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I am reading the first chapter of Aliprantis and Border's Hitchhiker guide and there is a passage on Russel's paradox that has me confused. It reads:

Russell's Paradox is a clever argument devised by Bertrand Russell as an attack on the validity of the proof of the Diagonal Theorem. It goes like this. Let $S$ be the set of all sets, and let $\varphi:\mathscr{S}\rightarrow\mathscr{S}$ be defined by $\varphi(A)=\{B \in \mathscr{S}: B \in A\}$ for every $A \in \mathscr{S} $. Since $\varphi(A)$ is just the set of members of $A$, we have $\varphi(A)=A$.

However, if we take $A=\{1,2\}$ then $\varphi(A)=\emptyset$ since $A$ does not contain any sets, which are the elements of $\mathscr S$. Am I missing something? Should $\varphi$ be defined from $2^{\mathscr S}$ to $2^{\mathscr S}$?

Thanks!

Arctic Char
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Condor5
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  • Your quote says "for every $A \in \mathscr{S}$" but your example for $A$ is not an element of $\mathscr{S}$ – Henry Jun 14 '21 at 17:53
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    In set theory $1={\varnothing}$ and $2={\varnothing,{\varnothing}}$. – Maxim Nikitin Jun 14 '21 at 17:54
  • @MaximNikitin That's true, but the bigger point is that sets only contain other sets. They can't contain "objects" that aren't sets. The natural numbers are nicknames for sets, and you've presented the von Neumann encoding. – Rushabh Mehta Jun 14 '21 at 18:17

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