Part 3 of Cantor’s
Über unendliche lineare Punktmannig-faltigkeiten
(On infinite linear point-sets)
• English Translation •

This is a new (2021) English translation of Part 3 of Cantor’s work Über unendliche lineare Punktmannig-faltigkeiten (On infinite linear point-sets). Also available online are English translations of Part 1, Part 2, and Part 5 (Grundlagen). (Footnote: Translator’s note: As published by Teubner, Leipzig 1883. This is the third part of a series 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. 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 3 of 6

In the two preceding Parts 1 and 2 we adhered strictly to the subject matter described in the heading and dealt exclusively with linear point-sets, that is, rule-defined specific point-sets that belong in an infinite continuous straight line. I had deliberately drawn this restriction for the presentation because, especially with regard to my prior results (Footnote: Georg Cantor: “Ein Beitrag zur Mannigfaltigkeitslehre” (A development in the theory of sets), Crelle’s Journal, Vol. 84, 1878, p. 242.   Translator’s note, see an online English translation of that article at A Contribution to the Theory of Sets.) in which planar, spatial and general n‑dimensional structures have been defined by a clear relationship to linear point-sets, it might be assumed from the outset that most of the properties and relationships that emerge from linear point-sets can also be demonstrated, with obvious modifications, for point-sets that are included in continuous surfaces, spaces or n‑dimensional regions. But I would like to clarify this generalization, since it is not only of interest of itself and with regard to applications in the theory of functions, but also because it provides new insights into the field of linear point-sets.

To begin with, the notions of derivatives (Footnote: Translator’s note: Here Cantor uses the term derivative to refer to a different concept than that used for the derivative as it is used in calculus.) that have occurred up to now are of a different order, the latter being determined not only by finite integers, but also being characterized by certain well-defined infinite symbols, and which are readily expandable to point-sets in continuous n‑dimensional regions. Here, too, the concept of derivation is based on the concept of a limit point of a given point-set P, which is defined by the fact that in every neighborhood of it, no matter how small that neighborhood may be, there exist points in the set P which differ from it, and where it is immaterial whether or not such a limit point itself belongs to the set P. The general theorem that every point-set consisting of an infinite number of points in an n‑dimensional extended continuous finite region has at least one limit point, was first stated and proved by C. Weierstrass, and extensively utilized by him in the theory of functions.

The set of all limit points of a set P forms a new point-set P′ which is generally different from P and which I call the first derivative of P. The iteration of this concept, either by finite repetition, or even by infinite repetition gives rise, by logical necessity, to the concept of higher-order derivatives. Here there is always the readily justified phenomenon that every derivative, with the exception of the first, is included as a subset of the preceding one, including the first derivative P′, while the original given point-set P in general contains points that are completely different to its derivatives. Likewise, the concept of everywhere-dense, which initially we only considered for linear point-sets, can easily be expanded to sets in higher dimensions; a point-set P in a continuous n‑dimensional region A is called everywhere-dense in the region a if a is a continuous sub-region of ​​A where every continuous sub-region a′ of a has points of the set P in its interior.

The first derivative P′ (and also all subsequent derivatives) of a point-set P that is everywhere-dense in a continuous region a includes the continuous region a itself together with all points of the boundary of the latter and, conversely, this property of the point-set P can also be used as a starting point for the definition of everywhere-dense for the region a.

The concept of cardinal-number, which includes the concept of whole number, and which is the foundation of the theory of sizes of sets, is of special significance and it should be considered to be the most genuine general landmark in the theory of sets. The concept is not in any way restricted only to linear point-sets, but it is rather a well-defined property of any set that might come under consideration, regardless of the theoretical nature of its elements.

I call a set (a multiplicity, a collection) of elements belonging to any conceptual sphere well-defined if, on the basis of the definition and based on the logical principle of the excluded middle, it must be regarded as internally determined - both as to whether any object within the same overall conceptual realm belongs or does not belong as an element of the set - as well as whether two objects that belong to the set are equal or are not equal, regardless of any differences in their formal definition.

In general, the methods and techniques available will not, in fact always be able to provide an authoritative and accurate determination, but that is not necessarily important. In actual cases where a determination is required, any such actual (external) determination is to be derived only from the internal determination, by the development of such additional means.

Here I would like to recall the definition of the set of all algebraic numbers, which can undoubtedly be understood in such a way that by the internal determination, it is definitive as to whether a certain arbitrary number η belongs to the algebraic numbers or not. Nevertheless, the problem of actually solving this decision for a given number η, as is well known, is often one of the most difficult problems. For example, it is still an open question of great interest as to whether the number π, which expresses the ratio of the circumference of the circle to the diameter, is an algebraic number, or as is most likely, a transcendent number. (Footnote: Translator’s note: The transcendence of π was first proven by Ferdinand Lindemann in Mathematische Annalen 20, 1882, pp 213-225, which in fact was the next issue after the one in which this article was published.) For the base number e of the natural logarithmic system, this problem was only solved eight years ago (Footnote: Charles Hermite, “Sur la fonction exponentielle” (On the exponential function), Gauthier-Villars, 1874.) where it is shown that the number e is not the root of any algebraic equation with integer rational coefficients.

If one is dealing with a geometric set, the elements of which can be not only points, but also lines, surfaces, or bodies, then, if it is well-defined, the question of its cardinal-number arises immediately here too, and this will either be the same cardinal-number as occurs in point-sets or it will be greater than all such cardinal-numbers.

In particular, regarding the point-sets contained in n‑dimensional continuous regions, I have shown rigorously (Footnote: Georg Cantor: “Ein Beitrag zur Mannigfaltigkeitslehre” (A development in the theory of sets), Crelle’s Journal, Vol. 84, 1878, p. 242.   Translator’s note, see an online English translation of that article at A Contribution to the Theory of Sets.) that their cardinal-number coincides with that of the linear point-sets. This fact can be understood to be a simple consequence of the theorem proved there, where an n‑dimensional continuous structure can be brought into a mutually unambiguous, completely regular and relatively simple relationship to a one‑dimensional continuous region, i.e: to the straight linear continuum. The question of the different cardinal-numbers of point-sets can thus be investigated by the use of linear point-sets, without limiting the generality in any way, as already emphasized at the end of the paper just mentioned.

I had borrowed the expression “mightiness” from J. Steiner (Footnote: Jacob Steiner, Vorlesungen über Synthetische Geometrie (Lectures on Synthetic Geometry), ed. Schröter, Part 2, Teubner, Leipzig and Berlin, 1867.) who uses it in a very special, though related sense, to express the concept that two structures are related to one another through a projective assignment in such a way that each element of one corresponds to only one element of the other. In the case of the absolute notion of cardinal-number that is meant here, the mutually unambiguous relationship is maintained where no restriction applies to any such rule of assignment, in particular that there is no restriction in relation to continuity and discontinuity, so that two sets are only given the same cardinal-number if they can be mutually and unambiguously assigned onto each other according to some rule. If the two sets are well-defined, then it is to be regarded as internally determined whether or not they have the same cardinal-number, but obtaining an actual decision on the matter is often one of the most difficult tasks in actual cases. It was only after many fruitless attempts eight years ago that I succeeded in showing that the linear continuum does not have the same cardinal-number as the set of all natural numbers, using a theorem that I previously proved (Footnote: Georg Cantor, “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, 1874, p. 260.   Translator’s note, see an online English translation of that article at On a Property of the Set of all Real Algebraic Numbers.) and which is also given in Part 1 of this publication.

Considering here only the mathematical aspects of the concepts here presented, leaving out for the time being any consideration of other conceptual realms, they encompass the areas of arithmetic, the theory of functions, and geometry, and they are all combined into a higher concept based on the notion of cardinal-number. In this way the discontinuous and the continuous are found to be observable from a single viewpoint and measurable by a measure that is common to both.

The smallest cardinal-number that can appear in any infinite set, that is in any set that consists of an infinite number of elements, is the cardinal-number of the set of all positive whole rational numbers. I refer to sets of this type as infinite countable sets, or simply countable sets; they are characterized by the fact that they take the form of a simple infinite, rule-based sequence (which can be of various types):

E₁, E₂, …, E_v, …

where the rule dictates that each element of the set stands at a certain position in the sequence and where the sequence does not contain any terms other than the elements of the given set. Every infinite subset of a countable set also forms an infinite countable set. If one has a finite collection of sets, or a countably infinite collection of sets (E′), (E′′), (E′′′), …, where each one is itself countable, then the resulting combination of all elements of all of (E′), (E′′), (E′′′), … is also countable.

These two simple and easily proved theorems form the basis for proving countability. From them one can immediately recognize, as I have already noted several times, that all sets that are given in the form of an n‑dimensional infinite sequence with the general term E_v1, _v2, …, _vn (where v₁, v₂, …, v_n all have positive integer values that are independent of one another) are countable sets, i.e., one can envisage them in the form of simple infinite rows. The set whose general term is of the form:

E_v1, _v2, …, _vμ,

where μ includes all positive integer values, also belongs to this class; a particularly remarkable case of the latter is the set of all algebraic numbers. (Footnote: Georg Cantor, “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, 1874, p 258.   Translator’s note, see an online English translation of that article at On a Property of the Set of all Real Algebraic Numbers.) Accordingly, both arithmetic and algebra offer an inexhaustible wealth of examples of countability, but geometry is no less productive in this aspect.

The following theorem, which allows for some elegant applications in number theory and function theory, should give an indication of ​​this:

In an n‑dimensional continuous region A which is extended everywhere to infinity, an infinite number of n‑dimensional continuous (Footnote: For every continuous structure the points of its boundary are regarded as belonging to it.) sub-regions (a) separated from one another and at most colliding at their boundaries, is defined, and this set of sub-regions is always countable.

It must be emphasized that no assumption is being made here regarding either the distribution or the volumetric size of the regions. They can be infinitely close to any point of A that does not belong to them, regardless of their size. This assertion has no exceptions, all that is required is that every sub-region a (all a are assumed to be n‑dimensional) has a certain (arbitrarily small) volume and that the different a coincide at most at their boundaries.

The proof of this theorem can be carried out as follows:

Let the n‑dimensional infinite region A be related, by means of reciprocal radius vectors, to an n‑dimensional extended structure B that is within an n + 1- dimensional infinite region A′, and which is defined by its points having a constant unit distance from a fixed point in the region A′. In the case of n = 1 this is a unit circle, in the case of n = 2 a unit sphere. Each n‑dimensional sub-region a of A corresponds to an n‑dimensional sub-region b of B which has a certain volume. If the countability of the set (b) can be demonstrated, then because of the mutually unambiguous assignment, the countability of the set (a) follows.

However, the set (b) is countable because the number of regions b, which are volumetrically larger than any given number γ, is necessarily finite, because their sum is smaller than the number 2ⁿπ, namely, smaller than the volume of the structure B, in which all of the b are contained. It follows from this that the regions b can be arranged in a simply infinite sequence according to their volumetric sizes, so that the smaller ones follow the larger ones and ultimately become infinitely small in this sequence. (Footnote: Translator’s note: Cantor added an amendment in the next published part (Part 4):
I would like to correct a value given in the previous part of this treatise, although the accuracy is not relevant for our purposes. On closer examination the n‑dimensional volume of the structure B is found to be:
 2π^n+1⁄₂    Γ(ⁿ⁺¹⁄₂) which only in cases n = 1 and n = 2 does it coincide with 2ⁿπ, otherwise it is smaller than 2ⁿπ.)

The case n = 1 yields the following theorem, which is essential for the further development of the theory of linear point-sets:

Every set of separate intervals (α … β ) which at most coincide at their end points and which are defined in an infinite straight line, is necessarily a countable set.

The same therefore also applies to the set of the endpoints α … β, but not always to the derivative of the latter set of points.

The case n = 2, which assigns the property of countability to every set of separate parts of surfaces in an infinite plane that collide at most at their boundaries, would appear to be of importance in the theory of functions of complex quantities. Here I might observe that it is not difficult to extend this theorem to include the set of separate parts of a surface which are defined in a region which covers the plane by m‑dimensional or even by countably infinitely many times.

As far as the countable point-sets are concerned, they present a strange phenomenon, which I would like to express by the following:

Let us consider some point-set (M ), which is everywhere-dense within an n‑dimensional, continuously connected region A and has the property of countability, so that the points that belong to (M ) are in the form of a sequence:

M₁, M₂, …, M_v, …

We may consider the set of all those points in our three‑dimensional space whose coordinates each have three algebraic numerical values in relation to an orthogonal coordinate system, for example x, y, z. If the countable point-set (M ) is removed from the region A and the remaining region is denoted by U, then there is the strange proposition that for n > 2 the region U does not cease to be continuously connected, in other words every two points N and N′ of the region U can always be connected by a continuous line, where all its points belong to the region U, so that not a single point of the set (M ) lies on it.

It suffices to recognize this theorem as correct for the case n = 2 and the proof essentially follows on from the theorem proved in Part 1 that for any rule-based sequence of real quantities:

ω₁, ω₂, …, ω_v, …

(among which the same entity can occur, which obviously does not change the essence of the sequence), in every given arbitrarily small interval (α … β ) real quantities η can be found that do not occur in that sequence.

Let A be some connected continuous region of the infinite plane. We assume that in A there is an everywhere-dense countable point-set (M ). Let N and N′ be any two different points of A not belonging to the set (M ) that are connected with each other by a some continuous line l running within A, and currently we are not concerned as to whether there are points of (M ) on that line. It can now be shown that the line l can be replaced by another continuous line l′, which also connects the points N and N′ with each other, and which also runs in the interior of A, but does not contain a single point of the set (M ).

In general, there will be an infinite number of points of the set (M ) on l, and hence they are a subset of (M ) on it, thus that subset is also a countable set. Consequently, according to the arithmetical theorem just stated, in every interval of the line l, however small, there are points which do not belong to (M ). We consider a finite number of points N₁, N₂, …, N_k , … on the line l in such a way that the straight lines N N₁, N N₁, …, N_k N′ also lie entirely in the interior of A. These lines can now always be replaced by arcs of a circle with the same endpoints, and which likewise run within A, and which do not contain a single point of the set (M ), and in their composition they form a continuous line l′ of the type described above.

It suffices to demonstrate this on one such path, so we take the first N N₁. The circles passing through the points N and N₁ form a simply infinite array, their centers lying on a certain straight line g. The position of such a center is determined by the distance u from a fixed point O of the straight line g, with the appropriate sign; an interval (α … β ) can always be assigned to the variable u as a length such that one of the two arcs connecting N and N₁ comes to lie entirely in the region A for each circle corresponding to such a u.

The centers of those circles lie in a straight line, and they are defined by the points:

M₁, M₂, …, M_v, …

of the set (M ), and on this straight line they constitute a countable point-set:

P₁, P₂, …, P_v, …

where the corresponding values of u are:

ω₁, ω₂, …, ω_v, …

If one assumes that there is a number η in the interval (α … β ), and which is not equal to any ω (according to the above propositions, this is always possible), one obtains by this assumption that:

u = η

defines one circle of the set of circles, where there is not a single point of the set (M ) on its circumference, and since α < η < β, it defines the required arc connecting the points N and N₁.

In this way it is shown that every two points N and N₁ of the region U which remains after subtracting an everywhere-dense, countable point-set (M ) from the region A can be connected by a continuous line l′ composed of a finite number of arcs, where all of its points belong to the region U, i.e: it does not contain a single point of the set (M ).

Incidentally, by the same means it would also be possible to establish a connection between the points N and N₁ by a continuous line which is entirely contained in the region U, and which is defined by a single analytical rule.

These theorems are linked to considerations about the nature of the three‑dimensional space in which the real world exists, for the development of a conceptual description and explanation of the phenomena that occur in it. As is well known, it is assumed to be universally continuous, both on account of the forms that appear in it and also with respect to the motions that take place within it. According to ongoing independent investigations of Dedekind (Footnote: Richard Dedekind, “Stetigkeit und irrationale Zahlen” (Continuity and irrational numbers), Viehweg, Braunschweig, 1872.) and this author (Footnote: Georg Cantor, “Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen” (On the expansion of a theorem from the theory of trigonometric series), Mathematische Annalen 5.1, 1872, pp 127-128.) this latter assumption does not entail anything else other then the notion that every point, whose coordinates x, y, z in relation to an orthogonal coordinate system are given by some definite real number (rational or irrational), is considered to actually belong to space. This does not involve any inner constraint and it must therefore be seen to be a free act of our intellectual constructive activity. The hypothesis of the continuity of space is nothing more than the inherently arbitrary presupposition of a complete, mutually unambiguous correspondence between the three‑dimensional, purely arithmetic continuum (x, y, z) and the space in which the observed real world exists. (Footnote: I think I can assume that it is generally known that a purely arithmetic theory of mathematics is possible, that is, one that is completely independent of all geometrical principles, and its basic features have already been developed. In this regard I refer to the cited essays by Dedekind and myself, which are of course only very brief, and to the excellent work of Herr Lipschitz “Fundamentals of Analysis”, Bonn 1877. Most of the fundamental difficulties that are found in mathematics appear to me to have their origin in the fact that this possibility of a purely arithmetic theory of quantities and sets is not understood. The errors of those authors, who consider the infinitely small to be a quantity rather than a mode of the variability of quantities, can be traced back to this. From the standpoint of pure arithmetical analysis, there are no infinitely small quantities, but only infinitely small variable quantities.)

But we can equally well extend our abstract reasoning to individual points of space, even if they occur everywhere-dense, and we can form the concept of a discontinuous three‑dimensional space U from the structure previously defined. The question then arises as to whether continuous movement can also be conceived in such discontinuous spaces; this must be answered in the affirmative by consideration of the preceding arguments, because we have shown that every two points of a structure U can be connected by an uncountable number of continuous, completely ordinary lines.

Strangely enough, it turns out that the mere fact of continuous movement does not permit any conclusion to be drawn regarding the universal continuity of the three‑dimensional concept of space used to explain the phenomena of movement.

It is therefore natural to attempt a modified theory of mechanics that is valid for type U spaces, in order to create a comparative analysis between the results of such an investigation and the actual facts, and this may provide the correct basis for the experience-based hypothesis of the universal continuity of space.

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