Oulun Lyseo Galois club 2013
Cardinal numbers of sets and Cantor’s paradise
By Teuvo Laurinolli
INTRODUCTION
A set (’joukko’) is a collection of objects which are called members or elements of the set.
Examples of sets are (1) the set of students in a classroom, (2) the set of books in a library, (3)
the set of atoms in the Milky Way galaxy, (4) the set of natural numbers up to 1000, (5) the set
ℕ of all natural numbers, (6) the set of rational numbers between 0 and 1, (7) the set of all
subsets1 of ℕ. The sets (1) – (4) are finite while (5) – (7) are infinite.
Cardinal number of a finite set is simply the number of elements in it as can be counted in a
usual way (one, two, three, …).
For example, the cardinal number of the set 𝐴 = {𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛 } is n or in symbols 𝐴̿ = 𝑛.
CARDINALITIES OF INFINITE SETS
If you want to check whether the cardinal numbers (or cardinalities) of two finite sets A and B
are equal you can use one of the following methods:
(i)
(ii)
count the elements of both sets and compare the answers,
see if it is possible to establish a one-one correspondence between the two sets
The latter method is more fundamental (and even more primitive) as in using it you don’t
have to know how to count. A German mathematician Georg Cantor (1845-1918) realized that
this method could be applied to infinite sets as well. For them (i) does not work because you
can never complete the counting. The results can be a bit surprising since, for example, one
can easily put the set ℕ of all natural numbers and its proper subset 2ℕ of all even natural
numbers into 1-1 correspondence (𝑘 ↔ 2𝑘). So these sets have the same cardinality2 even if
the latter contains only half of the elements of the former. This is not possible for finite sets!
Cantor concluded that this property (of allowing 1-1 correspondence with a proper subset) is
characteristic to infinite sets.
Cantor decided to introduce infinite numbers (he called them transfinite) for the cardinalities
of infinite sets. For the smallest transfinite number, the cardinality of ℕ, he gave the symbol
The set A is a subset of B, in symbols 𝐴 ⊂ 𝐵, if all elements of A are also elements of B. A
subset is proper if it does not contain all elements of the mother set.
2 This fact is often called Galilei’s paradox after an Italian scientist Galileo Galilei (1564-1642)
who observed it much earlier.
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̿ = ℵ0 . Cantor then set out to find larger
ℵ0 , the first letter (alef) in the Hebrew alphabet. So ℕ
transfinite cardinals by looking at numbers sets ℤ (integers), ℚ (rational numbers) and ℝ
̿ =ℕ
̿=ℚ
̿ = ℵ0 as the elements of each of these sets can
(real numbers). It turned out that ℤ
be written as infinite list (first, second, third, etc) which contain all elements of the set. Such a
list constitutes a 1-1 correspondence with ℕ.
CARDINALITY OF THE POWER SET
However Cantor was able to prove that the set ℝ cannot be put in 1-1 correspondence with ℕ
and therefore its cardinality (called cardinality of the continuum) must be greater than ℵ0 .
This fact can be proved by considering decimal expansions of real numbers between 0 and 1
but it follows from a more general theorem (Power set theorem) that we prove below.
For a given set A (finite or infinite) we define its power set 𝒫(𝐴) to be the set of all subsets of
A. In symbols
𝒫(𝐴) = {𝐵|𝐵 ⊂ 𝐴} = set of all B’s such that B is a subset of A.
Notice that every non-empty set A has at least two subsets: the empty set ∅ and A itself.
For exampler, if 𝐴 = {𝑎, 𝑏, 𝑐} is a set of three elements then its power set
𝒫(𝐴) = {∅, {𝑎}, {𝑏}, {𝑐}, {𝑎, 𝑏}, {𝑎, 𝑐}, {𝑏, 𝑐}, {𝑎, 𝑏, 𝑐}}
has eight elements and it is easy to see generally that the following theorem holds true.
Theorem 1:
Let A be a finite set with cardinality n. Then its power set 𝒫(𝐴) has cardinality 2𝑛 .
The next theorem is a general result for all sets, finite or infinite.
Theorem 2:
̿̿̿̿̿̿̿ > 𝐴̿ .
Let A be any set. Then 𝒫(𝐴)
̿̿̿̿̿̿̿ < 𝐴̿, (2) 𝒫(𝐴)
̿̿̿̿̿̿̿ = 𝐴̿ or (3) 𝒫(𝐴)
̿̿̿̿̿̿̿ > 𝐴̿.
Proof: There are three alternatives (1) 𝒫(𝐴)
Obviously (1) is impossible as 𝒫(𝐴) always contains a ”copy” of each element of A in the sense
that if 𝑎 ∈ 𝐴 then {𝑎} ∈ 𝒫(𝐴) . To prove the theorem it is therefore enough to show that (2) is
also impossible. To do this we apply a proof method known as reductio ad absurdum. We
assume that (2) is true and derive a logical contradiction.
̿̿̿̿̿̿̿ = 𝐴̿ . Then there exists a 1-1 correspondence (call it f) between A and
So assume that 𝒫(𝐴)
𝒫(𝐴). We can consider f as a function from A to 𝒫(𝐴). That is, for every 𝑥 ∈ 𝐴 we have 𝑓(𝑥) ∈
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𝒫(𝐴) or equivalently 𝑓(𝑥) ⊂ 𝐴. And conversely, for every subset 𝐵 ∈ 𝒫(𝐴) there is an element
𝑥 ∈ 𝐴 such that 𝑓(𝑥) = 𝐵.
Now define a subset 𝐶 ⊂ 𝐴 as follows
(*)
𝐶 = {𝑥 ∈ 𝐴|𝑥 ∉ 𝑓(𝑥)}
= set of all members of A which do not belong to their own ”image” subset
So to see if a given element 𝑥 ∈ 𝐴 belongs to 𝐶 you look at its ”image” subset 𝑓(𝑥) ⊂ 𝐴 and
check whether 𝑥 is in that subset or not.
Now because C is a subset of A it is a member of the power set 𝒫(𝐴). Then because f is 1-1
correspondence between A and 𝒫(𝐴) there must be an element 𝑐 ∈ 𝐴 such that 𝐶 = 𝑓(𝑐). We
will now ask if (i) 𝑐 ∈ 𝐶 or (ii) 𝑐 ∉ 𝐶. There are no other possibilities.
If (i) is true, that is 𝑐 ∈ 𝐶, then by the definition (*) we have 𝑐 ∉ 𝑓(𝑐). But 𝑓(𝑐) = 𝐶 and hence
𝑐 ∉ 𝐶. Therefore (i) cannot be true.
But if (ii) is true, that is 𝑐 ∉ 𝐶, then c satisfies the membership condition (*) and therefore 𝑐 ∈
𝐶. It follows that (ii) can neither be true.
We have landed in a contradiction either (i) or (ii) must be true, but neither of them can be
̿̿̿̿̿̿̿ =
true. The only way to resolve this absurdum is to conclude that our initial assumption 𝒫(𝐴)
𝐴̿ must be false. This proves the theorem.
QED
For finite sets Theorem 2 is, of course, a direct consequence of Theorem 1, since 2𝑛 > 𝑛 for all
natural numbers. But for infinite sets it has ”revolutionary” impact because it implies that
there are infinitely many transfinite cardinal numbers. Starting from the smallest transfinite
̿̿̿̿̿̿̿ which, in analogy with
̿ we can obtain a larger cardinal number 𝒫(ℕ)
cardinal number ℵ0 = ℕ
finite case, is denoted by an exponential notation 2ℵ0 , and repeat the power set operation to
obtain even higher cardinalities.
CARDINALITY OF THE CONTINUUM ℝ
It is not difficult to establish a 1-1 correspondence between the set ℝ of real numbers (or its
subinterval from 0 to 1) and the power set 𝒫(ℕ) of the set of natural numbers3. Therefore the
cardinality of the real number continuum is
̿̿̿̿̿̿̿ = 2ℵ0 .
̿ = 𝒫(ℕ)
ℝ
Binary expansion of a real number in the interval (0,1) is of type 0,101100010… which can
be understood as a coding for a subset {1,3,4,8, … } of ℕ according to the rule 1 = yes, 0 = no.
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̿ >ℕ
̿ or equivalently 2ℵ0 > ℵ0 . Cantor spent large part
From Theorem 2 it follows then that ℝ
of his life in trying to prove the following hypothesis
Cantor’s continuum hypothesis (v1): There are no cardinal numbers between ℵ0 and 2ℵ0 .
This hypothesis can also be expressed as follows:
Cantor’s continuum hypothesis (v2): 2ℵ0 = ℵ1.
̿ or 𝑆̿ = ℕ
̿.
Cantor’s continuum hypothesis (v3): For every subset 𝑆 ⊂ ℝ either 𝑆̿ = ℝ
A generalized version reads
Cantor’s generalized continuum hypothesis: For every set A and subset 𝑆 ⊂ 𝒫(𝐴) either
̿̿̿̿̿̿̿ or 𝑆̿ = 𝐴̿ . That is, there are no cardinalities between those of A and its power set
𝑆̿ = 𝒫(𝐴)
𝒫(𝐴).
These hypotheses have been shown (by Gödel 1938 and Cohen 1963) to be independent of
fundamental axioms of mathematics and can therefore be considered as (still) open problems.
CANTOR’S PARADISE?
In 1870’s Cantor had discovered a route to a fascinating new world – the world of transfinite
numbers, the world of alefs. Many mathematicians were skeptical towards this new world and
proclaimed it lunatic and full of hallucinations and contradictions. In the 1900 World
Congress of Mathematicians Cantor was vigorously defended by David Hilbert, one of the
most influential mathematicians of 20th century who said ”No one can ever expel us from
Cantor’s paradise.”
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