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Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountably, rather than countably, infinite. This proof differs from the more familiar proof that uses his diagonal argument. Georg Cantor's first proof was published in 1874, in an article that also contains a proof that the set of real algebraic numbers is countable, and a proof of the existence of transcendental numbers.

As early as 1930, some mathematicians have disagreed on whether Cantor's proof of the existence of transcendental numbers is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. A careful study of Cantor's article and its proofs will determine the nature of his proof. Cantor's correspondence shows the development of his ideas and reveals that he had a choice between two proofs: one uses the uncountability of the real numbers; the other does not. These proofs play an important role in the disagreement about his proof.

The title of Cantor's article is "On a Property of the Collection of All Real Algebraic Numbers." Historians of mathematics have studied the reasons why Cantor's article emphasizes the countability of the real algebraic numbers rather than the uncountability of the real numbers. They have discovered interesting facts about the article—for example, Cantor left out his uncountability theorem in the article he submitted, then added it during proofreading. They have also studied Richard Dedekind's contributions to the article and the article's legacy.

The article
Cantor's article is short, just $4 1⁄3$ pages. It begins with a discussion of the real algebraic numbers, and a statement of his first theorem: The collection of real algebraic numbers can be put into one-to-one correspondence with the collection of positive integers. Cantor restates this theorem in terms more familiar to mathematicians of his time: The collection of real algebraic numbers can be written as an infinite sequence 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. This theorem states: Given any sequence of real numbers x1, x2, x3, … and any interval [a, b], one can determine numbers in [a, b] that are not contained in the given sequence.

Cantor observes that combining his two theorems yields a new proof of the theorem: Every interval [a, b] contains infinitely many transcendental numbers. This theorem was first proved by Joseph Liouville.

He then remarks that his second theorem is:
 * the reason why collections of real numbers forming a so-called continuum (such as, all real numbers which are ≥ 0 and ≤ 1) cannot correspond one-to-one with the collection (ν) [the collection of all positive integers]; thus I have found the clear difference between a so-called continuum and a collection like the totality of real algebraic numbers.

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 say that this interval is an infinite set of larger cardinality than the set of positive integers. Cardinality and how to compare the cardinality of sets will appear in his next article, which was published in 1878.

Cantor does not explicitly prove his uncountability theorem, which follows easily from his second theorem. To prove it, we use proof by contradiction. Assume that the interval [a, b] can be put into one-to-one correspondence with the set of positive integers, or equivalently: The real numbers in [a, b] can be written as a sequence in which each real number appears only once. Applying Cantor's second theorem to this sequence and [a, b] produces a real number in [a, b] that does not belong to the sequence. This contradicts the original assumption, and proves the uncountability theorem.

Cantor's second theorem is constructive and separates the constructive content of his work from the proof by contradiction needed to establish uncountability. Cantor's uncountability theorem is just stated; it is not used in any proofs.

The proofs
To prove that the set of real algebraic numbers is countable, Cantor defines the height of a polynomial of degree n with integer coefficients as: n – 1 + |a0| + |a1| + ··· + |an|, where a0, a1, …, an are the coefficients of the polynomial. Then he orders the polynomials by their height, and orders the real roots 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.Using this ordering and placing only the first occurrence of a real algebraic number in the sequence produces a sequence without duplicates. Cantor obtains the same sequence by using irreducible polynomials: Next Cantor proves his second theorem: Given any sequence of real numbers x1, x2, x3, … and any interval [a, b], one can determine a number in [a, b] that is not contained in the given sequence. We simplify Cantor's proof by using open intervals. The open interval (a, b) is the set of real numbers > a and < b.

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 (a, b). Designate the smaller of these two numbers by a1, and the larger by b1. Similarly, find the first two numbers of the given sequence that are in (a1, b1). Designate the smaller by a2 and the larger by b2. Continuing this procedure generates a sequence of intervals (a1, b1), (a2, b2), … such that each interval in the sequence contains all succeeding intervals—that is, it generates a sequence of nested intervals. This implies the sequence a1, a2, a3, … is increasing, the sequence b1, b2, b3, … is decreasing, and ai < bj for all i and j.

Either the number of intervals generated is finite or infinite. If finite, let (aN, bN ) be the last interval. If infinite, let a∞ = limn → ∞ an and b∞ = limn → ∞ bn. Since ai < bj for all i and j, either a∞ = b∞ or a∞ < b∞. Thus, there are three cases to consider:







indent=1
 * Case 1: There is a last interval (aN, bN). Since at most one xn can be in this interval, every y in this interval except xn (if it exists) is not contained in the given sequence.
 * Case 2: a∞ = b∞. a∞ is not contained in the given sequence since for all n, a∞ belongs to (an, bn) but xn does not. Cantor states without proof that xn does not belong to (an, bn); this will be proved below.
 * Case 3: a∞ < b∞. Every y in [a∞, b∞] is not contained in the given sequence since for all n, y belongs to (an, bn) but xn does not.

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 observes that a∞ = b∞ when this construction is applied to the sequence of real algebraic numbers, which is the sequence he uses to construct transcendental numbers. More generally, a∞ = b∞ for any sequence that is dense in [a, b], which means that every subinterval (y, z) of [a, b] contains a term of the sequence.

We prove the contrapositive: if a∞ ≠ b∞, then the sequence is not dense in [a, b]. If a∞ ≠ b∞, then case 2 is false, so there are only two cases to consider:

indent=1
 * Case 1: There is a last interval (aN, bN), so at most one xn can be in this interval. If there is such an xn, then the interval (aN, xn) contains no terms of the sequence. Otherwise, the interval (aN, bN) contains no terms of the sequence.
 * Case 3: a∞ < b∞. The interval (a∞, b∞) contains no terms of the sequence.

The proof is complete since, in both cases, an open subinterval of [a, b] has been found that contains no terms of the sequence, which implies that the sequence is not dense in [a, b]. Therefore, if a sequence is dense in [a, b], then a∞ = b∞. The converse is false: applying Cantor's construction to the sequence −$\overline{2}$, $\overline{5}$, −$\overline{5}$, $\overline{2}$, … and the interval [−1, 1] produces an = −$1⁄2$ and bn = $1⁄2$, so a∞ = b∞ = 0. However, this sequence is not dense in [−1, 1] (or in any other interval).

Cantor's proofs are constructive. In fact, they have been used to write a computer program 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.

To prove Cantor's statement that xn does not belong to (an, bn), we analyze how his construction excludes the terms of the sequence: $1⁄3$, $1⁄3$, $1⁄n + 1$, $1⁄n + 1$, $1⁄2$, $1⁄3$, $2⁄3$, $1⁄4$, $3⁄4$, …. This sequence is obtained by ordering the rational numbers in (0, 1) by increasing denominators, and ordering those with the same denominator by increasing numerators. The table below shows the first five steps of the construction. Since this sequence is dense in [0, 1], a non-terminating sequence of intervals is generated.

Cantor's construction starts by finding the first two terms in the sequence belonging to (0, 1)—namely, $1⁄5$ and $2⁄5$, which form the interval ($3⁄5$, $4⁄5$). Next it finds the first two terms belonging to this interval—namely, $1⁄2$ and $1⁄3$. The interval ($1⁄3$, $1⁄2$) excludes the leading terms of the sequence up to $2⁄5$ with two exceptions: $3⁄7$ and $1⁄3$. These two terms are the first terms excluded by the next interval ($1⁄2$, $3⁄7$). Stopping the construction at this point, which is marked by a semicolon in the table, shows that the interval ($2⁄5$, $3⁄7$) excludes at least the leading terms up to $2⁄5$. In general, a non-terminating construction builds intervals (an, bn) that exclude some leading terms of the sequence, and each interval excludes at least two more leading terms than the preceding interval. This leads to the following lemma: For a non-terminating construction, there is a function g(n) such that for all n ≥ 1, g(n) ≥ 2n and x1, …, xg(n) ∉ (an, bn). Since g(n) ≥ 2n > n, this lemma implies that xn does not belong to (an, bn).

$3⁄7$

Since the sequence contains all the rational numbers in (0, 1), the construction generates an irrational number, which turns out to be √2 &minus; 1. $2⁄5$

The development of Cantor's ideas
The development leading to Cantor's article appears in the correspondence between Cantor and his fellow mathematician Richard Dedekind. 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 of the other?" Cantor added that collections having such a correspondence include the collection of positive rational numbers, and collections of the form (an 1, n2, . . ., nν ) where n1, n2,. . ., nν, 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.

On December 2, Cantor pointed out 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 that the set of real numbers is uncountable. Cantor starts by assuming the real numbers can be written as a sequence. Then he applies a construction to this sequence to produce a real not in the sequence, thus contradicting his original assumption. The letters of December 2 and 7 lead to a non-constructive proof of the existence of transcendental numbers.

On December 9, Cantor announced the theorem that allows him to construct transcendental numbers as well as prove the uncountability of the set of real numbers:
 * I show directly that if I start with a sequence
 * (I) ω1, ω2, …, ωn, …
 * I can determine, in every given interval [α, β], a number η that is not included in (I).

This is the second theorem in Cantor's article. It comes from realizing that his construction can be applied to any sequence, not just to sequences that supposedly enumerate the real numbers. So Cantor had a choice between two proofs that demonstrate the existence of transcendental numbers: one proof is constructive and the other is not. We now compare the proofs assuming that we have a sequence consisting of all the real algebraic numbers.

The constructive proof applies Cantor's construction to this sequence and the interval [a, b] to produce a transcendental number in this interval.

The non-constructive proof uses two proofs-by-contradiction:
 * 1) Assume that the real numbers in [a, b] can be written as a sequence. Applying Cantor's construction to this sequence and [a, b] produces a real number in [a, b] that does not belong to the sequence. This contradicts the original assumption. Therefore, the real numbers in [a, b] cannot be written as a sequence.
 * 2) Assume that there are no transcendental numbers in  [a, b]. This implies that all the real numbers in [a, b] are algebraic, which implies that they form a subsequence of the sequence of all real algebraic numbers. This contradicts what was proved in 1. Thus the assumption that there are no transcendental numbers in [a, b] is false. Therefore, there are transcendental numbers in this interval.

Cantor chose to publish the constructive proof, which not only constructs a transcendental number but also is shorter and avoids two proofs-by-contradiction.

The disagreement about Cantor's proof
Cantor never published the non-constructive reasoning found in his December 2 and 7 letters—it only appears in his correspondence, which was published in 1937. By that time, other mathematicians had rediscovered his reasoning and used it to produce the non-constructive proof discussed above. As early as 1921, this non-constructive proof was attributed to Cantor and criticized as a pure existence proof. In that year, Oskar Perron 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." Some mathematicians have attempted to correct this misunderstanding of Cantor's work. In 1930, 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 1977, 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 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. The program that uses his 1874 construction requires at least sub-exponential time.

The disagreement about Cantor's proof occurs because two groups of mathematicians are talking about different proofs: the constructive one that Cantor published and the non-constructive one that was rediscovered. The view that Cantor's proof is non-constructive appears in some books that were very successful as measured by the length of time new editions or reprints appeared—for example: Eric Temple Bell's Men of Mathematics (1937; still being reprinted), Godfrey Hardy 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 (1967; 2008 4th edition). None of these books mention that there is a constructive proof. On the other hand, the quotations above from Fraenkel and Kaplansky show that they knew both proofs. The disagreement about Cantor's proof shows no sign of being resolved: since 2014, at least two books appeared stating that Cantor's proof is constructive, and at least four appeared stating that his proof does not construct any (or a single) transcendental.

The non-constructive proof may appear in some books because it is a simple example of the power of non-constructive reasoning. In The Problems of Mathematics, Ian Stewart dramatizes the power of this reasoning:


 * … The set of real numbers is uncountable. There is an infinity bigger than the infinity of natural numbers! The proof is highly original. Roughly, the idea is to assume that the reals are countable, and argue for a contradiction. … Building on this, Cantor was able to give a dramatic proof that transcendental numbers must exist. … Cantor showed that the set of algebraic numbers is countable. Since the full set of reals is uncountable, there must exist numbers that are not algebraic. End of proof (which is basically a triviality); collapse of audience in incredulity. In fact Cantor's argument shows more: it shows that there must be uncountably many transcendentals! There are more transcendental numbers than algebraic ones; and you can prove it without ever exhibiting a single example of either.

Steward does not tell the full story about non-constructive proofs. Most mathematicians prefer constructive proofs over non-constructive ones. Also, some mathematicians have used non-constructive arguments to discover new proofs or theorems, but then find and publish constructive proofs. Cantor did this and so did Kurt Gödel. Cantor proved that the sets of algebraic numbers and real numbers have different properties (one is countable, the other is not), and hence there are real numbers that are not algebraic. Gödel proved that, in a sufficiently strong theory, the sets of provable statements and true statements have different properties (provability is expressible in the theory, but truth is not), and hence there are true statements that are not provable. Like Cantor, Gödel used a key idea from his non-constructive proof (in this case, the liar paradox) to produce the constructive proof that he published.

Asserting that Cantor proof gave a non-constructive proof 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 [Not every real number is algebraic] was at first rejected by many mathematicians, since it did not exhibit any specific transcendental number." Birkhoff and Mac Lane are talking about the non-constructive proof that Cantor never published. There was no reason to reject Cantor's published proof, which is constructive. Even Leopold Kronecker, 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 desired degree of accuracy. In this case, given a k, an n can be computed such that bn – an ≤ $3⁄7$ where (an, bn) is the n-th interval of Cantor's construction.

Why Cantor's article emphasizes the countability of the real algebraic numbers
Historians of mathematics have discovered several interesting facts about Cantor's article "On a Property of the Collection of All Real Algebraic Numbers":


 * 1) Cantor's uncountability theorem was left out of the submitted article. He added it during proofreading.
 * 2) 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.
 * 3) 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.
 * 4) The proof of Cantor's second theorem does not state why some limits exist. The proof he was using does.

To explain these facts, historians have pointed to the influence of Cantor's former professors, Karl Weierstrass and Leopold Kronecker. Cantor sent his results to Weierstrass on December 22, 1874. 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.

Cantor wanted his article to include his uncountability theorem, but followed Weierstrass' advice to leave it out. Weierstrass also said that he could add it during proofreading, which he did. It appears in a remark at the end of the article's introduction. Without the uncountability theorem, the article's most significant result is the theorem stating that the set of real algebraic numbers is countable. The article's title refers to this theorem.

Weierstrass probably convinced Cantor of the importance of applying his "ideas at first to a single case (such as that of the real algebraic numbers) …" This led Cantor to restrict his first theorem to real algebraic numbers. This restriction produces a pedagogically simpler article: since Cantor constructs transcendental numbers by using his second theorem (which works with sequences of real numbers), the article is simpler if his first theorem produces a real sequence rather than a complex sequence.

From his correspondence, it appears that Cantor only discussed his article with Weierstrass. However, he wrote to 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 was a member of the editorial board of Crelle's Journal. Kronecker had delayed publication of a 1870 article by Eduard Heine, and Cantor would send his article to this journal.

Kronecker's influence appears in Cantor's proof of his second theorem. Cantor used Dedekind's version of the proof except he did not state why the limits a∞ = limn → ∞ an and b∞ = limn → ∞ bn exist. In his private notes, Dedekind wrote: "… [my] version is carried over almost word-for-word in Cantor's article (Crelle's Journal, 77); of course my use of "the principle of continuity" is avoided at the relevant place …". Dedekind's principle of continuity (which is equivalent to the least upper bound property of the real numbers) is the reason why these limits exist. However, this principle comes from Dedekind's construction of the real numbers, a construction that Kronecker did not accept.

Kronecker's influence also appears in Weierstrass' advising Cantor to leave out his uncountability theorem. In his history of set theory, José Ferreirós states: "Had Cantor emphasized it [the uncountability result], as he had in the correspondence with Dedekind, there is no doubt that Kronecker and Weierstrass would have reacted negatively."

Dedekind's contributions to Cantor's article
Dedekind's previous work enabled him to understand and contribute quickly to Cantor's work. Since 1856, Dedekind had developed theories involving infinitely many infinite sets—for example: ideals, which he used in algebraic number theory, and Dedekind cuts, which he used to construct the real numbers.

One of Dedekind's contributions has already been mentioned: in his article, Cantor gives Dedekind's proof of his second theorem. In his December 7th letter, Cantor sent Dedekind a complicated proof (involving infinitely many sequences) that the interval [a, b] is uncountable. Dedekind replied with a simpler proof. Before Dedekind's letter arrived, Cantor wrote that he had found a simpler proof that did not use infinitely many sequences. So Cantor had two proofs, but preferred Dedekind's.

Dedekind's second 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 Ferreirós calls it "Dedekind's theorem." Their correspondence reveals what each mathematician contributed to the theorem.

In his November 29th letter, Cantor introduced the concept of countability and stated that he can prove the countability of the set of positive rational numbers and sets of the form (an 1, n2, . . ., nν ) where n1, n2,. . ., nν, and ν are positive integers. Cantor's second result uses indexed numbers: each set consists of the ranges of nν functions where each function maps k positive integer arguments to the real numbers for some k ≤ nν. His second result implies his first: let ν = 2, an 1 = n1, and an 1, n2 = $3⁄7$. The functions can be quite general—for example, an 1, n2, n3, n4, n5 = ($$)$$ + tan($1⁄k$).

Dedekind quickly replied with a proof of the theorem: the set of algebraic numbers is countable. To obtain this result from Cantor's theorem about indexed numbers, Dedekind had to remove the restriction to positive integer indices and realize that the ordering produced can order the polynomials.

In his next letter, Cantor did not claim to have proved Dedekind's result. He did indicate how he proved his theorem about indexed numbers: "Your proof that (n) [the set of positive integers] 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 n12 + n12 + ··· + nν2 = $$\mathfrak{N}$$ and order the elements accordingly." Cantor's ordering cannot handle indices that are 0.

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 handling of Dedekind's contributions adversely affected his relationship with Dedekind—for example, 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 concept of countability and the uncountability theorem. 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 Rn (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 natural numbers with his infinite ordinals. This extension was necessary for his work on the Cantor-Bendixson theorem. Cantor discovered other uses for the ordinals. Using sets of ordinals, he produced an infinity of sets having different infinite cardinalities. His work on infinite sets together with Dedekind's work with sets 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, which were published from 1899 to 1901.

In set theory, countable models are used. In 1922, Thoralf Skolem proved that if the axioms of set theory are consistent, then they have a countable model. In this model, the set of real numbers is countable. Skolem explained why this paradox does not produce a contradiction: The model views its set of real numbers as uncountable because it contains no one-to-one correspondence between this set and its set of positive integers. The one-to-one correspondence between these sets exists outside the model. In 1963, Paul Cohen used countable models to prove his independence theorems.