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Georg Cantor (left) and his remarkable 1874 publication “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” in Journal für die Reine und Angewandte Mathematik (1874).

The Nature of Infinity — and Beyond

An introduction to Georg Cantor and his transfinite paradise

Jørgen Veisdal
Dec 17, 2018 · 31 min read

How big is infinity?

Commonly thought unanswerable, Cantor in the 1870s, ‘80s and ‘90s introduced radical new ideas about the answer to this question which established set theory as a new branch of pure mathematics. This article hopes to introduce you to his most notable work, and its implications.


Early life (1845–69)

Georg Cantor was in a sense lucky to have been born when he was, in Saint Petersburg on the 3rd of March 1845. His parents were Danish. His mother Marie (family name Meyer) came from a family of musicians with Russian origins and his father Georg Woldemar was a very successful business man, first as a wholesaling agent in St. Petersburg, and later as a broker on the city’s stock market.

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Cantor’s father and mother, G.W. and Marie Cantor

Education (1860-69)

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Cantor’s grades at age 8, when he attended the St.Petri-Schule for German speaking people in St.Petersburg.
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Humboldt University of Berlin in 1850 (then Friedrich Wilhelm University)
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Georg Cantor around 1870

Early career (1870–73)

Some have argued that the antecedents of Cantor’s later groundbreaking work can be traced as far back as to his earliest post-graduate publications. In fact, in Cantor’s research devoted the theory of trigonometric series, one can indeed find traces of his early interest in the “continuum”. Following the influences of both Weierstrass in Berlin and Heine in Halle, Cantor’s first paper Über einen die trigonometrischen Reihen betreffenden Lehrsatz (“On a theorem concerning the trigonometric series”) was completed for publication in March of 1870 and positioned to “advance the understanding of the convergence properties of the representation of an arbitrarily given function by means of infinite trigonometric series”. Starting from the trigonometric series and the work on functions of a complex variable done by Riemann, Cantor in the paper showed the following theorem:

Cantor's Uniqueness Theorem (1870): Every function f: ℝ → ℝ can have at most one representation by a trigonometric series.
Cantor's definition of real numbers ℝ (1872): A real number is an infinite series of rational numbers: a₁, a₂, ..., aᵤ, ..such that for any given ε there exists an u₁ such that for u ≥ u₁ and for any positive integer v, |aᵤ₊ᵥ - aᵤ| < ε.

Correspondence with Richard Dedekind (1872–73)

Later in the same year, Cantor met Richard Dedekind for the first time, who at that point was professor of mathematics at the Technische Hochschule at Brunswick. Dedekind had previously published a paper that provided an axiomatic analysis of the structure of the set of real numbers ℝ. His definition was of the real numbers as a complete, ordered field. Cantor and Dedekind exchanged letters over a period of many years. The mathematical portions of their letters were later published by Noether and Cavailleès (1937) and are now kept at the University of Evansville in Indiana.

Halle, December 2nd 1873I was exceptionally pleased to receive your answer to my last letter. I put my question to you because I had wondered about it already several years ago, and was never certain whether the difficulty I found was subjective or whether it was inherent in the subject. Since you write that you too are unable to answer it, I may assume the latter. In addition, I should like to add that I have never seriously occupied myself with it, because it has no special practical interest for me. And I entirely agree with you when you say that for this reason it does not deserve much effort. But it would be good if it could be answered; e.g. if it could be answered with no, then one would have a new proof of Liouville's theorem that there are transcendental numbers.- G. Cantor
Halle, December 7th 1873"..In the last days I have had the time to pursue more thoroughly the conjecture I spoke to you about; only today do I believe myself to have finished with the thing; but if I should be deceiving myself, I should certainly find no more indulgent judge than you."
Halle, December 9th 1873I have already found a simplified proof of the theorem just proved, so that the decomposition of the sequence into (1),(2),(3), ... is no longer necessary. I show directly that if I start with a sequence(i)   ω₁, ω₂, ..., ωᵤ,then in every given interval (α ... β) I can determine a number η that is not contained in (i). From this it follows at once that the totality (x) cannot be correlated one-to-one with the totality (u); and I infer that there exist essential differences among totalities and value-sets that I was until recently unable to fathom.Now I must ask your forgiveness for having taken so much of your time with this question. Confirming the receipt of your friendly lines of the 8th of December, allow me to assure you that nothing can give me more pleasure than to have been lucky enough to arouse in you an interest for certain questions of analysis.- G. Cantor
Brunswick, December 7th 1873Cantor communicates to me a rigorous proof, found on the same day, of the theorem that the totality of all positive numbers ω < 1 cannot be one-to-one correlated with the totality (n).I answered this letter, received on the 8th of December, on the same day with congratulations for the fine success. At the same time, I rephrase much more simply the core of the proof (which was still quite complicated).- Richard Dedekind

Set Theory

Described by the Stanford Encyclopaedia of Philosophy as “one of the greatest achievements of modern mathematics”, set theory is widely acknowledged to have been founded by the paper that resulted from the work Cantor did in the period 1873–1884. In particular, the origins of set theory is traced back to a single paper published in 1874 by Cantor, entitled Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, (On a Property of the Collection of All Real Algebraic Numbers”). The fundamental and most consequential result it presents is the uncountability of the real numbers, and as consequence, the invention of a distinction between numbers that belong to “the continuum” and those that belong to “a collection like the totality of real algebraic numbers”. The paper appeared in Journal für die reine und angewandte Mathematik (“Crelle’s Journal”) just before Cantor turned 30 years old. As he wrote to Dedekind about two weeks after arriving at his proof:

Berlin, December 25th 1873"..Although I did not yet wish to publish the subject I recently for the first time discussed with you, I have nevertheless unexpectedly been caused to do so. I communicated my results to Herr Weierstrass on the 22nd; however, there was no time to go into details; already on the 23d I had the pleasure of a visit from him, at which I could communicate the proofs to him. He was of the opinion that I must publish the thing at least in so far as it concerns the algebraic numbers. So I wrote a short paper with the title: On a property of the set of all real algebraic numbers and sent it to Professor Borchardt to be considered for the Journal fur Math.As you will see, your comments (which I value highly) and your manner of putting some of the points were of great assistance to me."- G. Cantor
  1. In every interval [a,b] there are infinitely many numbers not included in any sequence; and as a consequence that
  2. The set of real numbers are uncountably infinite;

What is a set?

Cantor's Definition of a Set
By a set we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought. These objects are called the "elements" of M.

Countability

Countable sets
A set S is countable if there exists an injective function f from S to the natural numbers = {1,2,3, ...}. If such an f can be found that is also surjective (and therefor bijective), then S is called a countably infinite set, or denumerable.
For instance, for the set of even numbers (2n|n ∈ ℕ): 2 4 6 8 10 ... 2n
↓ ↓ ↓ ↓ ↓ ↓
1 2 3 4 5 ... n
We see that the elements of the two sets may be put in one-to-one correspondence with one other, and so we can determine that the set of even numbers is also countable.

The Countability of Rational Numbers (1873)

Cantor’s first published investigation into the countability of sets occurred in 1873 when he proved that the rational numbers ℚ (fractions/ratios) are countable. His rather elegant and intuitive proof went as follows:

Proof of the Countability of the Rational numbers ℚ
Let us first propose that the set of rational numbers is countable. To prove this assertion, let us arrange all the rational numbers (ratios of natural numbers) in an infinite table as such:
1/1 1/2 1/3 1/4 1/5 ...
2/1 2/2 2/3 2/4 2/5 ...
3/1 3/2 3/3 3/4 3/5 ...
4/1 4/2 4/3 4/4 4/5 ...
5/1 5/2 5/3 5/4 5/5 ...
... ... ... ... ...
Next, starting in the upper lefthand corner, move through the diagonals from left to right at 45 degrees, starting with 1/1, then 1/2 and 2/1, then 3/1, 2/2 and 1/3 and so on. Write down every new number you come across. You will obtain the following ordering:1/1, 1/2, 2/1, 3/1, 2/2, ...
1 2 3 4 5 ...
Which is not just a well-ordering, but also in one-to-one correspondence with the natural numbers in their natural order. This proves the countability of the rational numbers .

The Countability of Real Algebraic Numbers (1874)

A year later, in his 1884 paper, Cantor showed that the real algebraic numbers are countable. Real algebraic numbers are real numbers ω which satisfy equations of the form: aₒ ωᵘ + a¹ωᵘ⁻¹ + … + aᵤ= 0. That is to say, real algebraic numbers are roots of non-zero real polynomials. They are countable, i.e:

The Countability of Real Algebraic Numbers
The collection of all algebraic reals can be written as an infinite sequence.
Proof of the Countability of Real Algebraic Numbers (1874)
For each polynomial equation of the form

aωᵘ + aωᵘ⁻¹ + … + aᵤ = 0
with integer coefficients a, define its index to be the sum of the absolute values of the coefficients plus the degree of the equation:|a|+|a|+ ... +|aᵤ|The only equation of index 2 is ω = 0, so its solution, 0, is the first algebraic number. The four equations of index 3 are 2x = 0, x + 1 = 0, x – 1 = 0, and x2 = 0. They have roots 0, –1, 1, so he included the new values –1 and 1 as the second and third entries on his list of algebraic numbers. Observe that for each index there are only finitely many equations and that each equation only has finitely many roots. Listing the new roots by order of index and by increasing magnitude within each index, one establishes a systematic method for listing all the algebraic numbers. As with rationals, the one-to-one correspondence with the natural numbers proved that the set of algebraic numbers have to countably infinite.

The Uncountability of Real Numbers (1874)

Cantor’s most fruitful use of countability as a concept occurred in the third result of his 1874 paper when he demonstrated the uncountability of the real numbers — the first set shown to lack this property. A real number ℝ is a value of a continuous quantity that can represent a distance along a line. Any real number can be determined by a possibly infinite decimal representation, such as that of e.g. 8.632, 0.00001, 10.1 and so on, where each consecutive digit is measured in units one tenth the size of the previous one. The statement that the real numbers are uncountable is equivalent to the statement:

The Uncountability of Real Numbers
Given any sequence of real numbers and any interval [α ... β], one can determine a number η in [α ... β] that does not belong to the sequence. Hence, one can determine infinitely many such numbers η in [α ... β].
Proof of the Uncountability of the Real numbers ℝ (1874)
Suppose we have an infinite sequence of real numbers,
(i) ω₁, ω, ... ωᵥ, ...where the sequence is generated according to any law and the numbers are distinct from each other. Then in any given interval (α ... β) a number η (and consequently infinitely many such numbers) can be determined such that it does not occur in the series (i).To prove this, we go to the end of the interval [α ... β], which has been given to us arbitrarily and in which α < β. The first two numbers of our sequence (i) which lie in the interior of this interval (with the exception of the boundaries), can be designated by α', β', letting α' < β'. Similarly, let us designate the first two numbers of our sequence which lie in the interior of (α' ... β') by α", β" and let α" < β". In the same way, construct the next interval, and so on. Here, therefore, α', α" ... are by definition determinate numbers of our sequence (i), whose indices are continually increasing. The same goes for the sequence β', β", ...; Furthermore, the numbers α', α" ... are always increasing in size, while the numbers β', β", ... are always decreasing in size. Of the intervals [α ... β], [α' ... β'], [α" ... β"], .... each encloses all of those that follow. Here, only two cases are conceivable.In the first case, the number of intervals so formed is finite. In this case, let the last of them be (α ... β). Since its interior can be at most one number of the sequence (i), a number η can be chosen from this interval which is not contained in (i), thereby proving the theorem.In the second case, the number of constructed intervals is infinite. Then, because they are always increasing in size without growing into the infinite, the numbers α, α', α", ... have a determinate boundary value αʷ. The same holds for the numbers β, β', β", ... because they are always decreasing in size. Let their boundary value be βʷ. If αʷ = βʷ, then one easily persuades oneself, if one only looks back to the definition of the intervals that the number η = αʷ = βʷ cannot be contained in our sequence (i). However, if αʷ < βʷ, then every number η in the interior of the interval [αʷ ... βʷ] as well as its boundaries satisfies the requirement that it is not contained in the sequence (i).

Cantor’s Diagonal argument (1891)

Cantor seventeen years later provided a simpler proof using what has become known as Cantor’s diagonal argument, first published in an 1891 paper entitled Über eine elementere Frage der Mannigfaltigkeitslehre (“On an elementary question of Manifold Theory”). I include it here for its elegance and simplicity. Generalized, the now famous argument goes as follows:

Proof: Cantor’s diagonal argument (1891)
In his paper, Cantor considers the set M of all infinite sequences of the binary numbers m and w. Sequences such as:
E₁ = (m, m, m, m, m, ...),
E₂ = (w, w, w, w, w, ...),
E₃ = (m, w, m, w, m, ...),
E₄ = (w, m, w, m, w, ...),
E₅ = (m, m, w, w, m, ...)
Cantor asserts that there exists a set M that does not have the “breath” of the series E₁, E₂, E₃ … , meaning M is of a different size than the sum of each sequence En, i.e. that even though M is constructed of all the infinite sequences of the binary numbers m and w, he can always construct a new sequence E₀ which “is both an element of M and is not an element of M.”The new sequence E₀ is constructed using the complements of one digit from each sequence E₁, E₂, … En. A complement of a binary number is defined as the value obtained by inverting the bits in the representation of the number (swapping m for w and visa versa). So, the new sequence is made up of the complement the first digit from the sequence E₁ (m), the complement of the second digit from the sequence E₂ (w), the complement of the third digit from the sequence E₃ (m) and so on to finally the complement of the nth digit from the sequence En. From the example sequences above, the new sequence E₀ would then be:E₀ = (w, m, w, w, w, ...)By its construction, E₀ differs from each sequence En since their nth digits differ. Hence, E₀ cannot be one of the infinite sequences in the set M.
Proof of the Uncountability of the Real numbers ℝ
This proof is by contradiction, i.e. we will assume that the real numbers ℝ are countable and derive a contradiction. If the reals are countable, then they could be listed:
1. 657.853260...
2. 2.313333...
3. 3.141592...
4. .000307...
5. 49.494949...
6. .873257...
...
To obtain a contradiction, it suffices to show that there exists some real α that is missing from the list. The construction of such an α works by making its first decimal place different from the first decimal place of the first number of the list, by making the second decimal place different from the second decimal place of the second number, and in general by making the nth decimal place different from the nth decimal place of the nth number on the list.Even simpler, for our α we'll make the nth decimal place 1 unless it is already 1, in which case we'll make it 2. By this process, for our example list of numbers, we obtain:α = .122111...Which, by construction cannot be a member of the list we created. And so, by contradiction, our list of all reals cannot contain every number, and so must be uncountable.

There are more real numbers than there are natural numbers.


Halle, January 5th 1874"..Can a surface (say a square that includes the boundary) be uniquely referred to a line (say a straight line segment that includes the end points) so that for every point on the surface there is a corresponding point of the line and, conversely, for every point of the line there is a corresponding point of the surface? It still seems to me at the moment that the answer to this question is very difficult - although here too one is so impelled to say no that one would like to hold the proof to be almost superfluous."- G. Cantor
Halle, January 28th 1874"..When you get around to answering me, I should be grateful to hear whether you had the same difficulty as I in answering the question I sent to you in January about the correlation of a line and a surface, or whether I am deceiving myself. In Berlin a friend to whom I presented the same problem told me the subject was somewhat absurd, because it is self-evident that two independent variables cannot be reduced to one."- G. Cantor
Halle, June 20th 1877"..I should like to know whether you consider an inference-procedure that I use to be arithmetically rigorous.The problem is to show that surfaces, bodies, indeed even continuous structures of p dimensions can be correlated one-to-one with continuous lines, i.e. with structures of only one dimension—so that surfaces, bodies, indeed even continuous structures of p dimensions have the same power as curves. This idea seems to conflict with the one that is especially prevalent among the representatives of modern geometry, who speak of simply infinite, doubly, triply, . . . ρ-fold infinite structures. (Sometimes you even find the idea that the infinity of points of a surface or a body is obtained by as it were squaring or cubing the infinity of points of a line.)"- G. Cantor

Infinite sets

Definition of an infinite set
A set A is infinite if, and only if, there is a one-to-one correspondence between A and a set X which is a proper subset of A.
  1. The size of any whole must be greater than that of any of its parts;

Cardinal numbers

So just how does one go about studying the properties and differences of infinite sets? After his 1874 discovery of the existence on non-denumerable infinite sets, in 1878 Cantor turned to a more general study of what he called powers, or cardinal numbers — the study of the sizes of sets. The cardinality of the set A is usually denoted by |A|, sometimes card(A).

Cantor's Definition of Cardinal Numbers
We will call by the name 'power' or 'cardinal number' of M the general concept which, by means of our active faculty of thought, arises from the set M when we make abstraction from the nature of its various elements m and of the order in which they are given.
Theorem: The set ℝ² of all ordered pairs of real numbers (that is, the real plane) has the same size as ℝ.
Proof that |ℝ²| = |ℝ|
It suffices to prove that the set of all pairs (x,y), 0 < x,y < 1 can be mapped bijectively onto (0,1]. Consider the pair (x,y) and write x,y in their unique non-terminating decimal expansion as in the following example:
x = 0.3 01 2 007 08 ...
y = 0.009 2 05 1 0008 ...
Note that the digits of x and y have been separated into groups by always going to the next nonzero digit, inclusive. Now we associate to (x,y) the number z ∈ (0,1] by writing down the first x-group, after that the first y-group, then the second x-group, and so on. Thus, in our example, we obtain:z = 0.3 009 01 2 2 05 007 1 08 0008 ...Since neither x nor y exhibits only zeroes from a certain point on, we find that the expression for z is again a non-terminating decimal expansion. Conversely, from the expansion of z we can immediately read off the preimage (x,y) and the map is bijective.
Halle, June 29th 1877"..Please excuse my zeal for the subject if I make so many demands upon your kindness and patience; the communications which I lately sent you are even for me so unexpected, so new, that I can have no peace of mind until I obtain from you, honored friend, a decision about their correctness. So long as you have not agreed with me, I can only say: je le vois, mais je ne le crois pas."

“I see it, but I do not believe it”.


Infinite cardinal numbers

When Cantor in 1878 then turned to study infinite cardinal numbers, he was already aware of the existence of two such “powers” (Mächtigkeiten): point-sets (e.g. the natural numbers) and the continuum (e.g. real numbers). In his 1883 paper entitled Grundlagen einer allgemeinen Mannigfaltigkeitslehre (“Foundations of a General Theory of Manifolds”) he introduced a distinction between two infinities, the transfinite and the absolute:

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

The Continuum Hypothesis (1878)

There are no infinite cardinal numbers strictly in between the cardinality of the natural numbers ℵ₀ and the cardinality of the real numbers ℵ₁.

No introduction to Cantor would be complete without discussing the infamous hypothesis that has become forever linked with his life work, Cantor’s Continuum Hypothesis (CH). Much of his work on the conjecture was published in the six-part treatise Über unendliche, lineare Punktmannichfaltigkeiten (“On infinite, linear manifolds of points”) in the journal Mathematische Annalen between 1879 and 1884.

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Georg Cantor (left) and his six-part treatise Über unendliche, lineare Punktmannichfaltigkeiten in the journal Mathematische Annalen.
The question arises how the different parts of a continuous straight line, i.e. the different infinite manifolds of points that can be conceived in it, are related with respect to their powers. Let us divest this problem of its geometric guise, and understand by a linear manifold of real numbers, every conceivable totality of infinitely many, distinct real numbers. Then the questions arises, into how many and which classes do the linear manifolds fall, if manifolds of the same power are placed into the same class, and manifolds of different power into different classes?By an inductive procedure, whose more exact presentation will not be given here, the theorem is suggested that the number of classes of linear manifolds that this principle of sorting gives rise to is finite, and indeed, equals two.
c = || = ℵ₁

Attempted proofs

Cantor spent many of the remaining years of his life wrestling with providing a proof that the continuum hypothesis is true. His direct strategy was to use the derived sets P⁽ⁿ⁾ of a point set P to measure its cardinality. As Bertrand Russell put it:

Popularly speaking, the first derivative consists of all points in whose neighborhood an infinite number of terms of the collection are heaped up; and subsequent derivatives give, as it were, different degrees of concentration in any neighborhood. Thus, it is easy to see why derivatives are relevant to continuity; to be continuous, a collection must be as concentrated as possible in every neighborhood containing any terms of the collection.

Mental health

Cantor suffered his first serious mental breakdown in May of 1884, ten years after the publication of his first proof of the uncountability of the real numbers. Most historians believe the breakdown occurred as a result of an ongoing dispute Cantor had with Leopold Kronecker at the University of Berlin, coupled with the apparent intractability of the continuum hypothesis. As we can read from letters Cantor sent to Swedish mathematician Mittag-Leffler, Cantor’s first breakdown occurred just as he had returned from a joyous trip to Paris where he met, among other mathematicians, Henri Poincaré. Cantor writes that he liked Poincaré very much and was happy to learn that the great man understood his transfinite set theory and its applications. In addition, he writes that he spent time visiting galleries and museums, indulging in his love of the opera and theater. Cantor’s breakdown reportedly occurred shortly after he returned to Germany to attend to family affairs.

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Final years

After his 1884 hospitalization, there is no record that Cantor was admitted to any sanatorium again until 1899. That year, his youngest son died and Cantor reportedly lost his passion for mathematics. When in 1903 Julius König presented a paper which attempted to disprove the basic tenants of transfinite set theory, Cantor perceived it as a grave public humiliation. Despite Ernst Zermelo demonstrating the paper’s invalidity less than a day later, Cantor remained shaken and even momentarily began questioning the existence of God (Cantor was a devout Christian). The events preceded a series of additional hospitalizations at intervals of two to three years.

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Cantor in 1917

Paradise lost?

In 1900 German mathematician David Hilbert identified the continuum hypothesis as one of the 23 most significant problems to shape the future of mathematics in the 20th century. His prediction turned out accurate, as attempts by other mathematicians to prove or disprove Cantor’s conjecture led to some of the deepest work in set theory thus far.

“From the paradise, that Cantor created for us, no-one can expel us” — David Hilbert

Cantor’s Paradise

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