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In Jech's book, one of the very first exercises at the end of the first chapter is to show that there is no set $X$ that

$\mathscr{P}(X) \subseteq X$

With the axiom of regularity it's extremely easy, since $X \in \mathscr{P}(X)$ and therefore $X \in X$.

But it's so early he's hardly mentioned that axiom at this point, saving it for a later chapter. So I wonder if I'm missing something and that there's another reason this can't happen?

If there were some such set $X$, then it would have to contain $\varnothing$ and itself, and then also {$\varnothing, X$}, and then {$\varnothing, X,$ {$\varnothing, X$}}, and more in the same way.

It seems pretty clear no finite set could satisfy this, and I have no clue how to show a set like {$\varnothing, X,$ {$\varnothing, X$}$,$ $...$} exists, but I can't see how to show it doesn't exist either (without Regularity).

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No. Axiom of regularity does not need to prove this result. –  tetori Aug 11 '13 at 10:00
It can be shown provided you know there is no injection from the power set of X to X, since an inclusion is a specific kind of injection. –  coffeemath Aug 11 '13 at 10:01

3 Answers 3

up vote 3 down vote accepted

This seems to call for Cantor's diagonal argument: There is no surjective map $X\to \mathscr P(X)$, but there is a shortcut not even using the concept of map.

Using the Axiom Schema of Separation, we define $$S=\{\,x\in X\mid x\notin x\,\}.$$ In other words, we have $$ \forall x\colon(x\in S\leftrightarrow (x\in X\land x\notin x)),$$ hence specifically $$ S\in S\leftrightarrow (S\in X\land S\notin S).$$ By mere first order logic (namely: $p\leftrightarrow (q\land \neg p)$ is equivalent to $\neg p\land\neg q$, hence implies $\neg q$) we infer $S\notin X$. Since $S\in\mathscr P(X)$ by definition, this shows $\mathscr P(X)\not\subseteq X$.

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+1 for cuteness! –  Peter Smith Aug 11 '13 at 14:56

Nonexistence of this set can be shown that using the Cantor's theorem:

Cantor's theorem. There is no injection between $\mathcal{P}(X)$ to $X$.

and Cantor's theorem can be shown that without the axiom of regularity. Here is the proof:

Proof the Cantor's theorem. Let $r$ is the injective function between $\mathcal{P}(X)$ to $X$. Let we define $$R=\{x\in \mathcal{P}(X) \mid r(x)\notin x\}$$ then $R$ ($\in\mathcal{P}(X)$) is the set by axiom schema of specification and $R\in R$ or $R\notin R$ must be hold. If $R\in R$, then $R\notin R$ by definition of $R$. And if $R\notin R$ then $R\in R$ because $R$ satisfies $R\notin R$. So we get a contradiction. So there is no injective function between $\mathcal{P}(X)$ to $X$.

Proof of there is no set satisfying $\mathcal{P}(X)\subseteq X$. If it exists, then $i:\mathcal{P}(X)\to X$, $i(x)=x$ is injective. It contradicts with the Cantor's theorem.

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First of all, you are right it isn't easy to show that the set $S_X= \{\varnothing, X, \{\varnothing, X\}, ...\}$ exists or doesn't exist. In fact taking $X=\{\varnothing\}$ the existence of $S_X$ is the "Axiom of Infinity" and is known to be independent from all other axioms of ZFC. However, the existence of such a set, doesn't give you a counterexample.
Now to answer your question: If $\mathscr{P}(X) \subseteq X$, then there is a surjection $f:X \to \mathscr{P}(X)$ defined by $x \to x$ if $x \in \mathscr{P}(X)$ and $x \to \varnothing$ otherwise. Let $A=\{x\in X: x \notin f(x)\}$. Since $A \in \mathscr{P}(X)$, $f(A) = A$. By definition of $A$ we have $A \in A$ if and only if $A \notin A$ giving the desired contradiction.

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