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Some Infinities are Bigger Than Others – A Mathematical Proof

Updated on December 4, 2017
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Since long has the concept of infinity intrigued the humanity. It is uncertain when this idea first came out, but before Christ, Greeks and Indians have used it without too much mathematical accuracy. The first man to really use the notion of infinity as a number was Georg Cantor, in the 19th century.

Georg Ferdinand Ludwig Philipp Cantor is a Russian-born German mathematician, known for the development of the set theory. With the set theory, Cantor stablished the basis in which he would construct his controversial theory about the multiple infinities. Doing so, he was accused by some Christian Religious of defying the uniqueness of God. A big part of the mathematical community was also very critic regarding Cantor’s ideas.

A simple proof using the Cantor's argument

Defining Infinity

The first important set we need to know to understand Cantor’s theory is the set of the natural numbers, ℕ = {0,1,2,3,4…}. This set is important because it is the one we use to count elements. We say that the set of the natural numbers is infinite, but countable, which means that we can list its elements and that all these elements will be on this list (even if the list is infinite).

For example, we can count the set of the odd numbers by associating them with the set of the Natural numbers using a one-to-one correspondence, also known as a bijection:

0 -> 1

1 -> 3

2 -> 5

3 -> 7

4 -> 9

5 -> 11

and so on.

Another important example of a countable infinite set is the integers,

ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}

which is the same as the natural numbers but includes the negative numbers.

Intuitively, we could think that this set is bigger than the naturals, but we could make a simple one-to-one correspondence between them:

0 -> 0

1 -> -1

2 -> 1

3 -> -2

4 -> 2

5 -> -3

6 -> 3

and so on.

This proves that the set of natural numbers has the same number of elements as the set of integers, or that ℕ’s cardinality is equal to ℤ‘s cardinality.

So, the infinity here is defined using the cardinality of a set, which means the number of elements of this set.

Georg Cantor
Georg Cantor | Source

The Diagonal Argument

With all we have talked about, we must still show that there is a bigger set than the set of the natural numbers. To prove this, we should be able to construct a set of elements that could not be ordered. If we can list the elements of a set, there is a bijection between this set and ℕ, because we could simply associate 0 with the first element, 1 with the second and so on.

Cantor’s Diagonal Argument shows an example of set that can not be put into a list:

Let’s consider the set T of all the infinite lists of binary digits, so

T1 = (0,0,0,0,0,0, …)

T2 = (1,0,1,0,1,0, …)

T3 = (0,1,0,1,0,1, …)

T4 = (0,1,0,0,0,0, …)

T5 = (0,0,1,1,1,1, …)

T6 = (0,1,1,0,0,1, …)

are all elements of T.

Now, we suppose that we can take all the elements of T and put them into a list:

T1 = (0,0,0,0,0,0, …)

T2 = (1,1,1,0,1,0, …)

T3 = (0,1,0,1,0,1, …)

T4 = (0,1,0,0,0,0, …)

T5 = (0,0,1,1,1,1, …)

T6 = (0,1,1,0,0,1, …)

and so on.

Now, we take a diagonal element Tx = (0,1,0,0,1,1…) which is constructed taking the first digit of T1 , the second of T2 , the third of T3 and so on, for all the elements of the list.

We, then, create a new element switching from 0 to 1 and from 1 to 0, all elements of Tx. So,

Ty = (1,0,1,1,0,0, …).

Since the first digit of Ty is different from the first digit of T1 , the second is different from the second of T2 , the third is different from the third of T3 and so on, we can say that Ty is different from all the elements of the list. This means that it is impossible to list the elements of the set T.

If it is impossible to list the elements of the set, it is also impossible to make a one-to-one correspondence, or a bijection, between this set and ℕ. When we try to list the elements of T, there are always some that are left out. This means that T has more elements than ℕ, or that T’s cardinality is bigger than ℕ’s cardinality. In other words, there are some infinities bigger than others.

The Continuum Hypothesis and the Death of Cantor.

It is possible to show that the cardinality of the set of the Real Numbers, ℝ, or The Continuum, is equal to the cardinality of the set T we showed. Since the elements of the set T are formed by the digits 0 and 1 in an infinite sequence, we can see that there are elements in T, which means that there are 2 raised to the cardinality of ℕ elements in ℝ.

The Continuum Hypothesis states that the cardinality of the Continuum is the second highest infinite number:

Georg Cantor died in 1918 in a sanatorium in Halle. He could never prove his Hypothesis. In fact, some years later it was proved that it is actually impossible to prove the Continuum Hypothesis.

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