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.

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

Georg Cantor & Richard Dedekind | Wikipedia
The essence of mathematics lies in its freedom.
Given a set E of [0, 2π], does the convergence of a trigonometric series out of E imply that all coefficients are 0?
Left: Portrait of Georg Cantor by Erin Carmody | Right: Drawing Georg Cantor by Tim Brown

What Is Infinity?

Source: Zeno Paradox 1: Achilles and the Tortoise
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), …
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
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.

How did Cantor make these matches mathematically?

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, …}
How Cantor matched natural numbers and even natural numbers.
Z = {… -3, -2, -1, 0, 1, 2, 3, …}
How Cantor matched natural numbers and integers.
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.
Cantor’s diagonal argument
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…
. . . .
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..

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

Cantor’s paper in 1891 | Über eine elementere Frage der Mannigfaltigkeitslehre
The Blue Screen of Death in Windows 8, Windows 8.1, and Windows 10 (versions 1507–1511), which includes a sad emoticon and an Internet search for quick troubleshooting | Wikipedia

“From the paradise created for us by Cantor, no one will drive us out.”

David Hilbert, 1907 | Source: Wikimedia
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?

Is the set of all sets a set?

Russell in November 1957 | Source: Wikipedia

Hilbert’s Grand Hotel Paradox

Hilbert’s Grand Hotel
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?
Hilbert’s Grand Hotel | Infinitely many doors.
We can conclude this if we add an object to a countable set, that set is still countable.
Hilbert’s Grand Hotel | A bus with and infinite number of people.
Hilbert’s Grand Hotel | Finding room for infinitely many new customers.
Hilbert’s Grand Hotel | Finding room for infinitely many customers.
The Infinite Hotel Paradox — Jeff Dekofsky

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better.

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