Bertrand Russel Paradox
Set theory was a little confusing at early 1900s. Mathematicians were trying to build the set theory without any paradoxes. Are you wondering what they did? Keep reading!
First of all, we strongly reccomend you to examine the following article beforehand and continue this article to get further information about the paradox:
Later 1800s, Early 1900s
During later 1800s, one of the important discussions in mathematics was whether it was contradictory. In order to prove that it was not contradictory, they had to base this opinion on axioms. Every statement in mathematics had to be either proved or they had to be an axiom. At some point, intuitional statements were culpable of those contradictions. Thus, a paradox appeared. It was really heartbreaking for Frege and Hilbert because this paradox was like proof that mathematics was all about illusion. The paradox was stated by Bertrand Russel and shook the world at that time.
Bertrand Russel
He was a British philosopher, logician, writer, mathematician essayist and social critic. He was best known for his contributions to mathematics in mathematical logic. His work in logic mathematic includes his discovery of Bertrand-Zermelo Paradox, his development in the theory of types, his general theory of logical relations, his championing of logicism.
He was an anti-war activist and he supported anti-imperialism. Because of his pacifism during World War 1, he went to jail. Later, he concluded on the idea of fighting against Hitler was necessary.
Bertrand-Zermelo Paradox
It is also called as Bertrand-Zermelo Paradox because Zermelo had also discovered the problem, in 1902.
Let Y be the set of sets which do not include themselves. For example, the set of real numbers is not a real number so, the set of real numbers is one of the elements of Y. In mathematical language, x∉x ⇔ x∈Y. If we put Y in the sentence, we will have this one: Y∉Y ⇔ Y∈Y. It means that if Y does not include itself then by definition of the set Y, it includes itself. Conversely, if Y include itself, then it can’t be in the set Y, which gives us an obvious contradiction.
What Was The Solution? ZFC
Bertrand-Zermelo paradox had a shocking effect on mathematics by starting the studies for building axioms for set theory. There has been a lot of theories that are a candidate to build up the set theory excluding possible paradoxes such as Bertrand-Zermelo Paradox. Today, the most accepted theory is ZFC theory, Zermelo-Fraenkel Choice theory, which prevents set theory from paradoxes. It prevents it basically by not referring any definition to the words “set” and “to be an element of a set”. It builds the set theory with 8 axioms also it does not allow for the existence of a universal set.
For detailed information about ZFC, you can check out this link.
Resources:
1 Nesin, Ali. Sezgisel Kümeler Kuramı. İstanbul: nesin yayınevi, 2017.
2 Zermelo-Fraenkel Set Theory. Wikipedia. Web. 22.03.2020
3 Bertrand Russel. Wikipedia. Web. 22.03.2020
4 Bertrand Russel. Standford Encylopedia of Philosophy. Web. 23.03.2020
