
Cantor's first set theory article contains
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
's first theorems of transfinite
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, which studies
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only ...
s and their properties. One of these theorems is his "revolutionary discovery" that the
set of all
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s is
uncountably, rather than
countably
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his
diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real
algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the
topological notion of a set being
dense in an interval.
Cantor's article also contains a proof of the existence of
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classe ...
s. Both
constructive and non-constructive proofs have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has led to a misconception that Cantor's arguments are non-constructive. Since the proof that Cantor published either constructs transcendental numbers or does not, an analysis of his article can determine whether or not this proof is constructive. Cantor's correspondence with
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
shows the development of his ideas and reveals that he had a choice between two proofs: a non-constructive proof that uses the uncountability of the real numbers and a constructive proof that does not use uncountability.
Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted — he added it during
proofreading
Proofreading is the reading of a galley proof or an electronic copy of a publication to find and correct reproduction errors of text or art. Proofreading is the final step in the editorial cycle before publication.
Professional
Tradition ...
. They have traced this and other facts about the article to the influence of
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematic ...
and
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have recognized the role played by the uncountability theorem and the concept of countability in the development of set theory,
measure theory, and the
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
.
The article
Cantor's article is short, less than four and a half pages. It begins with a discussion of the real
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s and a statement of his first theorem: The set of real algebraic numbers can be put into
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
with the set of positive integers.
[. English translation: .] Cantor restates this theorem in terms more familiar to mathematicians of his time: The set of real algebraic numbers can be written as an infinite
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
in which each number appears only once.
[.]
Cantor's second theorem works with a
closed interval 'a'', ''b'' which is the set of real numbers ≥ ''a'' and ≤ ''b''. The theorem states: Given any sequence of real numbers ''x''
1, ''x''
2, ''x''
3, ... and any interval
'a'', ''b'' there is a number in
'a'', ''b''that is not contained in the given sequence. Hence, there are infinitely many such numbers.
[. English translation: .]
Cantor observes that combining his two theorems yields a new proof of
Liouville's theorem that every interval
'a'', ''b''contains infinitely many
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classe ...
s.
[
Cantor then remarks that his second theorem is:
This remark contains Cantor's uncountability theorem, which only states that an interval 'a'', ''b''cannot be put into one-to-one correspondence with the set of positive integers. It does not state that this interval is an infinite set of larger ]cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
than the set of positive integers. Cardinality is defined in Cantor's next article, which was published in 1878.
Cantor only states his uncountability theorem. He does not use it in any proofs.[
]
The proofs
First theorem
To prove that the set of real algebraic numbers is countable, define the ''height'' of a polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
of degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
''n'' with integer coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s as: ''n'' − 1 + , ''a''0, + , ''a''1, + ... + , ''a''''n'', , where ''a''0, ''a''1, ..., ''a''''n'' are the coefficients of the polynomial. Order the polynomials by their height, and order the real root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
s of polynomials of the same height by numeric order. Since there are only a finite number of roots of polynomials of a given height, these orderings put the real algebraic numbers into a sequence. Cantor went a step further and produced a sequence in which each real algebraic number appears just once. He did this by only using polynomials that are irreducible over the integers. The following table contains the beginning of Cantor's enumeration.
Second theorem
Only the first part of Cantor's second theorem needs to be proved. It states: Given any sequence of real numbers ''x''1, ''x''2, ''x''3, ... and any interval 'a'', ''b'' there is a number in 'a'', ''b''that is not contained in the given sequence.
To find a number in 'a'', ''b''that is not contained in the given sequence, construct two sequences of real numbers as follows: Find the first two numbers of the given sequence that are in the open interval (''a'', ''b''). Denote the smaller of these two numbers by ''a''1 and the larger by ''b''1. Similarly, find the first two numbers of the given sequence that are in (''a''1, ''b''1). Denote the smaller by ''a''2 and the larger by ''b''2. Continuing this procedure generates a sequence of intervals (''a''1, ''b''1), (''a''2, ''b''2), (''a''3, ''b''3), ... such that each interval in the sequence contains all succeeding intervals—that is, it generates a sequence of nested intervals. This implies that the sequence ''a''1, ''a''2, ''a''3, ... is increasing and the sequence ''b''1, ''b''2, ''b''3, ... is decreasing.
Either the number of intervals generated is finite or infinite. If finite, let (''a''''L'', ''b''''L'') be the last interval. If infinite, take the limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
s ''a''∞ = lim''n'' → ∞ ''a''''n'' and ''b''∞ = lim''n'' → ∞ ''b''''n''. Since ''a''''n'' < ''b''''n'' for all ''n'', either ''a''∞ = ''b''∞ or ''a''∞ < ''b''∞. Thus, there are three cases to consider:
:Case 1: There is a last interval (''a''''L'', ''b''''L''). Since at most one ''x''''n'' can be in this interval, every ''y'' in this interval except ''x''''n'' (if it exists) is not in the given sequence.
:Case 2: ''a''∞ = ''b''∞. Then ''a''∞ is not in the sequence since for all ''n'': ''a''∞ is in the interval (''a''''n'', ''b''''n'') but ''x''''n'' does not belong to (''a''''n'', ''b''''n''). In symbols: ''a''∞ ∈ (''a''''n'', ''b''''n'') but ''x''''n'' ∉
In mathematics, an element (or member) of a Set (mathematics), set is any one of the Equality (mathematics), distinct Mathematical object, objects that belong to that set.
Sets
Writing A = \ means that the elements of the set are the numbers 1, ...
(''a''''n'', ''b''''n'').
:
:Case 3: ''a''∞ < ''b''∞. Then every ''y'' in ∞, ''b''∞">'a''∞, ''b''∞is not contained in the given sequence since for all ''n'': ''y'' belongs to (''a''''n'', ''b''''n'') but ''x''''n'' does not.[. English translation: .]
The proof is complete since, in all cases, at least one real number in 'a'', ''b''has been found that is not contained in the given sequence.
Cantor's proofs are constructive and have been used to write a computer program
A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components.
A computer progra ...
that generates the digits of a transcendental number. This program applies Cantor's construction to a sequence containing all the real algebraic numbers between 0 and 1. The article that discusses this program gives some of its output, which shows how the construction generates a transcendental.
Example of Cantor's construction
An example illustrates how Cantor's construction works. Consider the sequence: , , , , , , , , , ... This sequence is obtained by ordering the rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s in (0, 1) by increasing denominators, ordering those with the same denominator by increasing numerators, and omitting reducible fractions. The table below shows the first five steps of the construction. The table's first column contains the intervals (''a''''n'', ''b''''n''). The second column lists the terms visited during the search for the first two terms in (''a''''n'', ''b''''n''). These two terms are in red.
Since the sequence contains all the rational numbers in (0, 1), the construction generates an irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, which turns out to be − 1.
Cantor's 1879 uncountability proof
Everywhere dense
In 1879, Cantor published a new uncountability proof that modifies his 1874 proof. He first defines the topological notion of a point set ''P'' being "everywhere dense in an interval":
:If ''P'' lies partially or completely in the interval �, β then the remarkable case can happen that ''every'' interval �, δcontained in �, β ''no matter how small,'' contains points of ''P''. In such a case, we will say that ''P'' is ''everywhere dense in the interval'' �, β
In this discussion of Cantor's proof: ''a'', ''b'', ''c'', ''d'' are used instead of α, β, γ, δ. Also, Cantor only uses his interval notation if the first endpoint is less than the second. For this discussion, this means that (''a'', ''b'') implies ''a'' < ''b''.
Since the discussion of Cantor's 1874 proof was simplified by using open intervals rather than closed intervals, the same simplification is used here. This requires an equivalent definition of everywhere dense: A set ''P'' is everywhere dense in the interval 'a'', ''b''if and only if every open subinterval (''c'', ''d'') of 'a'', ''b''contains at least one point of ''P''.
Cantor did not specify how many points of ''P'' an open subinterval (''c'', ''d'') must contain. He did not need to specify this because the assumption that every open subinterval contains at least one point of ''P'' implies that every open subinterval contains infinitely many points of ''P''.
Cantor's 1879 proof
Cantor modified his 1874 proof with a new proof of its second theorem: Given any sequence ''P'' of real numbers ''x''1, ''x''2, ''x''3, ... and any interval 'a'', ''b'' there is a number in 'a'', ''b''that is not contained in ''P''. Cantor's new proof has only two cases. First, it handles the case of ''P'' not being dense in the interval, then it deals with the more difficult case of ''P'' being dense in the interval. This division into cases not only indicates which sequences are more difficult to handle, but it also reveals the important role denseness plays in the proof.[Since Cantor's proof has not been published in English, an English translation is given alongside the original German text, which is from . The translation starts one sentence before the proof because this sentence mentions Cantor's 1874 proof. Cantor states it was printed in Borchardt's Journal. Crelle’s Journal was also called Borchardt’s Journal from 1856-1880 when Carl Wilhelm Borchardt edited the journal (). Square brackets are used to identify this mention of Cantor's earlier proof, to clarify the translation, and to provide page numbers. Also, "" (manifold) has been translated to "set" and Cantor's notation for closed sets (α . . . β) has been translated to �, β Cantor changed his terminology from to (set) in his 1883 article, which introduced sets of ordinal numbers (). Currently in mathematics, a ]manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
is type of topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
.
}
, -
, age 6
Age or AGE may refer to:
Time and its effects
* Age, the amount of time someone or something has been alive or has existed
** East Asian age reckoning, an Asian system of marking age starting at 1
* Ageing or aging, the process of becoming olde ...
br>
definitely occur ''within'' the interval �, β one of these numbers must have the ''least index,'' let it be ωκ1, and another: ωκ2 with the next larger index.
Let the smaller of the two numbers ωκ1, ωκ2 be denoted by α', the larger by β'. (Their equality is impossible because we assumed that our sequence consists of nothing but unequal numbers.)
Then according to the definition:
α < α' < β' < β,
furthermore:
κ1 < κ2;
and all numbers ωμ of our sequence,
for which μ ≤ κ2, do ''not'' lie in the interior of the interval �', β' as is immediately clear from the definition of the numbers κ1, κ2. Similarly, let ωκ3 and ωκ4 be the two numbers of our sequence with smallest indices that fall in the ''interior'' of the interval �', β'and let the smaller of the numbers ωκ3, ωκ4 be denoted by α'', the larger by β''.
Then one has:
α' < α'' < β'' < β',
κ2 < κ3 < κ4;
and one sees that all numbers ωμ of our sequence, for which μ ≤ κ4, do ''not'' fall into the ''interior'' of the interval '', β''">�'', β''
After one has followed this rule to reach an interval , the next interval is produced by selecting the first two (i. e. with lowest indices) numbers of our sequence (ω) (let them be and ωκ2ν) that fall into the ''interior'' of . Let the smaller of these two numbers be denoted by α(ν), the larger by β(ν).
The interval (ν), β(ν)">�(ν), β(ν)then lies in the ''interior'' of all preceding intervals and has the ''specific'' relation with our sequence (ω) that all numbers ωμ, for which μ ≤ κ2ν, ''definitely do not lie in its interior''. Since obviously:
and these numbers, as indices, are ''whole'' numbers, so:
κ2ν ≥ 2ν,
and hence:
ν < κ2ν;
thus, we can certainly say (and this is sufficient for the following):
''That if ν is an arbitrary whole number, the ealquantity ων lies outside the interval (ν) . . . β(ν)">�(ν) . . . β(ν)''
, german: eite 6br>
sicher Zahlen ''innerhalb'' des Intervalls (α . . . β) vorkommen, so muss eine von diesen Zahlen den ''kleinsten Index'' haben, sie sei ωκ1, und eine andere: ωκ2 mit dem nächst grösseren Index behaftet sein.
Die kleinere der beiden Zahlen ωκ1, ωκ2 werde mit α', die grössere mit β' bezeichnet. (Ihre Gleichheit ist ausgeschlossen, weil wir voraussetzten, dass unsere Reihe aus lauter ungleichen Zahlen besteht.)
Es ist alsdann der Definition nach:
α < α' < β' < β,
ferner:
κ1 < κ2;
und ausserdem ist zu bemerken, dass alle Zahlen ωμ unserer Reihe,
für welche μ ≤ κ2, ''nicht'' im Innern des Intervalls (α' . . . β') liegen, wie aus der Bestimmung der Zahlen κ1, κ2 sofort erhellt. Ganz ebenso mögen ωκ3, ωκ4 die beiden mit den kleinsten Indices versehenen Zahlen unserer Reihen ee note 1 belowsein, welche in das ''Innere'' des Intervalls (α' . . . β') fallen und die kleinere der Zahlen ωκ3, ωκ4 werde mit α'', die grössere mit β'' bezeichnet.
Man hat alsdann:
α' < α'' < β'' < β',
κ2 < κ3 < κ4;
und man erkennt, dass alle Zahlen ωμ unserer Reihe, für welche μ ≤ κ4 ''nicht'' in das ''Innere'' des Intervalls (α'' . . . β'') fallen.
Nachdem man unter Befolgung des gleichen Gesetzes zu einem Intervall gelangt ist, ergiebt sich das folgende Intervall dadurch aus demselben, dass man die beiden ersten (d. h. mit niedrigsten Indices versehenen) Zahlen unserer Reihe (ω) aufstellt (sie seien ωκ2ν – 1 und ωκ2ν), welche in das ''Innere'' von fallen; die kleinere dieser beiden Zahlen werde mit α(ν), die grössere mit β(ν) bezeichnet.
Das Intervall (α(ν) . . . β(ν)) liegt alsdann im ''Innern'' aller vorangegangenen Intervalle und hat zu unserer Reihe (ω) die ''eigenthümliche''
Beziehung, dass alle Zahlen ωμ, für welche μ ≤ κ2ν ''sicher nicht in seinem Innern'' liegen. Da offenbar:
und diese Zahlen, als Indices, ''ganze'' Zahlen sind, so ist:
κ2ν ≥ 2ν,
und daher:
ν < κ2ν;
wir können daher, und dies ist für das Folgende ausreichend, gewiss sagen:
''Dass, wenn ν eine beliebige ganze Zahl ist, die Grösse ων ausserhalb des Intervalls (α(ν) . . . β(ν)) liegt.'', label=none, italic=unset
, -
, age 7br>
Since the numbers α', α'', α, . . ., α(ν), . . . are continually increasing by value while simultaneously being enclosed in the interval �, β they have, by a well-known fundamental theorem of the theory of magnitudes ee note 2 below a limit that we denote by A, so that:
The same applies to the numbers β', β'', β, . . ., β(ν), . . ., which are continually decreasing and likewise lying in the interval �, β We call their limit B, so that:
Obviously, one has:
But it is easy to see that the case A < B can ''not'' occur here since otherwise every number ων of our sequence would lie ''outside'' of the interval , Bby lying outside the interval (ν), β(ν)">�(ν), β(ν) So our sequence (ω) would ''not'' be ''everywhere dense'' in the interval �, β contrary
to the assumption.
Thus, there only remains the case A = B and now it is demonstrated that the number:
η = A = B
does ''not'' occur in our sequence (ω).
If it were a member of our sequence, such as the νth, then one would have: η = ων.
But the latter equation is not possible for any value of ν because η is in the ''interior'' of the interval (ν), β(ν)">�(ν), β(ν) but ων lies ''outside'' of it.
, german: eite 7br>
Da die Zahlen α', α'', α, . . ., α(ν), . . . ihrer Grösse nach fortwährend wachsen, dabei jedoch im Intervalle (α . . . β) eingeschlossen sind, so haben sie, nach einem bekannten Fundamentalsatze der Grössenlehre, eine Grenze, die wir mit A bezeichnen, so dass:
Ein Gleiches gilt für die Zahlen β', β'', β, . . ., β(ν), . . . welche fortwährend abnehmen und dabei ebenfalls im Intervalle (α . . . β) liegen; wir nennen ihre Grenze B, so dass:
Man hat offenbar:
Es ist aber leicht zu sehen, dass der Fall A < B hier ''nicht'' vorkommen kann; da sonst jede Zahl ων, unserer Reihe ''ausserhalb'' des Intervalles (A . . . B) liegen würde, indem ων, ausserhalb des Intervalls (α(ν) . . . β(ν)) gelegen ist; unsere Reihe (ω) wäre im Intervall (α . . . β) ''nicht überalldicht,'' gegen die Voraussetzung.
Es bleibt daher nur der Fall A = B übrig und es zeigt sich nun, dass die Zahl:
in unserer Reihe (ω) ''nicht'' vorkommt.
Denn, würde sie ein Glied unserer Reihe sein, etwa das νte, so hätte man: η = ων.
Die letztere Gleichung ist aber für keinen Werth von v möglich, weil η im ''Innern'' des Intervalls (ν), β(ν)">�(ν), β(ν) ων aber ''ausserhalb'' desselben liegt., label=none, italic=unset
, -
, colspan="2" , Note 1: This is the only occurrence of "" ("our sequences") in the proof. There is only one sequence involved in Cantor's proof and everywhere else "" ("sequence") is used, so it is most likely a typographical error and should be "" ("our sequence"), which is how it has been translated.
Note 2: , which has been translated as "the theory of magnitudes", is a term used by 19th century German mathematicians that refers to the theory of discrete and continuous magnitudes. (.)
In the first case, ''P'' is not dense in 'a'', ''b'' By definition, ''P'' is dense in 'a'', ''b''if and only if for all subintervals (''c'', ''d'') of 'a'', ''b'' there is an ''x'' ∈ ''P'' such that . Taking the negation of each side of the "if and only if" produces: ''P'' is not dense in 'a'', ''b''if and only if there exists a subinterval (''c'', ''d'') of 'a'', ''b''such that for all ''x'' ∈ ''P'': . Therefore, every number in (''c'', ''d'') is not contained in the sequence ''P''.[ This case handles case 1 and case 3 of Cantor's 1874 proof.
In the second case, which handles case 2 of Cantor's 1874 proof, ''P'' is dense in 'a'', ''b'' The denseness of sequence ''P'' is used to recursively define a sequence of nested intervals that excludes all the numbers in ''P'' and whose ]intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
contains a single real number in 'a'', ''b'' The sequence of intervals starts with (''a'', ''b''). Given an interval in the sequence, the next interval is obtained by finding the two numbers with the least indices that belong to ''P'' and to the current interval. These two numbers are the endpoints of the next open interval. Since an open interval excludes its endpoints, every nested interval eliminates two numbers from the front of sequence ''P'', which implies that the intersection of the nested intervals excludes all the numbers in ''P''.[ Details of this proof and a proof that this intersection contains a single real number in 'a'', ''b''are given below.
]
The development of Cantor's ideas
The development leading to Cantor's 1874 article appears in the correspondence between Cantor and Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
. On November 29, 1873, Cantor asked Dedekind whether the collection of positive integers and the collection of positive real numbers "can be corresponded so that each individual of one collection corresponds to one and only one individual of the other?" Cantor added that collections having such a correspondence include the collection of positive rational numbers, and collections of the form (''a''''n''1, ''n''2, . . . , ''n''''ν'') where ''n''1, ''n''2, . . . , ''n''''ν'', and ''ν'' are positive integers.
Dedekind replied that he was unable to answer Cantor's question, and said that it "did not deserve too much effort because it has no particular practical interest." Dedekind also sent Cantor a proof that the set of algebraic numbers is countable.[. English translation: .]
On December 2, Cantor responded that his question does have interest: "It would be nice if it could be answered; for example, provided that it could be answered ''no'', one would have a new proof of Liouville's theorem that there are transcendental numbers."
On December 7, Cantor sent Dedekind a proof by contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...
that the set of real numbers is uncountable. Cantor starts by assuming that the real numbers in
''Translation: Theorem 68. There are transcendental numbers.''
If there were no transcendental numbers, then all numbers would be algebraic. Hence, the continuum would be identical to the set of all algebraic numbers. However, this is impossible because the set of all algebraic numbers is countable, but the continuum is not.
Cantor chose to publish the constructive proof, which not only produces a transcendental number but is also shorter and avoids two proofs by contradiction. The non-constructive proof from Cantor's correspondence is simpler than the one above because it works with all the real numbers rather than the interval 'a'', ''b'' This eliminates the subsequence step and all occurrences of 'a'', ''b''in the second proof by contradiction.[
]
A misconception about Cantor's work
Akihiro Kanamori, who specializes in set theory, stated that "Accounts of Cantor's work have mostly reversed the order for deducing the existence of transcendental numbers, establishing first the uncountability of the reals and only then drawing the existence conclusion from the countability of the algebraic numbers. In textbooks the inversion may be inevitable, but this has promoted the misconception that Cantor's arguments are non-constructive."[.]
Cantor's published proof and the reverse-order proof both use the theorem: Given a sequence of reals, a real can be found that is not in the sequence. By applying this theorem to the sequence of real algebraic numbers, Cantor produced a transcendental number. He then proved that the reals are uncountable: Assume that there is a sequence containing all the reals. Applying the theorem to this sequence produces a real not in the sequence, contradicting the assumption that the sequence contains all the reals. Hence, the reals are uncountable.[ The reverse-order proof starts by first proving the reals are uncountable. It then proves that transcendental numbers exist: If there were no transcendental numbers, all the reals would be algebraic and hence countable, which contradicts what was just proved. This contradiction proves that transcendental numbers exist without constructing any.][
]
The correspondence containing Cantor's non-constructive reasoning was published in 1937. By then, other mathematicians had rediscovered his non-constructive, reverse-order proof. As early as 1921, this proof was called "Cantor's proof" and criticized for not producing any transcendental numbers. In that year, Oskar Perron gave the reverse-order proof and then stated: "… Cantor's proof for the existence of transcendental numbers has, along with its simplicity and elegance, the great disadvantage that it is only an existence proof; it does not enable us to actually specify even a single transcendental number."
As early as 1930, some mathematicians have attempted to correct this misconception of Cantor's work. In that year, the set theorist Abraham Fraenkel stated that Cantor's method is "… a method that incidentally, contrary to a widespread interpretation, is fundamentally constructive and not merely existential." In 1972, Irving Kaplansky wrote: "It is often said that Cantor's proof is not 'constructive,' and so does not yield a tangible transcendental number. This remark is not justified. If we set up a definite listing of all algebraic numbers … and then apply the diagonal procedure …, we get a perfectly definite transcendental number (it could be computed to any number of decimal places)." Cantor's proof is not only constructive, it is also simpler than Perron's proof, which requires the detour of first proving that the set of all reals is uncountable.
Cantor's diagonal argument has often replaced his 1874 construction in expositions of his proof. The diagonal argument is constructive and produces a more efficient computer program than his 1874 construction. Using it, a computer program has been written that computes the digits of a transcendental number in polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
. The program that uses Cantor's 1874 construction requires at least sub-exponential time.
The presentation of the non-constructive proof without mentioning Cantor's constructive proof appears in some books that were quite successful as measured by the length of time new editions or reprints appeared—for example: Oskar Perron's Irrationalzahlen (1921; 1960, 4th edition), Eric Temple Bell's '' Men of Mathematics'' (1937; still being reprinted), Godfrey Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and E. M. Wright's ''An Introduction to the Theory of Numbers'' (1938; 2008 6th edition), Garrett Birkhoff and Saunders Mac Lane's ''A Survey of Modern Algebra'' (1941; 1997 5th edition), and Michael Spivak's ''Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
'' (1967; 2008 4th edition). Since 2014, at least two books have appeared stating that Cantor's proof is constructive, and at least four have appeared stating that his proof does not construct any (or a single) transcendental.
Asserting that Cantor gave a non-constructive argument without mentioning the constructive proof he published can lead to erroneous statements about the history of mathematics. In ''A Survey of Modern Algebra,'' Birkhoff and Mac Lane state: "Cantor's argument for this result ot every real number is algebraicwas at first rejected by many mathematicians, since it did not exhibit any specific transcendental number." The proof that Cantor published produces transcendental numbers, and there appears to be no evidence that his argument was rejected. Even Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
, who had strict views on what is acceptable in mathematics and who could have delayed publication of Cantor's article, did not delay it.[ In fact, applying Cantor's construction to the sequence of real algebraic numbers produces a limiting process that Kronecker accepted—namely, it determines a number to any required degree of accuracy.
]
The influence of Weierstrass and Kronecker on Cantor's article
Historians of mathematics have discovered the following facts about Cantor's article "On a Property of the Collection of All Real Algebraic Numbers":
* Cantor's uncountability theorem was left out of the article he submitted. He added it during proofreading
Proofreading is the reading of a galley proof or an electronic copy of a publication to find and correct reproduction errors of text or art. Proofreading is the final step in the editorial cycle before publication.
Professional
Tradition ...
.[.]
* The article's title refers to the set of real algebraic numbers. The main topic in Cantor's correspondence was the set of real numbers.
* The proof of Cantor's second theorem came from Dedekind. However, it omits Dedekind's explanation of why the limits ''a''∞ and ''b''∞ exist.
* Cantor restricted his first theorem to the set of real algebraic numbers. The proof he was using demonstrates the countability of the set of all algebraic numbers.[
To explain these facts, historians have pointed to the influence of Cantor's former professors, ]Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematic ...
and Leopold Kronecker. Cantor discussed his results with Weierstrass on December 23, 1873.[. English translation: .] Weierstrass was first amazed by the concept of countability, but then found the countability of the set of real algebraic numbers useful. Cantor did not want to publish yet, but Weierstrass felt that he must publish at least his results concerning the algebraic numbers.[
From his correspondence, it appears that Cantor only discussed his article with Weierstrass. However, Cantor told Dedekind: "The restriction which I have imposed on the published version of my investigations is caused in part by local circumstances …"][ Cantor biographer Joseph Dauben believes that "local circumstances" refers to Kronecker who, as a member of the editorial board of '']Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Au ...
'', had delayed publication of an 1870 article by Eduard Heine
Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician.
Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Le ...
, one of Cantor's colleagues. Cantor would submit his article to ''Crelle's Journal''.
Weierstrass advised Cantor to leave his uncountability theorem out of the article he submitted, but Weierstrass also told Cantor that he could add it as a marginal note during proofreading, which he did.[ It appears in a
remark at the end of the article's introduction. The opinions of Kronecker and Weierstrass both played a role here. Kronecker did not accept infinite sets, and it seems that Weierstrass did not accept that two infinite sets could be so different, with one being countable and the other not. Weierstrass changed his opinion later. Without the uncountability theorem, the article needed a title that did not refer to this theorem. Cantor chose "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"), which refers to the countability of the set of real algebraic numbers, the result that Weierstrass found useful.
Kronecker's influence appears in the proof of Cantor's second theorem. Cantor used Dedekind's version of the proof except he left out why the limits ''a''∞ = lim''n'' → ∞ ''an'' and
''b''∞ = lim''n'' → ∞ ''bn'' exist. Dedekind had used his "principle of continuity" to prove they exist. This principle (which is equivalent to the least upper bound property of the real numbers) comes from Dedekind's construction of the real numbers, a construction Kronecker did not accept.
Cantor restricted his first theorem to the set of real algebraic numbers even though Dedekind had sent him a proof that handled all algebraic numbers.][ Cantor did this for expository reasons and because of "local circumstances." This restriction simplifies the article because the second theorem works with real sequences. Hence, the construction in the second theorem can be applied directly to the enumeration of the real algebraic numbers to produce "an effective procedure for the calculation of transcendental numbers." This procedure would be acceptable to Weierstrass.
]
Dedekind's contributions to Cantor's article
Since 1856, Dedekind had developed theories involving infinitely many infinite sets—for example: ideals, which he used in algebraic number theory, and Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the ...
s, which he used to construct the real numbers. This work enabled him to understand and contribute to Cantor's work.
Dedekind's first contribution concerns the theorem that the set of real algebraic numbers is countable. Cantor is usually given credit for this theorem, but the mathematical historian José Ferreirós calls it "Dedekind's theorem." Their correspondence reveals what each mathematician contributed to the theorem.
In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (''a''''n''1, ''n''2, ..., ''n''''ν'') where ''n''1, ''n''2, ..., ''n''''ν'', and ''ν'' are positive integers. Cantor's second result uses an indexed family of numbers: a set of the form (''a''''n''1, ''n''2, ..., ''n''''ν'') is the range of a function from the ''ν'' indices to the set of real numbers. His second result implies his first: let ''ν'' = 2 and ''a''''n''1, ''n''2 = . The function can be quite general—for example, ''a''''n''1, ''n''2, ''n''3, ''n''4, ''n''5 = +
Dedekind replied with a proof of the theorem that the set of all algebraic numbers is countable.[ In his reply to Dedekind, Cantor did not claim to have proved Dedekind's result. He did indicate how he proved his theorem about indexed families of numbers: "Your proof that (''n'') ]he set of positive integers
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...
can be correlated one-to-one with the field of all algebraic numbers is approximately the same as the way I prove my contention in the last letter. I take ''n''12 + ''n''22 + ··· + ''n''''ν''2 = \mathfrak and order the elements accordingly." However, Cantor's ordering is weaker than Dedekind's and cannot be extended to n-tuples of integers that include zeros.
Dedekind's second contribution is his proof of Cantor's second theorem. Dedekind sent this proof in reply to Cantor's letter that contained the uncountability theorem, which Cantor proved using infinitely many sequences. Cantor next wrote that he had found a simpler proof that did not use infinitely many sequences. So Cantor had a choice of proofs and chose to publish Dedekind's.
Cantor thanked Dedekind privately for his help: "… your comments (which I value highly) and your manner of putting some of the points were of great assistance to me."[ However, he did not mention Dedekind's help in his article. In previous articles, he had acknowledged help received from Kronecker, Weierstrass, Heine, and Hermann Schwarz. Cantor's failure to mention Dedekind's contributions damaged his relationship with Dedekind. Dedekind stopped replying to his letters and did not resume the correspondence until October 1876.
]
The legacy of Cantor's article
Cantor's article introduced the uncountability theorem and the concept of countability. Both would lead to significant developments in mathematics. The uncountability theorem demonstrated that one-to-one correspondences can be used to analyze infinite sets. In 1878, Cantor used them to define and compare cardinalities. He also constructed one-to-one correspondences to prove that the ''n''-dimensional spaces R''n'' (where R is the set of real numbers) and the set of irrational numbers have the same cardinality as R.
In 1883, Cantor extended the positive integers with his infinite ordinals. This extension was necessary for his work on the Cantor–Bendixson theorem. Cantor discovered other uses for the ordinals—for example, he used sets of ordinals to produce an infinity of sets having different infinite cardinalities. His work on infinite sets together with Dedekind's set-theoretical work created set theory.
The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets. In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions. Building on the work of Borel and Baire, Henri Lebesgue created his theories of measure and integration
Integration may refer to:
Biology
* Multisensory integration
* Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technolo ...
, which were published from 1899 to 1901.
Countable models are used in set theory. In 1922, Thoralf Skolem proved that if conventional axioms of set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
are consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consisten ...
, then they have a countable model. Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox, and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the model. Thus the model considers its set of real numbers to be uncountable, or more precisely, the first-order sentence that says the set of real numbers is uncountable is true within the model. In 1963, Paul Cohen used countable models to prove his independence
Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the s ...
theorems.[.]
See also
* Cantor's theorem
Notes
Note on Cantor's 1879 proof
References
Bibliography
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* .
* . (Reprinted by Dover Publications, 2002.)
* .
* .
* .
* .
* .
* .
* .
{{Mathematical logic
Georg Cantor
History of mathematics
Real analysis
Set theory