Adonis Diaries

Probabilistic Approach to Mathematical Philosophy? Concerning the concept of Infinity.

Posted on: May 25, 2022

It’s really hard for someone who loves mathematics to get bored: maticians can play fun games …

You may think that we are unable to match natural numbers to rational numbers, and the density is enormous.

However, we can use one-to-one correspondence in all positive rational numbers.

Let us move forward using the image below. We wrote natural numbers going to infinity in the first row and column. Then, following a specific pattern, we wrote all rational numbers.

For example, first, we wrote all of the numerators as one and made it so that the denominators were getting larger.

We followed the same process except with two as the numerator in the next row. We then continue these steps to infinity. If we follow the below method for matching, we notice that one-to-one correspondence is, in fact, possible — 1 to 2, 2 to ½, ½ to ⅓, ⅓ to 3, etc.

Cantor’s diagonal argument

As you can see, we can match all natural numbers to positive rational numbers. If we wanted to, we could also use this logic to match all rational numbers to integers. Therefore, we can surmise that rational numbers are countable.

What about real numbers?

Real numbers are Not countable, and Cantor has provided very lovely proof for this as well.

When proving that real numbers are not countable, Cantor used the contradiction method to show that the interval between (0–1) is uncountably large.

First, he assumes that the distance between (0–1) is countable, and when he proves it wrong, he gets a contradiction.

First, Cantor writes all natural numbers from 1 to n, starting at the top left of an empty piece of paper he finds. He then assumes he writes all the numbers between (0–1) on their right, naming them as x₁, x₂, x₃, etc.1 → x₁ = 0.256173…
2 → x₂ = 0.654321…
3 → x₃ = 0.876241…
4 → x₄ = 0.60000…
5 → x₅ = 0.67678…
6 → x₆ = 0.38751…
. . . .
n → xₙ = 0.a₁a₂a₃a₄…aₙ…
. . . .

According to his first assumption, Cantor thinks he should not find any other number between (0–1). He also knows, however, that he must prove this mathematically. 

That is why he starts looking for a number he thinks is not between (0–1), b.

Using a straightforward approach, Cantor finds a number b. 
First of all, he takes the first number that he wrote x₁ and increases its first decimal place by one, and writes b in the first decimal place. 
Therefore he makes two into three and says b = 0.3….. He then says that b is different from x₁.

Then, he makes the second decimal place of x₂ one greater and writes b in the second decimal place. Therefore, he makes 5 into 6 and says that b = 0.36…. 
He then says that b is different than x₂.

Afterward, he raises the third decimal place of x₃ by one and puts b in place of the third decimal place. Therefore, 6 becomes 7, and he writes that b = 0.367…. He then says that the number b is different from x₃.

Cantor continues this pattern and finds a number between (0–1) different from all the numbers he has written before. He then accepts that his assumption is false.

Using the contradiction method, he surmises that real numbers are uncountable because many numbers are left unmatched when one-to-one correspondence is done.

He makes a note in history that real numbers are uncountable.

Cantor published this revolutionary proof in his article called Über eine elementere Frage der Mannigfaltigkeitslehre .’

In less mathematical terms, he would show the world what he had discovered while dealing with his friend’s question, the existence of infinite-element sets with different numbers of elements.

In even simpler terms, he said that “while both natural numbers and real number sets have an infinite number of objects, the real numbers set has more objects in it.”

There are many different kinds of Infinity...In which Infinity would you like to play the game?

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