Set theory

Set Theory and Georg Cantor

Georg Ferdinand Ludwig Phillipp Cantor, or Georg Cantor, was one of the groundbreaking mathematicians to approach the concept of infinity. He worked intensively with set theory, working with the cardinality of sets, one-to-one correspondence, transcendental numbers, and different types of infinity. Over the course of the study, we shall take a journey through Cantor’s life, works, and arguments. First, Richard Dedikind proposed the proposition of infinity. He, instead of constructing it, began to recognize it, avoiding an argument made by Gauss:

I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a Faҫon de parler, the real meaning being a limit which certain ratios have approached indefinitely near, while others are permitted to increase without restriction.

Georg Ferdinand Ludwig Phillipp Cantor was born in 1845 in Saint Petersburg, Russia. He was a talented violinist having inherited skills from his father and mother. His father worked in the Saint Petersburg stock exchange. Cantor lived in Russia until he turned eleven. He got sick that year and the family moved to Germany to experience warmer winters. Cantor graduated from Darmstadt in 1860; in 1862, he was enrolled in the Federal Polytechnic Institution in Zurich. When his father died, he received an inheritance that enabled him to attend the university ofg Berlin in 1862. He received his PhD in 1867 for his math paper on number theory.

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Cantor first began teaching at a girls’ school. He then moved to the University of Halle where he would be promoted to Extraordinary Professor in 1872 and full professor in 1879. He achieved this status at the young age of 32. Unsatisfied, he wanted to pursue a better job. But his colleague, Leopold Kronecker fundamentally disagreed with Cantor’s studies. He believed it was incorrect to propose a set with certain qualities without giving certain examples.

Georg Cantor suffered from his first bout with depression in 1884. Because of this he took a break from math and began to teach philosophy. He did begin to work with math again, but it was not of the same caliber as before. He tried to reconcile with Kronecker who enthusiastically accepted, but their views on mathematics and philosophy still opposed each other. Many people suggest that because of this conflict Cantor was depressed, but others think it was a cause of his bipolarity.

Cantor retired from mathematics in 1913 and suffered from poverty because of WW1. He died on January 1918 in the asylum where he spent his final years.

As a mathematician, Cantor contributed many things to the mathematical field. H developed Set theory. He developed countability, denumerability, and 1-to-1 correspondences between sets. He was the first mathematician to theorize different sizes of infinity. Back then infinity was more of a philosophical topic rather than a mathematical topic. Plus, he received a lot of criticism from Leopold Kronecker.

So how is a set defined? Cantor defined a set as, “a collection into a whole, of definite, well-distinguished objects (called the elements of the set) of our perception or of our thought”. For example, every even number from 1 to 100 can be considered a set. Every prime number from 1 to 1000 can be considered a set. Even the amount of vegetables in the world can be considered a set. A set is just a group.

In a set, order is not important, for the sets {1,2,3,4,5} and the sets {4,5,2,3,1} are considered equal. To write that set L is equal to set H, you could write L=H. For that to be true, all the elements in set L have to be in set H, and the elements would all have to be equal. If set L contained {1,2,3}, the set H must contain {1,2,3}. However, if L has only some of the elements of H, we call L a subset of H. To show that something is an element of L, we use the symbol “ϵ”. If mϵL, it represents “m is an element of set L”. To represent unions between sets, we use.

L M means the union of sets L and M. We use the symbol when describing an intersection between sets. We use this notation when trying to find an element between two sets. To get a better representation of the use, let O be the set of odd integers from 1-10 and let P be the set of prime integers from 1-10. When we see O P, the elements of that intersection would be {3, 5, 7}. If we make a union between the sets, the elements of the union would be {1, 2, 3, 5, 7, 9}. You can think of union and intersections in the form of a Venn diagram. An intersection would be only the area where the circles intersect. A union would be the entire thing: the middle and the sides. Other important facts about set theory are cardinality and ordinal numbers. The cardinal number of a set represents the amount of elements in a set. An elements ordinal number shows where the number is in a sequence. Sometimes in well-ordered sets you can have each element with its ordinal number.

Cantor developed the term enumerability. When a set is enumerable, it means that is cardinal number is the same size as the natural numbers or is the same size as a subset of the natural numbers. In a countable set, there exists an injective function. An injective function is when you can associate distinct values with distinct arguments. This is also referred to as a 1-to-1 function. In addition to injective functions, there is a surjective function where for the function f(x)= y, there exists more than one x value to one y value. Bijection is when for f(x) = y between sets, there exists one and only one value of y. a bijective function is different from an injective function because in an injective function, you can map all them elements from set A to set B with some elements in B left over when with a bijective function all the elements in set A must map over to set B with only one corresponding element.

So where does this all tie into Cantor’s work? Well, to start off he was the first one to actually work with set theory. Through his work, he was able to prove that the set of odd integers is equal to the set of integers overall. For this proof, let us assume that the amount of even integers is equal to the amount of odd integers. Now, people will think, “But aren’t the odd integers a subset of the integers?” True, but subsets can have the same cardinality as the whole set. The way Cantor proved this was through proving the odd integers equal to the number of integers with a bijective function: f (x) = y = 2x+1, where x is an element of the entire set of integers. This way, -3 would go to –5, -2 would go to -3, -1 would go to -1, and 0 would go to -1. Through this, Cantor made a groundbreaking discovery. It would lead on to understanding different kinds of infinity.

Cantor came up with two great theorems. The first one, Cantor’s Theorem showed that the power set of a set is larger than the set itself. A power set contains all the subsets of a set. Consider a set whose elements are {1, 2}. The power set of this set would be {{}, {1}, {2}, {1, 2}}. The cardinality of this power set is 4. 4 is greater than two. As we described before, we showed that two sets have the same cardinality if they have the same number of elements and there exists a 1 to 1 correspondence. He proved his theorem by finding a subset, B, that was not in A. Consider a set, A, and its power set P(A). The subset B would be represented by:

F(x) is a general bijective function that maps the elements of set A’s power set to the elements of set A. This shows that for any element x of A, x is an element of B if and only if x does not equal f(x). But then that would mean x is an element of B where x isn’t an element of f(x) and then x is not an element of B? Impossible!

One of the most famous proofs of set theory was the diagonal proof by Cantor. He applied it to show that the real numbers were more numerous than the naturals, therefore proving the existence of uncountable sets. To prove it, we will use contradiction. Consider a list of the real numbers that could be put into a 1-to-1 correspondence with the naturals.

1. 1 .5657678…

   2 .3364625…

   3 .2425364…

   4 .3544657…

   5 .3535465…

   6 .1324354…

   7 .2000000…

Because of their 1 to one correspondence, should we try to construct another element in the list of real numbers, it would already be accounted for. But what the diagonal argument did was it took the first digit of the fist element, the second digit of the second element and so on and so on, all the way to the nth digit and added one to each individual digit mod ten. What would happen is we would add one to the first digit 5 mod ten and get six. Then we would add 1 to the second digit 3 mod ten and get 4. The pattern of numbers follows a diagonal formation, such as the numbers highlighted below.

1. 1 .5657678…

   2 .3364625…

   3 .2425364…

   4 .3544657…

   5 .3535465…

   6 .1324354…

   7 .2000000…

The digits we would get are 6, , 3, 5, 5, 6, and 1. From these digits, we make a decimal with each digit in the spot respective to the element they were taken from. For example, 6 would be the first digit because it was taken from the 1st element. 4 would be the next one for it was taken from the second element, and so on and so on. Following that pattern, we would construct the number .6435561…. This beauty of this proof is we have just constructed a number that isn’t part of the list! Why? For example, if we looked at the mth digit of this new number and the mth digit of the mth element of the list, we would see that they differ by that one number, thereby having created a new number. What we have done here is just made a way to make an infinite list strictly larger than the naturals therefore proving the existence of uncountable sets. What makes this proof so much more amazing is that there are so many ways to represent it. I used decimals to represent it. However, other people might use two variables and just switch them when changing by one. Other people might only use 0 and 1. Cantor’s work became an important part of other mathematician’s work. It became an important part in Russell’s Paradox, Godel’s Incompleteness theorem, and Turing’s Entscheidungsproblem (German for “decision problem”)

Through Cantor’s groundbreaking work, mathematicians were finally able to approach the concept of infinity. No longer was the topic reserved for the philosophers. Infinity could be used as a mathematical field.