There is actually a pretty good illustration of it known as the Barber Paradox the Russell Paradox states that “A set which contains all sets that does not contain itself must contain itself.” The illustration uses the following example:
“If a barber is a man who shaves all men who do not shave themselves and only men who do not shave themselves, then who shaves the barber?” If the barber shaves himself, then he is not one who shaves “only men who do not shave themselves” but if he does not then he is not “one who shaves all men who do not shave themselves.”
Russell’s Paradox is like asking if a book that lists books not about themselves should include itself in the list. If it does, it shouldn’t; if it doesn’t, it should. It shows that some rules can lead to contradictions.
Mathematicians like proofs. Proofs should be rigorous, based on axioms and rules of logic. But sometimes mathematicians still use some intuitions. One of these intuitions was that we can always define a set of things by saying what are the properties of its elements. So a matematician would say “a set of all even numbers” as a part of their proof and they would not think if they really have a basis to say stuff like this.
Russel showed that you can’t use this intuitive approach. He postulated “a set that contains all sets that don’t contain itself”. This definition of a set is internally contradictory. This shows that you can’t just define any set and be sure that this definition makes sense. So even this basic concept of “set” has to be rooted in an axiomatic system.
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There is actually a pretty good illustration of it known as the Barber Paradox the Russell Paradox states that “A set which contains all sets that does not contain itself must contain itself.” The illustration uses the following example:
“If a barber is a man who shaves all men who do not shave themselves and only men who do not shave themselves, then who shaves the barber?” If the barber shaves himself, then he is not one who shaves “only men who do not shave themselves” but if he does not then he is not “one who shaves all men who do not shave themselves.”
Russell’s Paradox is like asking if a book that lists books not about themselves should include itself in the list. If it does, it shouldn’t; if it doesn’t, it should. It shows that some rules can lead to contradictions.
Mathematicians like proofs. Proofs should be rigorous, based on axioms and rules of logic. But sometimes mathematicians still use some intuitions. One of these intuitions was that we can always define a set of things by saying what are the properties of its elements. So a matematician would say “a set of all even numbers” as a part of their proof and they would not think if they really have a basis to say stuff like this.
Russel showed that you can’t use this intuitive approach. He postulated “a set that contains all sets that don’t contain itself”. This definition of a set is internally contradictory. This shows that you can’t just define any set and be sure that this definition makes sense. So even this basic concept of “set” has to be rooted in an axiomatic system.