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Part 2 of Cantor’s

*Über unendliche lineare Punktmannig-faltigkeiten*

(On infinite linear point-sets)

This is a new (2021) English translation of Part 2 of Cantor’s work *Über unendliche lineare Punktmannig-faltigkeiten* (On infinite linear point-sets). (Footnote: Translator’s note: As published by Teubner, Leipzig 1883. This is the second 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. (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 2 of 6**

In order to clarify the abbreviations used in the following, I will first set out some definitions.

The equality of two point-sets *P* and *Q* is expressed by the formula *P = Q*. If two sets *P* and *Q* have no common element, we say that they are *disjoint* sets.(Footnote:
Translator’s note: In the original: *Unrelated* sets
)
Where a set *P* is obtained from the combination of several sets *P*_{1}, *P*_{2}, *P*_{3}, …, where there may be finitely or infinitely many of them, and where they all are disjoint, we denote this by:(Footnote:
Translator’s note: In the original: ≡ rather than = .
)

*P* = {*P*_{1}, *P*_{2}, *P*_{3}, …}

If all points of one set *P* belong to another set *Q*, then we say that *P* is a *subset* of *Q*(Footnote:
Translator’s note: In the original: *P* is *contained* in *Q*, or *P* is a *divisor* of *Q*
)
and *Q* is a* superset* of *P*.(Footnote:
Translator’s note: In the original: *Q* is a *multiplicator* of *P*
)

If *P*_{1}, *P*_{2}, *P*_{3}, … are any point-sets, where there may be finitely or infinitely many of them, then there is a set given by all the points in all of *P*_{1}, *P*_{2}, *P*_{3}, … is called the *union*(Footnote:
Translator’s note: In the original: *Lowest Common Multiple*
)
of *P*_{1}, *P*_{2}, *P*_{3}, …, and is denoted by:(Footnote:
Translator’s note: In the original: 𝔐 (*P*_{1}, *P*_{2}, *P*_{3}, … )
)

**⋃**{*P*_{1}, *P*_{2}, *P*_{3}, … }

Similarly, the point-set common to all of *P*_{1}, *P*_{2}, *P*_{3}, … is called the *intersection*(Footnote:
Translator’s note: In the original: *Greatest Common Divisor *
)
and we denote this by:(Footnote:
Translator’s note: In the original: 𝔇 (*P*_{1}, *P*_{2}, *P*_{3}, … )
)

**⋂**{*P*_{1}, *P*_{2}, *P*_{3}, … }

For example, if *P′, P′′, P′′′, …* are the successive derivatives of a point-set *P* (see Part 1), we can say that *P′′* is a subset of *P′*, and that *P′′′* is a subset both of *P′′* and of *P′*. In general *P*^{ (v)} is a subset of *P*^{ (v - 1)}, *P*^{ (v - 2)} …, *P′*. On the other hand, *P′* is *in general* not a subset of *P*, unless *P* is itself the first derivative of a set *Q*, in which case *P′* is the subset of *P*.

It is also useful to have a symbol that expresses the absence of any points, and we use the symbol *O*,(Footnote:
Translator’s note: The modern notation might be thought to be the empty set symbol ∅, except that Cantor states in his next paragraph that the capital O symbol is to indicate the absence of the existence of a set rather than asserting that an empty set exists. The notion of an “empty set” as an existing set was a later development, see Akihiro Kanamori, ‘The Empty Set, The Singleton, And The Ordered Pair’, The Bulletin of Symbolic Logic, 9.3, Sept 2003.
)
so that *P = O* means that the set *P* does not contain even a single point, so that, strictly speaking, it itself does not exist. We can give here an example, where the *n* + 1^{th} derivative of a point-set of the first number-class and which is of the nth type, is such that:

*P*^{ (n + 1)} = *O*

but where on the other hand, *P*^{ (n)} is not the same as *O*. Two sets are said to be * disjoint* if their intersection is *O*.

If two point-sets, *P* and *Q*, have the same *cardinal-number*, i.e. if they belong to the *same* number-class (see Part 1), we call them *equivalent* and express this relationship by:

*P ~ Q*

And if it is the case that *P ~ Q* and *Q ~ R* then it is always the case that:

*P ~ R*

Furthermore, if *P*_{1}, *P*_{2}, *P*_{3}, … is a series of sets, where no two sets have elements in common, and *Q*_{1}, *Q*_{2}, *Q*_{3}, … is another series for which the same applies, then we have that *P _{1} ~ Q_{1}* and

*P*and

_{2}~ Q_{2}*P*and so on, and also:

_{3}~ Q_{3}{*P*_{1}, *P*_{2}, *P*_{3}, …} ~ {*Q*_{1}, *Q*_{2}, *Q*_{3}, …}

As we have just seen, the point-sets of the first number-class can be completely characterized by the theory of derivation, as I have developed it up to now. For point-sets of the second number-class that theory is not sufficient, and an extension of the theory becomes necessary, which, when understood in greater depth, presents itself as an inevitable development.

Note that in the series of derivatives *P′, P′′, P′′′, …* of a set *P*, each term is a subset of the previous one, i.e. every new derivative *P*^{ (v)} is formed from the previous *P*^{ (v - 1)} by the omission of certain points, without the addition of any new points. If *P* belongs to the second number-class, then *P′* will be composed of two essentially different point-sets *Q* and *R*, so that we have:

*P′* = {*Q, R*}

where *Q* consists of those points of *P′* which are progressively lost in the sequence *P′, P′′, P′′′, …*, while *R* comprises those points which are in every set of the sequence *P′, P′′, P′′′, …*, so *R* is defined by the formula:

*R* = **⋂**{*P′, P′′, P′′′, …* }

But we also obviously have:

*R* = **⋂**{*P′′, P′′′, P′′′′, …* }

and in general:

*R* = **⋂**{*P* ^{(n1 )}, *P* ^{(n2 )}, *P* ^{(n3 )}, … }

where *n*_{1}, *n*_{2}, *n*_{3}, … is a sequence of increasing positive integers.

This point-set *R*, which emerges from the set *P*, is denoted by the use of the symbol *ω*, viz: (Footnote:
Translator’s note: In the original: ∞ rather than *ω*. See Cantor’s footnote in § 11 of Part 5 of this work, which is: The sign ∞, which I used in Part 2 of this work (*Mathematische Annalen* Vol. 17, p. 357), I replace from now on with *ω*, because the sign ∞ is already often used to denote non-actual infinities.
)

*P*^{ (ω)}

and is called the derivative of *P* whose order is *ω*. We denote the first derivative of *P*^{ (ω)} by *P*^{ (ω + 1)}, the *n*^{th} derivative of *P*^{ (ω)} by *P*^{ (ω + n)}. But *P*^{ (ω)} will also have a derivative of order *ω*, generally not *O*, and we denote it by *P*^{ (2ω)}. By the continuation of these constructional terms we arrive at subsequent derivations that are denoted by:

*P*^{ (n0ω + n1)}

where *n*_{0} and *n*_{1} are positive integers. But we can also go beyond that by:

*R* = **⋂**{*P* ^{(ω)}, *P* ^{(2ω)}, *P* ^{(3ω)}, …}

and we denote this by *P*^{ (ω2 )}. By repeating the same operation and combining it with those obtained earlier, the more general concept emerges:

*P*^{ (n0ω2 + n1ω2 + n2 )}

and by continuing this procedure one comes to:

*P*^{ (n0ωv + n1ωv - 1+ … + nv )}

where *n*_{0}, *n*_{1}, …, *n*_{v} are positive integers. One arrives at further terms by increasing values of the variable *v*, which gives:

*P*^{(ωω )} = **⋂** {*P*^{(ω)}, *P*^{(ω2 )}, *P*^{(ω3 )}, … }

By subsequent progression we successively obtain the further terms:

*P*^{(n ωω )}, *P*^{(ωω + 1 )}, *P*^{(ωω + n )}, *P*^{(ωn ω )},
*P*^{(ωω n )}, *P*^{(ωωω )}, …

And we see in the above a logical generation of terms (Footnote:
I arrived at this conclusion ten years ago. I mentioned it in my article *Über die Ausdehnung eines Satzes aus der Theorie
der trigonometrischen Reihen* (On the generalization of a theorem in the theory of trigonometric series), Mathematische Annalen Vol. 5, pp. 123–132, 1872.
)
which always increases further and further, in an inevitable progression that is free from any arbitrariness.

For point-sets of the first cardinal-number, it follows from the definition of such that we have:

*P*^{ (ω)} = *O*

It is noteworthy that the reverse can also be proved, that every point-set for which that equation applies is of the first number-class, hence sets which have the first cardinal-number are completely characterized by that equation.

It is easy to give an example of a point-set of the second number-class, for which *P*^{ (ω)} consists of a point *p*. Consider the case of intervals that follow one upon another, which border to each other, are infinitely decreasing, and converging towards *p*, and where these intervals are point-sets of the first number-class; their ordinal-numbers increase beyond any limits as the corresponding intervals approach *p*. Hence when taken together as a set, they constitute the requisite example; this also serves to settle the question raised in Part 1 of this work, which is whether an interval, if it is *everywhere-dense*, must belong to a set of the second number-class; we can see from this example that this is *by no means* necessary.

It is equally easy to construct point-sets of the second cardinal-number, such as *P* ^{(ω + n)} or *P* ^{(2ω)} or more generally:

*P*^{ (n0ωv + n1ωv - 1+ … + nv )}

which consist of a specified point *p*.

All such sets are *not* everywhere-dense in any interval, and moreover belong to the *first* number-class; in these two relationships they resemble the point-sets of the first kind.

Footnotes: