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A Contribution to the Theory of Sets
Ein Beitrag zur Mannigfaltigkeitslehre

The following is an English translation Cantor’s 1878 paper regarding a one-to-one correspondence between points of a real number line and points of a plane or of higher dimensions:

Georg Cantor, “Ein Beitrag zur Mannigfaltigkeitslehre”, Journal für die reine und angewandte Mathematik (Crelles Journal), 1878, Vol.84 (1878) pp.242-258.

The original German text is viewable online at Cantor - Ein Beitrag zur Mannigfaltigkeitslehre, and another repository is at Cantor - Ein Beitrag zur Mannigfaltigkeitslehre.

 

English translation by James R Meyer, copyright 2022 www.jamesrmeyer.com

 

Note that the term “cardinality” for infinite sets was not in current usage at the time Cantor wrote this paper; he uses the term “Mächtigkeit”, which can have corresponding English meanings such as ‘thickness’, ‘width’, ‘mightiness’, ‘potency’, etc. I have used the term “magnitude” as a suitable translation.

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INDEX

 

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A Contribution to the Theory of Sets

 

If two well-defined sets M and N can be set to correspond to each other uniquely and completely, element for element (which, if it is possible in one way, can always be done in many other ways), then in what follows we may say that these sets are of equal magnitude, or that they are equivalent. By a subset of a set M we mean any other set M′ whose elements are also elements of M. If the two sets M and N are not of equal magnitude, then either M will have the same magnitude as some subset of N or N will have the same magnitude of some subset of M; in the first case we say that M has a smaller magnitude than N, in the second case we say that M has a larger magnitude than N.

 

If the sets under consideration are finite, that is, they are composed of a finite number of elements, it is easy to see that the concept of magnitude corresponds to that of number and moreover to the concept of whole positive number, since two such finite sets are of equal magnitude if and only if the number of their elements is the same. A subset of a finite set is always of smaller magnitude than the set itself; but this relationship disappears completely for infinite sets, that is, sets composed of an infinite number of elements. From the mere fact that an infinite set M is a subset of another N, or that it can be set in a unequivocal and complete correspondence to one, it should by no means be concluded that its magnitude is smaller than that of N. This conclusion is only justified if one knows that the magnitude of M is not equal to that of N; and the fact that N is a subset of M or can be set in an unequivocal and complete correspondence to such should not be regarded as sufficient to conclude that the magnitude of M is greater than that of N. To give a simple example, if M is the sequence of positive integers ν, and N is the sequence of positive even integers 2ν, then here N is a subset of M, but nevertheless M and N are of equal magnitude.

 

The sequence of positive integers ν offers, as can easily be shown, the smallest of all magnitudes that occur in infinite sets. Nevertheless the class of all sets which are of this smallest magnitude is an exceedingly rich and extensive one. This class includes, for example, all those sets which Herr R. Dedekind calls “finite fields” in his valuable and beautiful investigations into algebraic numbers⁠. (Footnote: See also “Dirichlet’s Lectures on Number Theory”, second edition, Braunschweig 1871, p.425f.(Vorlesungen über Zahlentheorie, zweite Auflage).) Furthermore, the sets that I first analyzed are included here, which I have called “Point sets of the νth kind ”⁠. (Footnote: See Mathematical Annals, Vol.5, p.129 (Mathematische Annalen von Clebsch und Neumann).) Every set that appears as a simple infinite sequence, with the general term αν, obviously belongs here; but also the double sequence and in general the n‑fold sequence with the general term αν1, αν2, … ανn (where each ν1, ν2, … νn has the domain of all positive integers independently of each other) belong to this class. On an earlier occasion it was also proved that the set (ω) of all real (and one could also add, all complex) algebraic numbers can be considered in the form of a sequence with the general term ων, which means only that the set (ω) as well as each infinite subset of it have the magnitude of the entire sequence of numbers.

 

With regard to sets of this class, the following easy-to-prove theorems hold:

“If M is a set of the magnitude of the sequence of positive integers, then every infinite subset of M has the same magnitude as M.”

“If M′, M′′, M′′′ is a finite or simple infinite sequence of sets, each of which has the magnitude of the sequence of positive integers, then the set M which results from the union of the subsets M′, M′′, M′′′ has also the same magnitude.”

In the following, the so-called continuous, n‑fold spaces will be analyzed with regard to their magnitudes.

 

The assumptions which Riemann⁠ (Footnote: See Riemann’s collected mathematical works. Leipzig 1876. p.254f (Riemanns gesammelte mathematische Werke).) and Helmholtz⁠ (Footnote: See Helmholtz, “About the factual foundations of geometry”. Heidelberg yearbook 1868, no.46 and 47 (Über die tatsächlichen Grundlagen der Geometrie. Heidelberger Jahrbücher), and
“On the facts underlying geometry”. Goettinger News 1868, no.9 (Über die Tatsachen, welche der Geometrie zugrunde liegen. Nachr. Ges. Wiss. Göttingen, Math.-physik. Kl. 1868, Nr.9), and
Popular lectures by the same author, Vol 3, Braunschweig 1870, p.21f (desselben Verfassers populäre Vorträge, H.3, S.2lf. Braunschweig 1876).)
and others since⁠ (Footnote: See J. Rosanes, “About the latest investigations into our view of space”, Breslau, 1871, p.13 (Über die neuesten Untersuchungen in betreff unserer Anschauung vom Raume), and
O. Liebmann, “On the analysis of reality”. Strassburg, 1876, p.58 (Zur Analysis der Wirklichkeit), and
B. Erdmann, “The axioms of geometry”, Leipzig, 1877, p.45 (Die Axiome der Geometrie).)
made about the hypotheses on which geometry is based began, as is well known, from the concept of an n‑fold extended, continuous space where its essential characteristic is the fact that its elements depend on n independent, real, continuous variables x1, x2, … xn , so that for each element of the space there is a possible system of values x1, x2, … xn, but also conversely, every admissible system of values x1, x2, … xn includes its own extent of variability. As can be seen from the course of those investigations, the assumption was also made, generally implicitly, that the underlying correspondence of the elements of the space to the system of values x1, x2, … xn is a continuous one, so that for each infinitely small change of the system of values of x1, x2, … xn there is an infinitely small change of the corresponding element and conversely every infinitely small change of an element corresponds to a change in the value of its coordinates. For now, it may be left as an open question as to whether this assumption is to be regarded as sufficient, or whether it should be supplemented by even more specialized conditions, so that the intention behind the concept of an n‑fold, continuous space, such that it can be considered to be secure against any contradiction, is intrinsically consistent.⁠ (Footnote: The answer to this question, to which we shall return on another occasion, does not seem to me to encounter any difficulties worth mentioning.)

 

Here it will be shown that if that requirement is dropped (i.e: if no restriction is made regarding the correspondence between a space and its coordinates), then that requirement which was considered by its authors to be essential (according to which an n‑fold continuous space is such that its elements are determined by n real, continuous coordinates that are independent of one another) becomes completely obsolete.

 

As our analysis will show, it is even possible to uniquely and completely determine the elements of an n‑fold extended continuous space by a single, real continuous coordinate t. It then follows that, if no assumptions are made regarding the nature of the correspondence, the number of independent, continuous, real coordinates that can be used for a unique and complete determination of the elements of an n‑fold extended continuous space can be defined to be any arbitrary number, and therefore that number is not to be regarded as an invariable characteristic of a given space. By asking myself the question whether a continuous space of n dimensions can be mapped uniquely and completely to a continuous set of one and only one dimension, such that each element corresponds to one and only one element of the other, it was found that this question must be answered in the affirmative.

 

Accordingly, a continuous surface can be clearly and completely related to a continuous line, and the same applies to continuous bodies and continuous structures with any number of dimensions.

 

Using the terminology introduced above, we can therefore say that the magnitude of any continuous n‑fold space is equal to the magnitude of a singly-extended continuous set, such as a finite continuous straight segment.

 

§ 1

Since two continuous structures that have the same number of dimensions can be clearly and completely related to each other by means of analytic functions, the purpose we are here pursuing (namely to demonstrate the possibility of unambiguous and complete associations of continuous structures with different numbers of dimensions), as one can easily see, is completely dependent on the proof of the following theorem:

(A)If x1, x2, … xn are independent, variables, each of which can assume any real number values ≥ 0 and ≤ 1, and if t is another variable with the same domain of values (0 ≤ t ≤ 1), then it is possible to associate the variable t to a system consisting of n variables x1, x2, … xn in such a way that for each specific value of t there is a specific system of values x1, x2, … xn and vice versa, that for every such specific system of values x1, x2, … xn, there is a corresponding t.  ◆

 

As a consequence of this theorem there is another that comes to mind:

(B)A continuous space that extends over n dimensions can be mapped uniquely and completely to a continuous set of one dimension; two continuous spaces, one of n dimensions, and the other of m dimensions, where nm, have equal magnitude; the elements of a continuous space extended in n dimensions can be uniquely determined by a single continuous, real coordinate t, but they can also be uniquely and completely determined by a system of continuous coordinates t1, t2, … tm.  ◆

 

§ 2

To prove Theorem (A) we start from the well-known theorem that every irrational number e, 0 < e < 1, can be set in a completely determined manner in the form of an infinite continued fraction:

where αν are positive integers.

 

For every irrational number e, 0 < e < 1, there is a specific corresponding infinite sequence of positive integers αν and conversely every such sequence determines a specific irrational number e, 0 < e < 1.

 

If e1, e2, … en are n are variables independent of each other, where each of which can take any irrational numerical value in the interval (0…1), and each of these only once, then we set:

e1 = (α1, 1, α1, 2 , … α1, ν , …),

. . . . . . . . . . . . . . . . . . . . . . .

eμ = (αμ, 1, αμ, 2 , … αμ, ν , …),

. . . . . . . . . . . . . . . . . . . . . . .

en = (αn, 1, αn, 2 , … αn, ν , …).

 

These n irrational numbers uniquely determine an n + 1th irrational number d, 0 < d < 1:

d = ( β1 , β1 , … β1 , …),

by creating the following relationship between the numbers α and β:

 

But also vice versa: if one starts with an irrational number d, 0 < d < 1, then this determines the sequence of βν, and by virtue of (1) also the sequence of αu,v (i.e: d uniquely determines the system of n irrational numbers e1, e2, … en). The following theorem results from this consideration:

(C)If e1, e2, … en are n variables independent of each other, where each of which can assume any irrational numerical value in the interval (0…1), and where d is another variable with the same domain, then it is possible for that single variable d and the system of n variables e1, e2, … en to correspond uniquely and completely to each other.  ◆

 

§ 3

Following the proof of Theorem (C) in the previous section, it is now incumbent upon us to prove the following theorem:

(D)A variable e, which can assume any irrational numerical value in the interval (0…1), can be set to correspond uniquely to a variable x, whose domain is all real (i.e: rational and irrational) values where 0 ≤ x ≤ 1, so that for every irrational value of e, 0 < e < 1 there is one and only one real value of x, 0 ≤ x ≤ 1 and vice versa for every real value of x there is a specific corresponding irrational value of e.  ◆

 

Once this Theorem (D) has been proved, then following upon that and Section 2, we have that e1, e2, … en and d denote n + 1 variables that correspond uniquely and completely to the other variables x1, x2, … xn and  t, where each one of these variables can take every real value ≥ 0 and ≤ 1 without restriction. Since in Section 2 an unambiguous and complete correspondence is established between the variable d and the set of n variables e1, e2, … en, in this way one obtains an unambiguous and complete correspondence of the single continuous variable t and the set of n continuous variables x1, x2, … xn, by which the validity of Theorem (A) will be proven.

 

Hence, in the following we only have to deal with the proof of Theorem (D); for the sake of brevity, we will use a simple terminology, which we will first describe.

 

By a linear set of real numbers we want to understand any well-defined set of real, differing, (i.e: not equal) numbers such that any given number does not occur more than once as an element in a linear set.

 

The real valued variables encountered in the course of this inquiry are all such that the domain of each of them (i.e: the set of values which it can take) is a given linear set. We therefore do not want to particularly emphasize the assumption, which is made implicitly throughout the following. For two given variables a and b we shall say that they have no connection if no value that a can take is equal to some value of b (i.e: the two sets of values which the variables ab can take have no elements in common), and it can be said that a and b are unconnected⁠. (Footnote: Two sets M and N either have no connection if they have no common element; or they are connected by a certain third set P, namely by the set of the elements they have in common.)

 

If one has a finite or infinite sequence a′, a′′, a′′′, … , a(v), … of well-defined variables or constants that have no pairwise connection, then a variable a can be defined by the fact that its domain arises from the combination of the domains of a′, a′′, a′′′, … , a(v), …. Conversely, a given variable a can be decomposed in several ways into other a′, a′′, … which have no pairwise connection; in both of these cases we express the relation of the variable a to the variables a′, a′′, a′′′, … , a(v), … by the following formula:

a ≡ {a′, a′′, a′′′, … , a(v), …}

 

Hence the existence of this formula includes the following:

  1. every value which any one of the variables a(v) can assume is also a corresponding value of the variable a, and
  2. every value which a can assume is also assumed by one and only one of the variables a(v).

 

To explain this formula, for example, if φ is a variable containing all rational numerical values which are ≥ 0 and ≤ 1, e a variable containing all irrational numerical values of the interval (0…1), and finally x a variable which can assume all real (rational and irrational) numerical values that are ≥ 0 and ≤ 1, then we have that:

x ≡ {φ, e}

 

If a and b are two variables of such a kind that it is possible for them to correspond uniquely and completely to one another, in other words, if their domains are of equal magnitude, then we shall call a and b equivalent to one another and we indicate this by one of the two formulas:

a ~ b   or   b ~ a

According to this definition of the equivalence of two variables, it easily follows that a ~ a; furthermore that if a ~ b and b ~ c, then it is also always the case that a ~ c.

 

In the following analysis, the following theorem, the proof of which we may omit because of its simplicity, will be applied at various points:

(E)If a′, a′′, a′′′, … , a(v), … is a finite or infinite sequence of variables or constants which have no pairwise connection, and b′, b′′, b′′′, … , b(v), … another sequence of the same type, to each variable a(v) of the first sequence corresponds a definite variable b(v) of the second sequence, and these corresponding variables are always equivalent to one another, that is a(v) ~ b(v), and it is also always the case that:

a ~ b

if

a ≡ {a′, a′′, a′′′, … , a(v), …}

and

  b ≡ {b′, b′′, b′′′, … , b(v), …}   ◆

 

 

§ 4

Our analysis has now progressed to the point that we are only left with the proof of Theorem (D) in Section 3. To achieve this goal, we suppose that all rational numbers which are ≥ 0 and ≤ 1 are in the form of a simple infinite sequence:

φ1, φ2, φ3, …, φν, …

where the general term is φν. That this can be the case is shown quite simply as follows:

If pq is the irreducible form for a rational number that is ≥ 0 and ≤ 1 (i.e: where p and q are non-negative integers with the greatest common divisor 1), then let p + q = N. Then a definite, integral, positive value of N belongs to every number; conversely, only a finite number of numbers pq belongs to such a value of N. If the numbers pq are considered to be ordered that those belonging to smaller values ​​of N precede those belonging to greater values ​​of N, and furthermore, for any single value of N, the numbers pq follow one another according to their magnitude, the larger following the smaller, then every one of the numbers pq is in a very specific position in a simple infinite sequence, the general term of which is denoted by φν. This theorem can also be concluded from an article I recently published⁠ (Footnote: See G. Cantor, “On a property of the set of all real algebraic numbers”, this journal, Vol.77, p.258f (Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Crelles Journal)   See an online English translation at On a Property of the Set of all Real Algebraic Numbers.) according to which the essential attribute (ω) of all real algebraic numbers is in the form of an infinite sequence:

ω1, ω2, …, ων, …

with the general term ων . This property of the attribute (ω) is carried over to the attribute of all rational numbers that are ≥ 0 and ≤ 1, because this set is a subset of that set. Now let e be the variable occurring in Theorem (D), which has to assume all real numerical values of the interval (0…1), with the exception of the numbers φν.

 

Suppose further that in the interval (0…1) there is an infinite sequence of irrational numbers εν which is bound only by the conditions that in general εν < εν + 1 and that lim εν = 1 for ν = ∞; for example, let

εν = 1 - 22ν

We denote by f a variable that can take any real value in the interval (0…1), with the exception of the values εν and by g another variable that can take any real value in the interval (0…1), with the exception of the values εν and φν. We claim that:

e ~ f

In fact, according to the terminology of Section 3:

e ≡ { g, εν }

f ≡ { g, φν }

and since g ~ g, εν ~ φν, then by Theorem (E) we conclude that:

e ~ f.

 

Theorem (D), which is to be proved, is therefore reduced to the following proposition:

(F)A variable f, which can take any value in the interval (0…1), except for the values of a given sequence εν and which is bound by the conditions that εν < εν + 1 and that lim εν = 1 for ν = ∞ can be set to correspond uniquely and completely to a variable x, which can take any value ≥ 0 and ≤ 1; in other words, f ~ x.  ◆

 

 

§ 5

We base the proof of Theorem (F) on the following Theorems (G), (H) and (J):

(G)If y is a variable which can take any value in the interval (0…1) with the exception only of 0, and if x is a variable which can take any value in the interval (0…1) without exception, then:

y ~ x.  ◆

 

The simplest way of proving this Theorem (G) is to consider the curve in the diagram below, whose abscissas from 0 represent the magnitude x, and whose ordinates represent the magnitude y, and where the curve consists of the infinitely many parallel segments

a b, a′ b′, … , a(ν)b(ν), …

which become infinitely smaller as ν increases to infinity, and of the isolated point c, which those lines approach asymptotically. Here, however, the end points a, a′, ... , a(ν), ... belong to the curve, whereas the end points b, b′, ... . ,b(ν), ... are to be considered as excluded from it.

Diagram for Theorem G

 

The lengths represented in the figure are:

Op = pc = 1

Ob = bp = Oa = 12

a(ν)d(ν) = d(ν)b(ν) = bv-1bv = 12v+1

One can determine that while the abscissa x takes all values from 0 to 1, the ordinate y takes all these values with the exception of the single value 0.

 

Following on from this proof of Theorem (G), by the application of the transformation formulas:

one obtains the following generalization of Theorem (G):

(H)A variable z, which can take all values of an interval (α … β), where αβ, with the exception of the one final value α, is equivalent to a variable u, which can take all values of the same interval (α … β) without exception.  ◆

 

From here we then arrive at the following theorem:

(J)If w is a variable which has to assume all values of the interval (α … β) with the exception of the two final values α and β of that interval, and if u is the same variable as in (H), then:

w ~ u.  ◆

 

Indeed, if we let γ be some value between α and β, and we introduce four new variables ω′, ω′′, u′′ and z, where:

z is to be the same variable as in Theorem (H),

ω′can take assume any value in the interval (α … γ), with the exception of the two final values α and γ,

ω′′ receives all values of the interval (γ … β) with the exception of the single endpoint value β,

u′′ is a variable which has to assume all values of the interval (γ … β) including both endpoint values.

 

then we have:

ωω′, ω′′

zω′, u′′

 

However, as a consequence of Theorem (H) we have that:

ω′′ ~ u′′

and so we can conclude that :

ω ~ z.

Also by Theorem (H) we have that:

z ~ u

and consequently it follows that:

ω ~ u

which proves Theorem (J).

 

 

We can now prove Theorem (F) as follows: By the definition of the variables f and x in the statement of (F), we introduce certain auxiliary variables:

f ′, f ′′, f ′′′, … , f(v), …

and

x′, x′′′′, … , x(2v), …

where:

f ′ is a variable whose domain includes all values of the interval (0 … ε1) with the exception of the single final value ε1,

f(v) for all ν > 1 is a variable whose domain includes all values of the interval (εν - 1 εν) except for the two end values εν - 1 and εν,

x(2v) is a variable whose domain includes all values of the interval (ε2ν - 1 ε2ν) without exception.

 

If one adds the constant number 1 to the variables f ′, f ′′, f ′′′, … , f(v), …, then all these variables taken together have the same domain as f, that is, one has:

f ≡ { f ′, f ′′, f ′′′, … , f(v), … , 1}

We can also determine that:

x ≡ { f ′, x ′′, f ′′′, x ′′′′, … , f(2v - 1), x(2v), … , 1}

 

According to Theorem (J) however:

f(2v) ~ x(2v)

and furthermore:

f(2v - 1) ~ f(2v - 1) and 1 ~ 1

therefore, because of Theorem (E) in Section 3 we have that:

f ~ x

which was the conjecture to be proved.

 

 

§ 6

I now want to give a shorter proof of Theorem (D). The reason for not using only this proof is because the auxiliary lemmas (F), (G), (H), (J), which were used in the more complex proof, are of interest in themselves.

 

Let x, as before, be a variable which can take any real value in the interval (0…1), including the endpoint values; Let e be a variable which has the domain only of irrational values in the interval (0… 1). The conjecture to be proved is that x ~ e.

 

We consider the rational numbers ≥ 0 and ≤ 1 as in Section 4, in sequence form with the general term φν where ν traverses through the sequence of numbers 1, 2, 3, … . Furthermore, we assume some arbitrary infinite sequence of only irrational and mutually different numbers in the interval (0…1) and we let the general term of this sequence be ην  (e.g: ην = 22ν).

 

Let h be a variable which can take on any value in the interval (0…1) with the exception of φν and ην . As in Section 3 the terminology used is:

and

e ≡ { h, ην }

We can also write this last formula as follows:

We can note that:

h ~ h ;    η2ν ~ η2ν - 1 ;    φν ~ η2ν

and applying Theorem (E) from Section 3 to the two formulas (1) and (2) we have that:

x ~ e

which was the conjecture to be proved.

 

 

§ 7

For the proof of Theorem (A), the idea suggests itself of choosing the infinite decimal representation format instead of the sequence formula that is used. Although it might appear that this method would lead more quickly to the required goal, it nevertheless involves a difficulty which I wish to draw attention to here, and which was the reason why I refrained from using decimal representation in this analysis.

 

For example, if you have two variables x1 and x2 and set:

x1 = α1 ⁄10 + α2 ⁄100 + … + αν ⁄10ν + …

x2 = β1 ⁄10 + β2 ⁄100 + … + βν ⁄10ν + …

with the proviso that the numbers αν and βν are integers ≥ 0 and ≤ 9, and from a certain ν onwards are not always a value of 0 (except when x1 or x2 are themselves equal to zero), then these representations of x1 and x2 must be uniquely determined in all cases (i.e: x1 and x2 determine the infinite number sequence αν and βν and vice versa). If you derive from x1 and x2 a number:

t = γ1 ⁄10 + γ2 ⁄100 + … + γν ⁄10ν + …

by letting:

γ2ν - 1 = αν and γ2ν = βν for ν = 1, 2, …

this does define a unique relationship between the system of x1 and x2 and the single variable t since a single set of values for x1 and x2 gives a specific value of t. However the fact to be considered here is that the variable t does not take on all the values of the interval (0…1); it is restricted in its variability, while x1 and x2 are not subjected to any restriction within that same interval. All values of the series sum:

γ1 ⁄10 + γ2 ⁄100 + … + γν ⁄10ν + …

where from some ν > 1 onwards all of γ2ν - 1 or all of γ2ν have the value zero must be considered to be excluded from the domain of t because such would lead to excluded decimal representations, namely finite representations of x1 or x2 .

 

 

§ 8

Having completed the intended analysis in the preceding paragraphs, there is now an opportunity to expand somewhat on the above. Theorem (A) and also Theorem (B) can be generalized, where continuous spaces of an infinitely large number of dimensions have the same magnitude as continuous sets of one dimension. However, this generalization is essentially reliant on one assumption, namely that such infinitely many dimensions themselves form a set which has the magnitude of the complete positive number sequence.

 

Instead of Theorem (A) we have the following:

(A′)If x1, x2, … , xμ , … is a simple infinite sequence of mutually independent variable real magnitudes, each of which can assume any value that is ≥ 0 and ≤ 1, and if t is another variable with the same domain of values (0 ≤ t ≤ 1), then it is possible for the single variable t to correspond completely and uniquely to the sequence of infinitely many x1, x2, … , xμ , ….  ◆

 

By the help of Theorem (D) in Section 3 this proposition (A′) is reduced to the following:

(C′)If e1, e2, … , eμ , … is a simple infinite sequence of mutually independent variable real magnitudes, each of which can assume any irrational numerical value of the interval (0…1), and if d is another variable with the same domain of values, then it is possible for the single variable d to correspond completely and uniquely to the sequence of infinitely many e1, e2, … , eμ , ….  ◆

 

The easiest way to prove Theorem (C′) is to use the continued fraction expansion as in Section 2, where:

eν = (αν, 1, αν, 2 , … αμ, ν , …) for ν = 1, 2, …

and

d = (β1, β2 , … βλ, …)

establish a relationship between the integer positive numbers α and β, namely:

αμ, ν = βλ

where

It is easily shown that the above function has the remarkable property that it represents all positive integers once and one only once if μ and ν can take the values of every positive integer value independently of one another.

 

With Theorem (A′), however, it now seems that a limit has been reached as to the possibility of any further generalization of Theorem (A) and the consequent conclusions.

 

Since it has been demonstrated that for an extraordinarily rich and wide range of spaces that they have the property that they can be uniquely and completely made to correspond to a bounded, continuous straight line or any part of it (by a part of a line we mean any subset of its points), then the question arises as to how the various parts of a continuous straight line (i.e: the various infinite subsets of points within it) are related as regards their magnitude. We can detach this problem from its geometric associations and consider, as already discussed in Section 3, that a linear set of real numbers defines every conceivable example of an infinite number of real numbers that are different from one another, so the question arises as to how many and into which classes the linear sets can be broken down, if sets of equal magnitude belong to a single class, and sets of different magnitudes are in different classes. By means of an inductive procedure, the presentation of which we shall not go into in detail here, this leads to the proposition that the number of classes of linear sets resulting from this principle of division is finite, namely that it is equal to two.

 

According to the above, the linear sets would consist of two classes:⁠ (Footnote: That these two cases are in fact different follows from what is proved in § 2 of the previously cited work (This Journal Vol.77 p.258f ), according to which, if there is a valid infinite sequence ω1, ω2, …, ωv, …, then in any given interval (αβ) there can always be numbers μ found that do not occur in the given sequence.   See an online English translation of that article at On a Property of the Set of all Real Algebraic Numbers.)

the first of which includes all sets which can be given by a function with a free variable ν, where ν has the domain of all positive integers,

while

the second class embraces all those sets which can be given by a function with a free variable ν, where ν has the domain of all real values ≥0 and ≤1.

 

Corresponding to these two classes, therefore, only two magnitudes occur for the case of the infinite linear sets; we defer the detailed examination of this question to a later occasion.

 


 

Footnotes:

 


 

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