′′ and the larger be β′′, which gives us:

α′ < α′′ < β′′ < β′

x₂ < x₃ < x₄

when it can be seen that all of the numbers ω_μ of our sequence, where μ ≤ x₄, cannot lie in the interior of the interval [α′′, β′′ ].

After one has arrived at an interval [α^{(v  - 1)}, β ^{(v  - 1)}] by following the same rule, the following interval is similarly obtained by taking the first two numbers (i.e: the two with the lowest indices) of our sequence (ω), which lie in the interior of [α^{(v  - 1)}, β ^{(v  - 1)}], and we indicate them by ω_x2v - 1 and ω_x2v. The smaller of these two numbers we call α^(v), and the larger β ^(v). The interval [α^(v), β ^(v) ] then lies in the interior of all previous intervals and has the special relationship to our sequence (ω) that all numbers ω_μ for which μ ≤ x_2v applies, are certainly not in its interior. It is evident that:

x₁ < x₂ < x₃ < … x_2v - 2 < x_2v - 1 < x_2v …

and the indices of these numbers are integers, so that:

x_2v ≥ 2v

and therefore:

v < x_2v

we can therefore say with certainty, and this is sufficient for the following:

If v is an arbitrary integer, the point ω_v lies outside the interval [α^(v), β ^(v) ].

Since the numbers α′, α′′, α′′′, …, α^(v), … increase continuously, but are included in the interval [α, β ], then, according to a well-known fundamental theorem of analysis, there is a limit which we denote by A, where:

A = Lim α^(v)     for v = ∞

The same applies to the numbers β′, β′′, β′′′, …, β^(v), …, which continuously decrease and also lie in the interval [α, β ]. We call the limit B, where:

B = Lim β ^(v)     for v = ∞

It is evident that:

α^(v)  <  A  ≤  B  <  β ^(v)

But it is easy to see that the case A < B cannot occur here, since otherwise every number ω_v in our sequence would lie outside of the interval [A, B], since ω_v is outside the interval [α(v), β(v)]. Hence our sequence (ω) would not be everywhere-dense in the interval [α, β ], contrary to the assumption that A < B.

Therefore only the case A = B remains and we can now show that the number

η = A = B

does not appear in our sequence (ω). If η were a member of our sequence, for example the v^th, one would then have η = ω_v. This is not possible for any value of v, because η lies in the interior of the interval [α(v), β(v)], but ω_v lies outside it.

Go to Part 2     Go to Part 3     Go to Part 5 (Grundlagen)