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

In five short pages, Cantor’s paper presents three important results:

The set of real algebraic numbers is countable; and In every interval [a,b] there are infinitely many numbers not included in any sequence; and as a consequence that The set of real numbers are uncountably infinite;

The rest of this article is devoted to explaining the implications of the third result, on the uncountability of real numbers. For this, we begin with a few fundamental concepts.

What is a set?

“A set is a Many which allows itself to be thought of as a One” — Georg Cantor

A set is a collection of elements. The set consisting of the numbers 3,4 and 5 is denoted by {3, 4, 5}. For larger sets and the sake of simplicity, an ellipsis is often used if the reader can easily guess what the missing elements are. Cantor’s original definition of an “aggregate” (set), translated, went as follows:

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

A countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

The property of countability is an important one in set theory. An intuitive interpretation of countability is “listability”, that the elements of a set may be written down in a list. The most inherently countable set is the natural numbers ℕ, in that the elements of ℕ are the counting numbers themselves (1,2,3, …). As we know, they are infinite in number, and so termed “countably infinite”, or “denumerable”. For other sets, formally, by stating that a set is countable one means that the elements of the set can be put in one-to-one correspondence with elements of the set of natural numbers ℕ, i.e. that:

Countable sets

A set S is countable if there exists an injective function f from S to the natural numbers ℕ = {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 property makes it possible to compare sets in terms of the number of elements they contain without actually counting anything, and in this way make inferences about the relative sizes of both finite and infinite sets. For practical reasons let us illustrate the finite case by imagining a classroom with 100 seats. Filled with students, one can make an inference about the size of the set of students in relation to size of the set of seats. If seats are vacant, the set of seats is larger than the set of students. If no seats are vacant and some students are standing, the size of the set of students is larger than that of seats, and so on.

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 ℚ 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 ℚ.

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.

Cantor showed it in his 1874 paper by the following proof:

Proof of the Countability of Real Algebraic Numbers (1874)

For each polynomial equation of the form



aₒωᵘ + a₁ωᵘ⁻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: |aₒ|+|a₁|+ ... +|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].

As we’ve seen from his letter exchanges with Dedekind in 1873, we know how Cantor worked towards the momentous result. His original proof (Cantor’s First Uncountability Proof) went as follows, and is based on the Bolzano-Weierstrass theorem:

Proof of the Uncountability of the Real numbers ℝ (1874)

Suppose we have an infinite sequence of real numbers, (i) ω₁, ω₂, ... 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 (αᵛ ... βᵛ). 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.

Applied to prove the uncountability of the real numbers ℝ:

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.

The conclusions of both proofs (1874 and 1891) are the same — although both the natural numbers and the real numbers are infinite in number and so go on forever, there “aren’t enough” natural numbers to create a one-to-one correspondence between them and the real numbers. Cantor’s brilliant discovery, in other words, showed rigorously that infinity comes in different sizes, some of which are larger than others.