Weierstrass`s Construction of the Irrational Numbers

Weierstrass’s Construction of the Irrational
Numbers
J. Christopher Tweddle
January 6, 2012
Abstract
We present an overview of the development of the irrational numbers due to Karl Weierstrass. This construction was first presented
during lectures in the 1860s in Berlin. Weierstrass never published his
construction. Several of his students (Kossak, Horwitz, von Dantscher
and Pincherle, to name a few) gave accounts in lecture notes from the
courses. However, these notes were not published under the direction
of Weierstrass.
1
INTRODUCTION
Our goal is to explore the development of the system of real numbers. In
particular, we will focus on the rigorous construction of the real number system that was presented in the latter part of the nineteenth century by Karl
Weierstrass. He never published his construction in its entirety. Lecture
notes containing his construction were published by his students, including
Kossak (1872) [15], Hurwitz (1880) (found in [24]), von Dantscher (1908) [23]
and Pincherle (1880) [19]. These notes differ slightly in the details; moreover,
none of these works present a thorough exposition accessible to the modern
reader. During the latter half of the nineteenth century there was a movement to establish a rigorous, logically sound foundation for mathematics.
One subject that received attention was the development of the theory of
irrational numbers. Since it is best to start at the beginning, we shall turn
to the ancient Greeks.
1
Historical Background
The Pythagoreans first discovered the existence of the irrational numbers.
The classical Greeks reserved the word number for the positive integers
greater than one. What we today call the rational numbers were known to
the Greeks as commensurable ratios. The Greeks were aware of the existence
of magnitudes that are not commensurable; for example, the length of the diagonal of a square is not commensurable with that of the side. Eudoxus gave
a systematic development, done geometrically, of the “real numbers,” which
the Greeks represented as geometric segments, called magnitudes, according
to Wilder [25, p. 142]. Kline [14, Chap. 3] discusses Eudoxus’ development
of a new theory of magnitude and proportion. Under his interpretation,
a magnitude was not a number, but instead it represented an entity, such
as a line segment or an area. Unlike the Greek’s “numbers,” which moved
discretely from one value to the next, the magnitudes (time, for example)
varied continuously. In continuing his treatment of magnitudes, Eudoxus developed the notions of a ratio of magnitudes and of a proportion (an equality) of ratios, which allowed for the treatments of both commensurable and
incommensurable ratios. Eudoxus clearly distinguished between magnitudes
and “numbers;” hence his treatment avoids irrational numbers as “numbers.”
(For more on Eudoxus’ treatment, and the presentation in Euclid’s Elements,
see Kline [14, Chaps. 3 & 4].) His formalization, as presented in Book V of
Euclid’s Elements, is not without its short-comings though. Bourbaki [2,
p. 150] comments:
As admirable as the construction of Eudoxus was, and not leaving
anything to be desired from the point of view of coherence and
rigor, it must be admitted that it lacked suppleness, and was
hardly favorable for the development of numerical calculations
and especially of algebraic calculations.
Morris Kline [14, Chap. 41] adds that the logic in this presentation is flawed
since an incommensurable ratio is never defined. In Plato’s dialogue Theaetetus, Theodorus of Cyrene is credited with showing that the squares whose
areas are three through seventeen square units (except those of four, nine
or sixteen) have sides whose lengths are not commensurable with the unit
length. This demonstration is known as the Spiral of Theodorus. According
to Kline [14, Chap. 3], Theaetetus, a student of Theodorus, further devel2
oped the theory of irrational numbers, but only those that arose in geometric
construction.
Even with its deficiencies, the ancient Greek’s interpretation of the real
numbers was used by mathematicians for centuries to follow. In the sixteenth
century, Rafael Bombelli, in his Algebra, offered a theory of numbers based
on ratios of positive integers. However, Bourbaki [2] asserts that he did not
develop this theory beyond the consideration of radicals. Wilder [25] reports
that in 1585, Simon Stevin published La Thiende, giving a detailed discussion
of the decimal representation of “fractions” (quotations marks from Wilder).
In the following centuries, the development of the theories of infinitesimal
calculus and series further highlighted the lack of a rigorous foundation for
the real numbers. Bourbaki [2] notes that the theories of infinitesimal calculus
were not in accord with the Archimedean postulate (that given two distinct
positive real numbers, a and b, there exists an integer n such that na > b),
raising concerns regarding the real numbers as they were understood at the
time.
As a consequence of the development of non-Euclidean geometries at the
turn of the nineteenth century, Euclidean geometry was no longer held to
be an a priori truth (see Kline [14, Chap. 36] for the evolution of nonEuclidean geometries). Kline [14, Chap. 41] cites an 1817 letter to Heinrich
Olbers, and another in 1830 to Friedrich Bessel, in which Karl Friedrich
Gauss asserted that arithmetic, and not geometry, was a priori, and only its
laws were necessary and true. Thus there was motivation to construct the
real numbers on a rigorous (i.e., arithmetic) foundation. Previous definitions
of irrational numbers were given in terms of geometrical considerations, such
as those that might arise when finding the length of a leg of a right triangle.
This procedure was however logically insufficient. Bertrand Russell [22] states
that irrational numbers must be defined independently of geometry. If only a
geometrical definition were possible, no such arithmetic entities would exist.
Thus if we wish to be assured of the existence of irrational numbers, we must
find a more sound basis for their construction.
In the early part of the nineteenth century, it was assumed as “obvious” that the real numbers were continuous (complete, in modern language);
hence, it was assumed that two curves could not cross without intersecting
at a point. This assumption is revealed in Euclid’s construction of an equilateral triangle (found in Book I of Elements). In this construction, given a
line segment, one may construct by compass, two arcs with a radius equal
to the length of the given line segment. According to the construction, these
3
arcs intersect at a point, which then became the third vertex of the triangle.
Also known at the time was Augustin-Louis Cauchy’s criterion for convergence of a sequence: a sequence converges when given any arbitrarily small,
fixed amount, we may find a term in the sequence beyond which any two
terms of the sequence differ by less than the given amount. Ebbinghaus, et.
al. [9], state that the completeness of the real numbers and Cauchy’s criterion
were used in many proofs of the day; including those of Gauss, Joseph-Louis
Lagrange, and Bernhard Bolzano’s proof of the Intermediate Value Property
(for more on Bolzano see Bottazzini [1, sec. 3.3]). According to Ferreriós
[10, p. 118], Bolzano was aware of the need for a rigorous characterization of
the real numbers in order to provide a sound proof of the intermediate value
theorem (presented in 1817). Thus some of the early proofs of analysis were
constructed on an unstable foundation.
William Rowan Hamilton (of quaternion fame) offered the first modern
construction of the irrational numbers in two papers from 1833 and 1835
(published together as “Algebra as the Science of Pure Time” [11]). Hamilton based his construction of numbers on time, considered to be a basic
intuition in Immanuel Kant’s philosophy. However, time does not serve as a
logical foundation for mathematics. Hamilton then began to develop a theory
of separations of the numbers, not unlike Richard Dedekind’s cuts; Hamilton never completed his work on this topic (see Machamer and Turnbull [17]
for analysis of Hamilton’s presentation). Apart from Hamilton, Kline [14,
Chap. 41] indicates that other pre-Weierstrassian constructions of the period attempted to define the irrational numbers as the limit of a sequence of
rational numbers. The problem with this approach is that the limit, if irrational, does not logically exist until the irrational numbers have been defined.
Georg Cantor [4] suggests that this error went unnoticed because it did not
lead to inconsistencies with the “usual” properties of the real numbers, as
understood at the time.
Karl Weierstrass first publicly lectured on his construction as a part of
a course on the general theory of analytic functions, taught in the winter of
1863–1864, in Berlin, that avoided this logical problem. According to Dugac
[7], Weierstrass defined the irrational numbers in terms of certain collections
of rational numbers said to have a finite value (the definition of finite value for
a collection of rational numbers is given below). Although he never published
his presentation, many of his students have offered hints of the construction.
E. Kossak [15] claimed to present Weierstrass’s theory. Additionally, Hurwitz [24] and Pincherle [19] published notes from Weierstrass’s lectures. (For
4
more on the authenticity of these presentation see Dugac [7].) Von Dantscher
attended Weierstrass’s lecture given during the summer 1872; he gives his
interpretation of the presentation in [23], carefully denoting the (few) passages that come directly from Weierstrass. Pringsheim [20] (from the French
translation [21, footnote on p. 149]) emphasizes that theses publications were
not made under the direction of Weierstrass. Dugac’s 1973 article, “Eléments
d’Analyse de Karl Weierstrass” [7], is based on these presentations; Dugac
[6, 8] gives comments comparing Weierstrass’s construction to Dedekind’s
and Mèray’s. P. E. B. Jourdain [13] also comments on these presentations
of Weierstrass’s construction. However, these articles are as much about the
history and philosophy of the development of the irrational numbers as the
construction itself. Moreover, none give an argument that the resulting real
numbers are in fact complete, as we shall do below.
Cantor, one of Weierstrass’s students, realized the importance of the theory of irrational numbers, and published a related construction based on
Cauchy sequences of rational numbers in 1872. Cantor’s first paper [3] on
the topic was expounded upon by Eduard Heine [12], another of Weierstrass’s
students. Independent of the Germans’ work (this is around the time of the
Franco-Prussian War), Charles Méray [18] (published in French in 1869) gave
a development parallel to Cantor’s in terms of limits (either rational or “fictitious”) of Cauchy sequences of rational numbers.
Upon reading the results of Heine [12] and Cantor [3], Dedekind published
his own construction in 1872 (in translation as [5]), based on separations of
the rational numbers that he had formulated in 1858. Although Dedekind’s
theory was logically sound, Cantor [4] criticized it on the basis that these
separations do not come about naturally in analysis. Dedekind uses intuitive ideas from geometry to provide motivation for his development and
thus there are similarities to Euxodus’ treatment of the real numbers. However, Dedekind avoided the cumbersome nature of the Greek’s definitions.
Furthermore, Bottazzini [1, p. 269] states that unlike the Greek treatment of
incommensurable magnitudes, Dedekind’s theory clearly indicates a notion of
the continuum, whose “essence is precisely given by the axiom of continuity.”
Moreover, Dedekind [5, p. 1] states that
Even now such resort to geometric intuition in a first presentation
of the differential calculus, I regard as exceedingly useful, from
the didactic standpoints, and indeed indespensable, if one does
not wish to lose too much time. But that this from of introduction
5
into the differential calculus can make no claim to being scientific,
no one will deny.
Thus, while Dedekind’s cuts have a geometric feel, his theory is soundly
established in an arithmetic manner.
The common trait in all of these definitions of irrational numbers is a welldefined collection of rational numbers. The difference, according to Cantor
[4], lies in the “generative moment,” where the conditions that the collections must satisfy are established, and where these collections are associated
with the numbers they define. According to Bottazzini [1], this difference
stems from their motivations: Dedekind established a rigorous foundation
for differential calculus; Cantor was concerned with developing a uniqueness
theorem for the representation of a function by trigonometric series; and
Weierstrass saw the formulation of the real number system as essential to
the development of his theory of analytic functions.
2
WEIERSTRASS’S CONSTRUCTION
It is the intention here to briefly present the underlying ideas of the construction of Weierstrass. We will establish the necessary definitions and criteria
to give the formulation of the real numbers, without proving that the construction satisfies the ordered field axioms.
Our goal is to establish the real numbers as the collection of equivalence classes of certain collections of rational numbers. Weierstrass began
his construction with a rather informal treatment of the natural numbers
and integers. We shall develop our basic building block, defined in terms of
positive integers, and construct the rational and real numbers on this basis.
Definition. Given any positive integer n, the aliquot part of n is the expression 1/n.
The aliquot parts have the “usual” properties that we associate with the
rational number 1/n. Namely,
( )
1
1
1
1
+ + ··· + = 1 = n
.
n
n
n
n
|
{z
}
n times
For simplicity, we will use the convention of writing k repetitions of the
aliquot part 1/n by k/n, for k < n. We now turn our attention to collections
6
of these aliquot parts, which we shall call aggregates (to borrow from Dugac,
although Weierstrass himself avoided such terms). We have chosen this term
(instead of sets), since we wish to allow the repetition of an aliquot part
within a collection; in programming, collections of these types are called
multisets. To avoid confusion, we will use square brackets when we wish to
explicitly display an aggregate; for example, [1/2, 1/3, 1/4]. While aggregates
may be either finite or infinite, it is reasonable to limit our discussion to those
that are at most countable. Weierstrass presented his construction in the
context of analytic functions, a setting involving countable series. Moreover,
if the aggregate is to have finite value (as defined below; essentially, if the
series is to be bounded), then it may contain at most countably many positive
terms.
In order to compare two aggregates, Weierstrass established the following
transformations (found in Dugac [8]):
1. n elements of the form 1/n may be replaced by 1, similarly k aliquot
parts of the form 1/kl can be replaced by 1/l; and
2. any number can be replaced by its aliquot parts (i.e., 1 could be replaced by p(1/p)).
These transformations allow us to develop an order on the collection of aggregates.
Definition (Order). Let a and b be two aggregates. We define a ≤ b if, and
only if, for all proper subaggregates a′ of transformations of a that contain
finitely many aliquot parts, we can transform a′ to a′′ so that every aliquot
part in a′′ occurs in b (which may likewise need to be transformed), including
any multiplicities.
The phrase “proper subaggregate” means a collection of elements from
the aggregate, but not the entire aggregate. We note that in the previous
definition, it does not matter whether the aggregate a contains a finite or
infinite number of aliquot parts, since our comparison is based on a finite
subaggregate.
Proposition. The relation “≤” is transitive.
7
Proof. Suppose that a ≤ b and b ≤ c, for aggregates a, b and c. Let a′ be
any finite, proper subaggregate of a. Since a ≤ b, we can transform a′ to
a′′ , so that each aliquot part occurring in a′′ also occurs in b, with proper
multiplicities. Since b ≤ c, we can transform any finite, proper subaggregate
of a′′ to a′′′ , whose aliquot parts are contained in c. Thus a ≤ c and the
relation is transitive.
Next we define equality and observe that since “≤” is a reflexive, transitive relation, the definition of equality gives an equivalence relation of the
collection of all aggregates.
Definition (Equality). Let a and b be two aggregates. Then a = b if, and
only if, a ≤ b and b ≤ a.
We will use the equivalence relation “=” to partition the collection of
aggregates into equivalence classes later.
Example. As an example, we will illustrate how one would go about showing
1/3 = .333 . . . We will use “≡” to relate aggregates to the letter we have
chosen to represent it. Let
[ ]
[
]
1
3 3
3
a≡
and
b≡
,
,
,...
3
10 100 1000
be aggregates representing 1/3 and .333 . . ., respectively.
Let a′ be a proper, finite subaggregate of any transformation of a. Let
[a a a
]
1
2
3
′′
a ≡
,
,
,...
10 102 103
be an infinite decimal representation of a′ , as described in the lemma below,
where ai ∈ {0, 1, 2 . . . , 9}. We consider three possibilities:
1. If a1 > 3, then
a1
1 3a1 − 10
1
= +
> .
10
3
30
3
′′
′′
Thus a ̸≤ [1/3], a contradiction, as a is a transform of a proper
subaggregate of a. Thus a1 ≤ 3.
2. Suppose that a1 < 3. Then
∞
∞
∑
∑
9
1
an
≤
=
.
n
10
10
10
n=2
n=2
8
Thus
[
] [ ]
a1 1
3
a ≤
,
≤
≤ b.
10 10
10
′′
Therefore a ≤ b as desired.
3. If a1 = 3, then we consider a2 . Using an argument similar to (1),
a2 ≤ 3. If the inequality is strict, then a modification of (2) gives,
a ≤ b as desired. If a2 = 3, we then consider a3 .
Continuing this argument, we either eventually reach some n for which ai = 3
for i < n and an < 3, or there is no such n and ai = 3 for all positive integers
i. In the first case, we have shown that a ≤ b as desired. In the latter (which,
in fact cannot be, since a′ was a proper subaggregate), a′′ is identical to b
and thus a = b. In any event, we have a ≤ b.
To show that b ≤ a, let b′ be any proper, finite subaggregate of b. Since
′
b is has finitely many parts, it contains a smallest member 3/10n (that is to
say, it is a finite decimal expansion). Transform a to
[
]
3 3
3
1
,
,..., n,
;
10 102
10 3 · 10n
we note that
1
1 ∑ 3
=
−
.
3 · 10n
3 i=1 10i
n
Thus every part of b′ occurs in (a transformation) of a. Therefore b ≤ a.
Hence a = b as we set out to show.
We need one more definition before giving a formulation of the real numbers.
Definition. An aggregate a has finite value if, and only if, there exists an
aggregate b, containing finitely many elements, such that a ≤ b.
Note that if an an aggregate a in an equivalence class α has finite value,
then all aggregates in α will have finite value as well, since any aggregate in
α can be transformed into a. We now have the necessary language to present
Weierstrass’s definitions.
Definition. A positive real number is an equivalence class of aggregates
which have finite value.
9
Definition. A real number is rational if, and only if, there is a finite aggregate in its equivalence class.
Although Weierstrass did not use this language, it may be useful to think
of a positive real number as a (possibly infinite) sum of aliquot parts. Hence
if an equivalence class is a rational number, we consider it to be the sum of
a finite aggregate in that class.
Note that the aggregate a ≡ [1, 1/2!, 1/3!, . . .] has finite value, since a ≤
[3]. However a represents∑
the irrational number e, as you will recall from
∞ xn
Taylor’s expansion ex =
n=0 n! . Thus, there is no finite aggregate (an
aggregate consisting of finitely many aliquot parts) that is equivalent to a.
We will not rigorously verify the ordered field axioms are satisfied for
Weierstrass’s theory of real numbers. However, we will give definitions of the
principle operations.
Definition. The sum a+b is the aggregate of all aliquot parts of a, together
with all the parts of b, each having multiplicity equal to the number of
occurrences in a, together with the number in b.
The sum of two aggregates is the multiset union of those aggregates. For
example, if we want to add the aggregate [1/3, 1/4] to the aggregate [1/4],
we would get the following
[1/3, 1/4] + [1/4] = [1/3, 1/4, 1/4] = [4/12, 3/12, 3/12] = [10/12].
Definition. The product ab, is the aggregate consisting of all possible pairwise products ai bj , where ai ∈ a and bj ∈ b.
That is to say, the product of two aggregates behaves as a (possibly
infinite) distribution property of multiplication over addition. When viewed
in this regard, the definitions of sum and product given by Weierstrass are
consistent with our modern interpretation of (absolutely) convergent series.
The following theorem is based on a result from Lightstone [16]; however,
in that presentation the author does not include the “discard” step that is
necessary to ensure the constructed upper bound is indeed least. First, we
need a lemma.
Lemma. Each positive real number contains in its equivalence class a “standard infinite decimal” representative. That is to say, each real number may
be written as an aggregate
]
[ a a a
2
3
1
a0 , , 2 , 3 , . . . ,
10 10 10
10
where a0 is an integer, and an ∈ {0, 1, . . . , 9} for all positive integers n and,
for every n there is a positive integer m > n with am ̸= 0.
For example the real number 1/5 would be expressed as an “infinite decimal,” [1/10, 9/100, 9/1000, . . .]. Furthermore, each representative may be
constructed so that for each n, a term with denominator 10n occurs precisely
one time in the aggregate. We will use these “infinite decimal” representatives to construct the least upper bound.
Proof. Let b be an aggregate positive rational numbers,
]
[
b1 b2 b3
b ≡ b0 , , , , . . . ,
n1 n2 n3
where ni is a natural number and bi is a nonnegative integer. We proceed by
induction on the number of terms in an aggregate.
To begin, consider the rational number
p
b1
= b0 + .
q
n1
Let a0 be the integer part of pq and denote the fractional part of pq by pq11 .
1
We may now define a1 to be the integer part of 10p
. Let pq22 denote the
q1
1
2
fractional part of 10p
. We continue by letting a2 be the integer part of 10p
q1
q2
p3
pi
and q3 the fractional part. Note that qi will be a proper fraction, so that
ai ∈ {0, 1, 2, . . . , 9} for i = 1, 2, 3 . . . Continuing (we are repeatedly dividing
by 1/10), we have
]
p [ a1 a2
≡ a0 , , 2 , . . . .
q
10 10
If it so happens that this aggregate contains on finitely many terms, that is
p [ a1 a2
ak ]
≡ a0 , , 2 , . . . , k ,
q
10 10
10
we may rewrite it as
[
]
a1 a2
ak − 1
9
9
p
≡ a0 , , 2 , . . . ,
,
,
,... .
q
10 10
10k 10k+1 10k+2
Thus we have transformed b into
]
[
b2 b3
a1 a2
′
b ≡ a0 , , 2 , . . . , , , . . . .
10 10
n2 n3
11
For the inductive step, suppose we have transformed b into
[
]
a1 a2
bk bk+1
′
b ≡ a0 , , 2 , . . . , ,
,... .
10 10
nk nk+1
Proceeding as above, we transform the rational number
bk
nk
to get
[
]
′
′
bk
′ a1 a2
≡ a0 , , 2 , . . . .
nk
10 10
Thus we have
[
]
a1 a2
bk bk+1
b ≡ a0 , , 2 , . . . , ,
,...
10 10
nk nk+1
[
]
′
′
a1 a2
bk+1
′ a1 a2
= a0 , , 2 , . . . , a 0 , , 2 , . . . ,
,...
10 10
10 10
nk+1
]
[
′
′
bk+1
′ a1 + a1 a2 + a2
,
,...,
,... .
= a0 + a0 ,
10
102
nk+1
′
It is possible that ai + a′i > 9. If this is the case, we express
ai + a′i
c1
c2
ci
= c0 +
+ 2 + ... + i,
i
10
10 10
10
where cj ∈ {0, 1, 2, . . . , 9} for j = 0, 1, . . . , i; repeat for each power of 10 in
the aggregate, as necessary. Now we may regroup the corresponding powers
of 10. Should we again have a numerator greater than 9, we may repeat this
process (possibly countably many times). As a result, we have expressed
[
]
a1 a2
bk+1
b = a0 , , 2 , . . . ,
,... ,
10 10
nk+1
completing the inductive step. It follows by induction that any aggregate
may be transformed into a “standard infinite decimal” as desired.
Theorem (Completeness). A non-empty bounded set of positive real numbers
has a least upper bound.
Proof. Let S be a non-empty bounded set of positive real numbers, each
given by its standard infinite decimal representation. Since S is a bounded
set, there is a largest integer a0 in some aggregate. Discard all the aggregates
12
in S whose 0th term is not a0 , and denote the set of remaining aggregates S0 .
Since there are only finitely many possibilities for a1 , we may find a maximal
first term a1 /10 in S0 . We now discard all members of S0 not having a1 /10
as a first term, and call the set of remaining aggregates S1 . Continuing in
such a fashion, we construct the aggregate
[ a a a
]
1
2
3
a ≡ a0 , , 2 , 3 , . . . ,
10 10 10
which we claim is a least upper bound.
Clearly a is an upper bound, for given any b ∈ S, with b ̸= a, written
[
]
b1 b2 b3
b ≡ b0 , , 2 , 3 , . . . ,
10 10 10
we may find a least integer i such that bi < ai . It follows that b < a. That
is to say that either b0 < a0 , in which case b < a, or b0 = a0 and we then
compare the terms b1 and a1 , etc.
It remains to show that a is the least upper bound. Suppose that it is
not. That is, suppose
[ c c
]
c3
1
2
c ≡ c0 , , 2 , 3 , . . .
10 10 10
is an upper bound for S, and that c < a. Then there is a least integer n,
such that cn < an . Since n is the least such integer, we see that
c0 = a0 , c1 = a1 , . . . , cn−1 = an−1 .
That is, c ∈ Sn−1 . But an was chosen to be the maximal nth term among
the elements of Sn−1 . Hence c is not an upper bound. Therefore, a is the
least upper bound as desired.
3
Conclusion
In modern language, one could consider Weierstrass’s real numbers to be
bounded (possibly infinite) sums of positive rational numbers. This is fitting, as Weierstrass included this construction as part of a series of lectures
on analytic functions. In fact, in the case that an aggregate contains finitely
many terms (thus representing a rational number), the aggregate corresponds
13
to the rational number that is the sum of its parts. Moreover, this is consistent with the manner in which the ancient Egyptians represented rational
numbers as finite sums of unit fractions—a practice continued into the Middle Ages. In this light, the definitions of addition and multiplication of real
numbers behave in the expected way; also the infinite decimal representation
used in the proof of completeness is natural.
Furthermore, it is easy to see the connection between Weierstrass’s characterization and Cauchy sequences of rational numbers, used by Méray, Cantor and Heine. Indeed, there is a correspondence between Cauchy sequences
and the sequence of partial sums of the series. A Weierstrassian real number
gives rise to a convergent, and hence Cauchy, sequence of partial sums. The
converse is a little more subtle. First, we recall that Cantor (as well as Méray
and Heine) defined real numbers as equivalence classes of Cauchy sequences.
It can be shown that each equivalence class contains a strictly increasing
Cauchy sequence {bn }. We can then define an aggregate [a0 , a1 , . . .] of positive rational numbers by letting a0 = b0 , a1 = b1 − b0 , a2 = b2 − b1 , etc. The
relation between Cauchy sequences and convergent sums is apearant enough
that Cantor [4, p. 80] remarked that his presentation “externally bears a
certain resemblance to the Weierstrass definition.”
4
Acknowledgements
I would like to thank Professors Neal L. Carothers and V. Frederick Rickey
for their support during preliminary work on this project. I also wish to
thank Professor James T. Smith for his feedback on earlier drafts of this
work, as well as his insight into the work of Pincherle [19].
References
[1] Umberto Bottazzini. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag, New York,
1986. Translated from the Italian by Warren Van Emend.
[2] Nicolas Bourbaki. Elements of the History of Mathematics. SpringerVerlag, Berlin, 1994. Translated from the 1984 French original by John
Meldrum.
14
[3] Georg Cantor. Über die Ausdenhung eines Satzes aus der Theorie der
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