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.
One of the significant losses in this day and age is how unexcited we are about everything. Many people seem not to be affected by discoveries and information, which is one thing that Georg Cantor was very diligent in avoiding. 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 at the time, Steve Jobs was merely vitamin in any given orange.
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.
While many people debate over if 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 across from it, his renowned diagonal method. Lastly, we see on the bottom left Cantor’s famous saying:
The essence of mathematics lies in its freedom.
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 for thousands of years people thought to be one infinity was more than one. You see, everything started from there. 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 for mathematics, and while they were supported by proof, many would see them to 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, who’s psychological health was already bad, into the hospital, halting his working. 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. That is why the single thing that has plagued humanity for so long, jealousy, was the reason why Cantor and his ideas were rejected. Wasn’t it jealousy that led to the first murder anyway?
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 mathemetician. 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. 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 brains. How can it be possible to add infinite numbers? Fortunately, a mathemetician who lived around 100 years before Cantor, named 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.
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 the concept 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 concept 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, …, ωHowever, 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), …
However, 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. Up until then, sets were finite things made up of objects. Cantor decided to objectify infinity using sets.
Georg Cantor first had to define the concept of sets. 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}
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. 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 actually 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.
However, 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. As you know, 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 up 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.
Therefore, we can say that the two sets have an equal number of objects. While it may seem to be counterintuitive that we can do this, knowing that within the natural numbers set there are odd numbers. However, as we can still clearly match them, there is no paradox to solve.
Using the same method, we can match all natural numbers with integers as well. 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.
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) in 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. It is because our life span is limited. You can read my article about this topic by visiting the article below.
You may think that we are unable to match natural numbers to rational numbers. 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. In the first row and column, we wrote natural numbers going to infinity. 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 the denominators are getting larger and larger. In the next row, we followed the same process except with two as the numerator. 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.
As you can see, we can match all natural numbers to positive rational numbers. If we wanted to, we could use this logic to match all rational numbers to integers as well. Therefore, we can surmise that rational numbers are countable.
What about real numbers, however? Sadly, real numbers are not countable. Cantor has provided a 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. That means that 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 that 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 that 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 it’s 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 that 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.”
This finding of Cantor’s, the representation of infinity as a number, caused an earthquake effect in the deep philosophy world. That is because every revolution creates chaos in the established order. The first reaction of those who heard of Cantor’s finding must have been ‘Jesus Christ.’ For example, Tobias Dantzig wrote, “Cantor’s proof of this theorem is a triumph of human ingenuity.” in his book ‘Number, The Language of Science’ about Cantor’s “algebraic numbers are also countable” theory.
Don’t be upset if you didn’t understand what Cantor said above. That is because, in the time of Georg Cantor, many of the leading mathematicians also didn’t understand him and, for lack of a better example, blue screen of death as Microsoft Windows does when it fries its circuits. Even further, it caused some mathematicians to panic and say that “mathematics is getting out of hand.” Almost all mathematicians had a defensive reflex to the incoming modern mathematics and outright refused Cantor’s findings.
What they missed, however, was that Cantor and his ideas were of utmost importance. They refused to understand this, however. On the other hand, the genius of that time, Hilbert, was among the ones who understood him. That is because he had an incredible knack for seeing the future. In the year 1900, at a conference in Paris, Hilbert asked 23 questions and chose a question about Cantor’s continuity hypothesis to be his first. Hilbert was an amazing man and would lastly say,
“From the paradise created for us by Cantor, no one will drive us out.”
Here is the list of Hilbert's questions:Problem 1 — Cantor’s problem of the cardinal number of the continuum.
Problem 2 — The compatibility of the arithmetic axioms.
Problem 3 — The equality of two volumes of two tetrahedra of equal bases and equal altitudes.
Problem 4 — Problem of the straight line as the shortest distance between two points.
Problem 5 — Lie’s concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. (i.e., are continuous groups automatically differential groups?)
Problem 6 — Mathematical treatment of the axioms of physics.
Problem 7 — Irrationality and transcendence of certain numbers.
Problem 8 — Problems (with the distribution) of prime numbers.
Problem 9 — Proof of the most general law of reciprocity in any number field.
Problem 10 — Determination of the solvability of a diophantine equation.
Problem 11 — Quadratic forms with any algebraic numerical coefficients.
Problem 12 — Extension of Kronecker’s theorem on abelian fields.
Problem 13 — Impossibility of the solution of the general equation of the 7th degree.
Problem 14 — Proof of the finiteness of certain complete systems of functions.
Problem 15 — Rigorous foundation of Schubert’s calculus.
Problem 16 — Problem of the topology of algebraic curves and surfaces.
Problem 17 — Expression of definite forms by squares.
Problem 18 — Building space from congruent polyhedra.
Problem 19 — Are the solutions of regular problems in the calculus of variations always necessarily analytic?
Problem 20 — The general problem of boundary curves.
Problem 21 — Proof of the existence of linear differential equations having a prescribed monodromic group.
Problem 22 — Uniformization of analytic relations by means of automorphic functions.
Problem 23 — Further development of the methods of the calculus of variations.
Is the set of all sets a set?
Right after Cantor divided the mathematics world, Bertrand Russell used Cantor’s set concept to ask this question:
Is the set of all sets a set?
This question divided the mathematics world one more time. In order to get rid of Russell’s paradox, they should give up from set theory for many of the mathematicians.
As I talked about above, at first, according to Cantor’s identification of a set, for an object to be a set, it didn’t have to meet any requirements. Any object that came to one’s mind could be considered a set. Cantor didn’t know that when he put forth his theory of sets, it would cause discord in the mathematics world. For a while, mathematicians considered all objects to be sets. That is until Bertrand Russell asked that inevitable question, Is the set of all sets a set?
Good thing Bertrand Russell found a way to solve the paradox that he put forth. In 1908, Russell put fourth the “type theory,” ranking all sets. For example, to identify a set in the third level, you would need to use sets from the first two levels. With this rule, Russell’s set of all sets would not be considered a set. This theory has been a difficult theory to work for mathematicians, however. That is why they would later change this theory with one that was simpler.
Cantor had brought forth many other problems, and therefore the mathematics world had gone into a deep-running crisis. Years later, that period would be coined “The Crisis in the Foundations of Mathematics.” The only way to surface from the crisis was to throw out the classical methods of mathematics. And that is precisely what happened. Mathematicians decided that the “set theory” was axiomatic and started to build mathematics on top of it. Right after that, they mathematicized the definition of infinity.
I would like to include a very intriguing detail here. While the development of analysis and geometry took hundreds of years, thanks to the genius of Cantor, what was coined as modern mathematics, the set theory, only took a few years to develop. That is why the thing that Cantor presents to humanity is of utmost importance.
Hilbert’s Grand Hotel Paradox
After Cantor’s death, in his lecture named “Über das Unendliche” in 1924, David Hilbert asked his famous hotel question. This question would be called in literature as the Infinite Hotel Paradox or Hilbert’s Grand Hotel. Hilbert’s problem would be popularized by George Gamow in his book One Two Three … Infinity. The infinite hotel paradox that most people have a hard time grasping can be explained using the above mentioned transfinite theory by Cantor.
Hilbert’s question was as follows:
A hotel has a countable number of infinite rooms, and each room is occupied. One night when a new customer approaches, the hotel manager has to find a way not to lose the customer. But how?
For this instance, let’s assume that Georg Cantor is the hotel manager. Cantor would find a room for the customer using a straightforward method. He would request that all the customers move to the next room. Meaning the customer a room number 1 would go to number 2, 2 would go to 3, and the customer at room number n would go to n + 1. Therefore, room number 1 would be left empty.
We can conclude this if we add an object to a countable set, that set is still countable.The next night, while all the rooms are full, a bus full of 100 people approaches. Cantor quickly asks all of his customers to move 100 rooms over. Meaning the person in number 1 moves to 101, the person in 2 moves to 102, the person in 102 moves to 202, and so on. Therefore, the first 100 rooms have been emptied.
That means that if Cantor receives a countable number of customers, he will be able to find a solution to the problem.
Something exciting happens the next night, however. A bus with an infinite number of people approaches.
Without a problem, Cantor matches the infinite number of customers as he did prior, which means that he asks all of his customers to move over double their room number. Therefore, all odd-numbered rooms are emptied, and he can host another infinite number of new customers.
Something exciting happens later, however. An infinite number of buses holding an infinite number of people approach the hotel. After some thinking, Cantor pictures the hotel as a pyramid.
That means that in the first row, there is only 1. In the second row, there are 2 and 3. In the third row, there are 4, 5, and 6. He continues this forever. Cantor starts painting the doors of the rooms according to his drawing.
Then Cantor moves all of the customers present in the hotel to the yellow door rooms. Afterward, he puts the customers of the first of the infinite buses in the red door rooms. The second of the infinite buses of customers go to the now emptied blue door rooms. Therefore, if Cantor has a countable infinite number of customers, he can place another countable infinite number of new customers in their rooms.
Therefore, our hotel manager has effectively solved Hilbert’s hotel paradox.
You can also watch Jeff Dekofsky’s video about Hilbert’s hotel paradox!
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