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Archive for April 23rd, 2022

There are many Infinities: they vary by their magnitude

The Father of Infinity and Modern Mathematics: Georg Cantor

“In mathematics, the art of proposing a question must be held of higher value than solving it.”

This analysis belongs to the undervalued genius of his time, Georg Cantor.

Georg Cantor was affected by discoveries and information, He seldom lost his excitement for anything. Prime examples of this include how one day, when he was in deep thought, he realized that any line segment’s points match all points of three-dimensional space. Realizing this, he immediately sent a note to one of the only people that could correctly understand this, his friend Julies Dedekind, saying:

“Je le vois, mais je ne le crois pas!” Which, when translated, means “I see it, but I don’t believe it.”

When we put it like that, our minds go straight to him texting Dedekind. We have to realize that Steve Jobs was merely a vitamin in any given orange at the time.

We will have done an injustice if we talk so highly of Cantor without speaking of Dedekind. That is because the reason why Cantor was such an abstract thinker was because of Dedekind. Cantor and Dedekind spent many years passing letters to and from each other, speaking deeply of intense mathematical equations and problems.

Georg Cantor & Richard Dedekind | Wikipedia

While many people debate whether Cantor is or was one of the fathers of modern mathematics, I see it fit to name him the father of modern mathematics due to his founding of the fundamentals of math, Set Theory.

Cantor worked his entire life at Halle-Neustadt. 

Today, in front of that university stands a cube-shaped monument dedicated to him. On one of the sides of the monument, we see Cantor. We see written at the top right corner: “‘Georg Cantor mathematician, founder of set theory.” On the left side, we notice Cantor’s famous formula and his renowned diagonal method across from it. Lastly, we see on the bottom left Cantor’s famous saying:

The essence of mathematics lies in its freedom.
Image source 1 | Image source 2 | Image source 3

Cantor spent almost his entire life thinking and making discoveries. The more new things he discovered, the more enchanted his intrigue in mathematics became.

He never ceased to rock the world of mathematics with his brilliant discoveries. One day, when Cantor was hanging out in his room, his colleague Eduard Heine asked Cantor the question that would change the known fundamentals of mathematics entirely. These sorts of things always start with questions that keep us up at night anyway. 

The question was:

Given a set E of [0, 2π], does the convergence of a trigonometric series out of E imply that all coefficients are 0?

While in deep thought regarding this question, Cantor comes across an incredible discovery. Rational numbers cannot be matched with irrational ones. 

The matching of two infinite sets must mean that the magnitude of their infinities is different.

Cantor had discovered that what people thought to be one infinity for thousands of years was more than one. Cantor would publish every one of his ideas as articles to explain them mathematically from that point on.

As you will read in a little bit in detail, Cantor’s sayings were a whole new approach to mathematics, and while they were supported by proof, many would see them be dangerous.

If what Cantor said was accurate, then the entirety of mathematics would have to be redefined.

Starting with Leopold Kronecker and Henri Poincaré, many mathematicians argued fiercely against Cantor’s ideas and instead held Aristotle’s infinity concept to be true over Cantor’s.

After a while, Poincaré and his close friends started to throw insults at Cantor. The things he went through would regularly put Cantor, whose psychological health was already bad, into the hospital, halting his work.

At the time, to make ends meet, Cantor would apply to work for a different university in Berlin, but due to Leopold’s rejection of Cantor’s ideas, he would ultimately get rejected there as well.

In my opinion, the reason behind Cantor and his ideas being cast out was jealousy. We can’t say that people of genius status cannot understand a clear idea that has proof supporting it. Maybe the level Cantor achieved greatly surpassed the imagination limits of some. What has plagued humanity for so long, jealousy, was the reason why Cantor and his ideas were rejected.

Left: Portrait of Georg Cantor by Erin Carmody | Right: Drawing Georg Cantor by Tim Brown

Another reason for the mass rejection of Cantor was that the Church had anathematized him.

While Cantor was one of the prime counterstereotypes of the belief that “scientists are nonbelievers,” the Church still ostracized him.

Cantor was a genius and a religious mathematician. He saw and wrote about similarities between mathematical infinity and divine infinity for a large part of his life, especially towards its end.

Cantor’s ideas had made the Church uncomfortable, however. When Georg Cantor had put forth the idea that “infinities are also infinite,” Christian theologists believed that this was in direct opposition to the belief that “the only infinite is God.” They put forth that Cantor’s ideas of infinity were linked to pandeism.

What Is Infinity?

What was the concept that Cantor furiously hunted down, that he racked thousands of times through his brain, that led so many to pursue him in increasingly dishonorable ways? And what became of it anyway?

For someone in our day and age, whether that be a kid with limitless imagination or an average adult, the concept of infinity is equal to a very large amount.

To who is it a large amount? What is it a large amount compared to? For example, $1000 is a large amount for me, but $1000 is not much for someone like Jeff Bezos, who doesn’t pay a penny in taxes. 

The world is enormous compared to the neighborhood we live in, but it is tiny compared to the Space in which the world resides. 

According to the Big Bang theory, however, Space is finite and continuously expanding. Something that is expanding must have an end. In the grand scheme of things, our inability to reach the end of something doesn’t make it infinite. I can go from America to Turkey, but neither my lifespan nor my fuel will be enough to go from one planet to another.

Therefore, saying that a specific planet is infinitely far from me is nothing but an exaggeration, and it is not a grounded saying. We can never reach infinity.

Zeno of Elea was the first to introduce a worthwhile idea regarding infinity. While thinking about the concept of infinity, he continued down the famous Achilles and Turtle paradox and put forth that humans are unable to move. That is because, according to Zeno, every time Achilles reached the Turtle, it would move a little further, and this would repeat indefinitely.

Although it sounds very reasonable at first, overthinking Zeno’s idea fries one’s brain. How can it be possible to add infinite numbers? Fortunately, a mathematician who lived around 100 years before Cantor, Leonard Euler, showed us how to add infinite series.

In the end, by figuring out that the addition of an infinite series equates to a finite solution, he solved Zeno’s paradox.

Source: Zeno Paradox 1: Achilles and the Tortoise

Later, Aristotle would propose what many thinkers accept to this day:

“Infinity is like the horizon line; it is something that does not exist that we use for the ease of understanding. We use this concept in place of “no boundaries.” If something has the potential to grow greater than a predetermined magnitude or size, we say that it is going on forever.”

That is why to escape this uncertainty, the concept of infinity had to be defined. What better tool to use to describe it than math?

It is the only one that can describe such vast concepts without leaving question marks. In other words, the idea of infinity can only be defined based on countability presented by Cantor in 1884.

How did Cantor count infinity, however? I will try to explain very rudimentarily.

It is widely accepted that N = {0, 1, 2, 3, …} represents the natural numbers set. Cantor first added an “infinite number” to the end of 0, 1, 2, 3, … and represented it with ω (omega):
0, 1, 2, 3, …, ω 
However, Cantor did not stop here and continued to add numbers:
1, 2, 3, …, ω, ω +1, ω +2, ω +3, …
He continued adding numbers as such until 2ω: 
1, 2, 3, …, ω, …, 2ω.
Cantor realized that he could continue adding numbers in this fashion and reached the numbers below:
1 , 2 , 3 , … , ω, …, 2ω, , 2ω+1 , 2ω+2 , 2ω+3 , …
1 , 2 , 3 , … , ω, …, 2ω, …, 3ω, …, 4ω, …, 5ω, …
1 , 2 , 3 , … , ω, …, 2ω, …, ω², …, ω³, …, ω⁴, …, ω⁵, …
1 , 2 , 3 , … , ω, …, 2ω, …, ω², …, ω³, …, …, ω^ω
1 , 2 , 3 , … , ω, …, 2ω, …, ω², …, ω³, …, ω^ω, …, ω^ωω, …,
1 , 2 , 3 , … , ω, …, 2ω, …, ω², …, ω^ω, …, (ω^ω)^(ω)^(ω), …

After a while, Cantor started to think that the concept of infinity didn’t mean anything by itself.

After all, infinity was just the opposite of finite. The idea of finite didn’t have much meaning either; it was just the opposite of infinity. 

That is why Cantor defined “infinity” with a correlation to the “sets” concept. Until then, sets were finite things made up of objects, and Cantor decided to objectify infinity using sets.

Georg Cantor first had to define the concept of sets, and he decided to approach the problem with pure mathematical seriousness.

In the article he published in 1874, he described sets as follows:

A set is any collection into a whole of definite and separate objects of our intuition or of our thought.For instance:{x: x is an odd positive integer}
{x: x is a prime number less than 9,999}
Passage of Georg Cantor’s article with his set definition | Source: Wikipedia

Simply put, a set is a collection of objects.

While in mathematics, when we say “objects,” we think of numbers or sets, according to Cantor, for an object to be defined as a set, it didn’t have to meet specific requirements, and any object that came to mind could make a set.

After defining the “set,” Cantor started brainstorming what it meant for two “sets” to be identical in size. He then discovered one of his fundamental ideas, the one-to-one correspondence idea.

Using this method, Cantor would prove whether or not two sets had the same amount of objects. Cantor’s logic was very straightforward.

Two people would scratch a line on the wall for every animal they hunted. Cantor used the same approach, matching the concept of infinity, like real numbers, with other infinite sets. The technique used to understand infinity that Cantor used is this one-to-one correspondence technique.

According to Cantor, if we can match every object in set A with all objects in set B, and both sets do not have any unmatched objects, they are of equal size. A simple example of this would be matching the fingers on our left hand with the ones on our right hand.

One-to-one correspondence is very different from counting.

When we talk about the objects in each of the sets, we do not count them one by one but instead match them. Instead of saying these two sets have this number of objects, we say they have an equal number of objects.

Cantor’s most brilliant idea was not only using this one-to-one correspondence technique for finite sets but also for infinite sets.

After this method was introduced, sets would be divided into finite and infinite ones, and infinite ones by their magnitude as well, which means that there would now be more than one infinite set, each of them unique.

For example, Cantor would show us that the infinity of natural numbers (N) is equal to the infinity of rational numbers (Q) and that the infinity of real numbers (R) is greater than the infinity of natural numbers (N). At the end of the day, Cantor would do what no one else had done and prove that infinity was Not singular.

How did Cantor make these matches mathematically?

Let us first ponder on the natural numbers set.

Natural numbers go like 1, 2, 3, 4, 5, 6, … and we assume that they go on forever. According to Cantor, the natural numbers set is countable, and therefore if we can match another set with it, that set is also countable. 

In my opinion, this approach is one of the most significant mathematical events in the world.

First of all, let’s try to match the natural numbers set with the double natural numbers set.

Let N represent the natural numbers set. N= {1, 2, 3, 4, 5, 6, 7, …}
Let E represent the double natural numbers set. E= {2, 4, 6, 8, …}

As you can see below, all N and E elements can be matched with each other using the rule: n → 2n.

How Cantor matched natural numbers and even natural numbers.

Therefore, we can say that the two sets have an equal number of objects. While it may seem counterintuitive that we can do this, knowing that there are odd numbers within the natural numbers set. However, as we can still clearly match them, there is no paradox to solve.

Using the same method, we can also match all natural numbers with integers. This time Z will represent integers.

Z = {… -3, -2, -1, 0, 1, 2, 3, …}

If we match 1 with 0, we see that all numbers after this can be matched with first a positive and then a negative number. We see that these two sets have a one-to-one correspondence.

How Cantor matched natural numbers and integers.

If you notice, we have only been pairing two sets that do not have spaces in between them up to now. That means that we know there are no spaces between two consecutive numbers in the set of natural numbers and integers. For example, there are no natural numbers between 1 and 2.

Will this method work in the rational and irrational sets in which there are infinite elements in between numbers?

This question is intriguing because rational numbers are fascinating. For example, we can put any number of objects (numbers) between 1 and 2. However, the simplest method is that the average of the two numbers will be somewhere in the middle of them.

There is 3/2 exactly in between 1 and 2
There is 5/4 exactly in between 1 and 3/2
There is 9/8 exactly in between 1 and 5/4
There is 17/16 exactly in between 1 and 9/8.

As you may have guessed, we can continue to do this operation indefinitely. Of course, only in our minds, because our life span is limited. You can read my article about this topic by visiting the article below.




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