Cantor's Naive Set Theory and Russell's Paradox

Cantor's Naive Set Theory

Analysis of many proofs in Cantor's set theory shows that almost all theorems he proved can be derived from the following three axioms:

1. Extension axiom: any two sets are equal if and only if all elements in them are the same
2. Abstract axiom: any given property has a set of objects that satisfy the property
3. Choice axiom: every set has a choice function

The problem arises in abstract axioms, Russell discovered: "a collection of all objects of the nature of elements that do not belong to itself"

Russell's Paradox

Russell constructed a set S: S consists of all sets that do not belong to itself. Then Russell asked: Does S belong to S?
$$S=\{A|Aissettogether,andA\notin A\}$$
According to the law of excluded middle, an element either belongs to a certain set or does not belong to a certain set. Therefore, for a given set, it makes sense to ask whether it belongs to itself. But the answer to this seemingly reasonable question will be caught in a dilemma. If S belongs to S, according to the definition of S, S does not belong to S; conversely, if S does not belong to S, S belongs to S according to the same definition. It is contradictory anyway.

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