Russell's paradox
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Russell's paradox is a paradox in naive set theory that is named after Bertrand Russell, who described the paradox in 1901. German mathematician Ernst Zermelo discovered it in 1898 but did not publish the idea. The paradox highlights some problems that set theory had at the time. When Georg Cantor developed the theory in the 1880s, he already suspected that his theory would lead to such problems.
Consider the set of sets that do not contain themselves. The paradox appears when we consider whether this set should contain itself. If it should, then the set no longer satisfies its condition. If it should not, then it should contain itself. Thus, a contradiction arises. This paradox, stemming from a simple set definition, exposed a flaw in set theory and led mathematicians to introduce restrictions on set creation, as described in axiomatic set theory.
About 20 years later, Russell came up with the Barber paradox, which is a way of describing the same problem in a different way. The Barber paradox says: "Suppose there is a barber who shaves all adult men who do not shave themselves." In this case, answering the question of whether the barber shaves himself is impossible. If he does, then he would not, because he "shaves all men who do not shave themselves." Similarly, if he does not, then he would. This leads to a conundrum.
Informal presentation
[change | change source]Most sets we normaly talk about are not members of themselves.
- We call a set "normal" if it is not a member of itself.
- We call a set "abnormal" if it is a member of itself.
Every set must be either normal or abnormal.
Example 1: The set of all squares on a flat plane. This set is not itself a square, so it is not inside itself. That means it is a normal set.
Example 2: the set of everything that is not a square on the plane. This set is not a square either, so it belongs to itself. That makes it an abnormal set.
Now think about the set of all normal sets and call it: R.
We ask: Is R normal or is it abnormal?
- If R is normal: then it should be a inside the set of all normal sets - wich means inside itself. But that would make it abnormal
- if R is abnormal: then its not inside the set of all normal sets - wich means not inside itself. but that would make it normal
So R would have to be both normal and abnormal, wich would be impossible. This contradiction is called for Russels paradox.