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On a Property of the Set of All Real Algebraic Numbers by G. Cantor at Halle a. S. (Translation of a memoir published by Borchardt's Journal, vol. 77, p. 238.)

A real number ω is generally called a real algebraic number if it is a root of an equation of the form:

(1)   a0&thinsp;ωn + a1&thinsp;ωn − 1 + … + an = 0

where n, a0, a1, …, an are integers. We can assume that the numbers n and a0 are positive, that the coefficients a0, a1, …, an have no common divisor, and that the equation (1) is irreducible. These assumptions being made, it follows from the fundamental theorems of arithmetic and algebra that equation (1) having a definite real algebraic number for a root is a fully determined equation; conversely, to an equation of the form (1) corresponds at most as many real algebraic number roots of this equation as its degree n. These assumptions being made, it follows from the fundamental theorems of arithmetic and algebra that equation (1) having for a root is a fully determined equation; conversely, to an equation of the form (1) corresponds at most as many real algebraic number roots of this equation as its degree n.

The real algebraic numbers constitute in their totality a set of numbers that we designate by (ω). As follows from elementary considerations, this set (ω) is of such nature that there exists an infinity of numbers from (ω) whose difference with an arbitrary number α is less than a given number however small. This observation makes all the more striking, at first glance, the following property: the numbers of the set (ω) can be corresponded one-to-one to the numbers ν belonging to the set of positive integers, which will be designated by (ν), in such a way that to each real algebraic number ω corresponds a definite positive integer ν, and that inversely to each positive integer ν corresponds a fully determined real algebraic number ω. In other words, we can imagine the numbers of the set (ω) placed in an infinite sequence following a certain rule

(2)   ω1, ω2, …, ωn, …

in which all the numbers of the set (ω) appear, each of them in the sequence (2) at a definite place indicated by the corresponding index. Once a rule has been found permitting the placing of the numbers of (ω) in this way, others will be deduced from it by modifications that will be freely chosen. Thus, it will be sufficient for us to indicate, as we do in § 1, the placement method that appears to us to rest on the least number of considerations.

To give an application of this property of the set of all real algebraic numbers, I add to § 1 the § 2, in which I show that, when we consider as given any sequence of real numbers in the form (2), we can determine, in every interval [α, β] given in advance, numbers not contained in this sequence (2). By combining the propositions contained in §§ 1 and 2, we thus obtain a new proof of the following theorem, proved for the first time by Liouville (Journ. de Math. ed. by Liouville, series I, vol. XVI, 1851): in every interval [α, β] given in advance, there are an infinity of transcendental numbers; that is, numbers that are not algebraic reals. Furthermore, the theorem of § 2 gives the reason why we cannot correspond one-to-one the integers of the set (ω) with the real numbers forming a continuous set of numbers; that is, for example, all the real numbers that are ≥ 0 and ≤ 1. Thus, I have found in a clear way the essential difference that there is between a continuous set of numbers and a set of numbers of the type that is formed by the set of all real algebraic numbers.

§ 1.

We return to equation (1), which is satisfied by a real algebraic number and which, after the assumptions made above, is a fully determined equation. Let us call the height of the number ω, the sum of the absolute values of the coefficients of the equation, plus the number n − 1, n being the degree of the equation. Designating this height by N and using a well-known notation to designate the absolute values of numbers, we have:

(3)   N = n − 1 + |a0| + |a1| + … + |an|.

This height N is, therefore, for each real algebraic number, a specific positive integer; conversely, to a given positive integer N corresponds only a limited number of real algebraic numbers having height N. Let φ(N) be this number; we have, for example, φ(1) = 1, φ(2) = 2, φ(3) = 4. The numbers of the set (ω), that is, all the real algebraic numbers, can then be placed in the following order: we take as the first number ω1, the only number of height 1. Following it, we write in order of increasing size the two real algebraic numbers of height N = 2, and we designate them by ω2, ω3; then after them and in order of increasing size, we write the four numbers of height N = 3. In general, after we have in this way enumerated and ordered the numbers of the set (ω) up to a specific height N = N1, we place after them and in order of increasing size the real algebraic numbers of height N = N1 + 1. Therefore, we obtain the set of all the real algebraic numbers in the form:

ω1, ω2, …, ων, …

and we can, by reference to this classification, speak of the ν-th real algebraic number, without omitting any number of the set (ω).

§ 2.

When we have an infinite sequence of real numbers, different from one another and succeeding one another according to some well-defined law

(4)   u1, u2, …, uν, …

we can, in every interval [α, β] given in advance, determine a number η that is not in the sequence (4); there exist, therefore, an infinity of such numbers. Here is the proof of this theorem.

We start from the interval given in advance [α, β] and let α < β; let us designate by α', β', the first two numbers of the sequence (4) different from each other, which are distinct from α, β and which are in the interior of this interval, and let α' < β'. Similarly, designate by α, β, the first two numbers of our sequence different from each other, which are in the interior of the interval [α', β'] and let α < β ; according to this same rule, we form a following interval [α, β] , and so on. According to this definition, the numbers α', α, … are well-defined numbers uk 1 , uk 2 , …, uk ν of our sequence (4) whose indices kν increase constantly, and the same thing happens for the numbers β', β , …. In addition, the numbers α', α, … are of increasing size, the numbers β', β , … of decreasing size, and each of the intervals [α, β], [α', β'], [α, β] , … include all those that follow it. We can then conceive of two cases.

Either the number of intervals that can be formed in this way is finite: let [α(ν), β(ν)] be the last of them. Since at most one number of the sequence (4) is in the interior of this interval, we can take from this interval a number η that does not belong to the sequence (4), and the theorem is thus proved in this case.

Or the number of intervals that can be formed in this way is infinite: then, since the numbers α, α', α'', … increase constantly without growing to infinity, they have a definite limit α(∞). Similarly, the numbers β, β', β'', … , which decrease constantly, have a definite limit β(∞). If α(∞) = β(∞) (which always occurs in applying this method to the set (ω) of real algebraic numbers), we easily check by returning to the definition of the intervals, that the number η = α(∞) = β(∞) cannot be contained in our sequence; for if this number η was contained in our sequence, we would have η = up, p being a definite index; but this is not possible because up is not in the interior of the interval [α(p), β(p)], while η is in it according to its definition. If, on the contrary, α(∞) < β(∞), all numbers η included in the interior of the interval [α(∞), β(∞)] or equal to one of the limits, fulfill the required condition of not belonging to the sequence (4).

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The theorems that we have just proved can be generalized in various ways. We only state here the following proposition:

"Let v1, v2, …, vν, … be a finite or infinite sequence of linear independent numbers, that is, numbers such that there exists between them no equation of the form

a1&thinsp;v1 + a2&thinsp;v2 + … + an&thinsp;vn = 0,

the coefficients a1, a2, …, an being integers that are not all zero at the same time. Let us imagine the set (Ω) of all numbers Ω that can be represented by rational functions with integer coefficients from the given numbers v1, v2, …, vν, … ; then, in every interval [α, β], there are an infinity of numbers that are not contained in the set (Ω)."

In fact, we see, with considerations analogous to those that have been employed in § 1, that the numbers of the set (Ω) can be placed in a sequence of the form

Ω1, Ω2, …, Ων, …,

from which the theorem in question follows according to the proposition proved in § 2.

M. B. Miningehade proved, by a reduction to the principles of Galois, a very particular case of the theorem that we have just stated; namely, the case in which the numbers v1, v2, …, vν are finite in number and in which the degree of the rational functions, which are used to form the numbers of the set (Ω), is given in advanced. (See Math. Annalen of Clebsch and Neumann, vol. III, p. 497.)

Berlin, December 23, 1873

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Notes on English translation: Translated "système" as "set." The standard closed interval notation [α, β] has been used rather than Cantor's notation (α . . . . β), which can be confused with the modern open interval notation (α, β). The French "dans l'intervalle" has been translated "in the interior of the interval" since:  Translating it as "in the interval" would make "up ne trouve pas dans l'intervalle" read: "up is not in the interval [α(p), β(p)]" which can be false. Let u1 = 1/3, u2 = 1/2 and [α, β] = [0, 1], then [α', β'] = [1/3, 1/2], thus u1 is in [α', β'] = [α(1), β(1)]. The original German article uses "im Innern von (α', … β') liegen" [lies in the interior of (α', … β')] which was translated to the French "se trouvent dans l'intervalle (α', … β')". This and other similar examples seem to imply that the French translator was using "dans" in the sense of "in the interior of."  