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- Oct 2024
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Thus if P isthe set of all sets, we can apparently form the set Q = {Ae P| A ¢ A}, leading tothe contradictory Oe Q iff O¢€ Q. This is Russell’s paradox (see Exercise 1A)and can be avoided (in our naive discussion) by agreeing that no aggregate shallbe a set which would be an element of itself.
Russell's paradox (1901) in set theory can be stated as:
If $$P$$ is the set of all sets, one can form the set $$Q = {A \in P | A \notin A}$$ which can lead to the contradiction $$Q \in Q$$ iff $$Q \notin Q$$.
This can be done by dividing P into two non-empty subsets, $$P_1 = {A \in p | A \notin A}$$ and $$P_2={A \in P | A \in A}$$. We then have the contradiction $$P_1 \in P_1$$ iff $$P_1 \notin P_1$$.
The paradox happens when we allow as sets A for which $$A \in A$$. It can be remedied by agreeing that no collection can be a set which would be an element of itself.
Relation to Groucho Marx's quote (earliest 1949) about resigning membership of a club which would have him as a member: https://hypothes.is/a/3_zAfITjEe-H5-PlfOlK8A
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The basis for our intuitive set theory is the Zermelo—Fraenkel set theory developedby Zermelo (Untersuchungen tiber die Grundlagen der Mengenlehre J) andstrengthened by Fraenkel (Zu den Grundlagen der Cantor—Zermeloschen Mengen-lehre). Their work rests on the researches of Cantor in the 1870’s which first putmathematics firmly on a set-theoretic base. Zermelo’s work, in particular, was adirect response to the Russell paradox.
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