When is $ A \subseteq \mathcal{P} (A)$?

I have a problem determining when is $ A \subseteq \mathcal{P} (A)$ and why, also I can't answer why and when is the other way around : $ \mathcal{P} A \subseteq A$

Can you please help me answer?

Solution 1:

Recall that $A$ is a transitive set when every $B\in A$ satisfies $B\subseteq A$. In particular, it means that all the elements of $A$ are themselves sets.

Proposition. $A$ is transitive if and only if $A\subseteq\mathcal P(A)$.

Proof. Suppose that $A$ is transitive, then if $B\in A$, then $B\subseteq A$, and therefore $B\in\mathcal P(A)$. Therefore $A\subseteq\mathcal P(A)$. In the other direction, suppose that $A\subseteq\mathcal P(A)$, then if $B\in A$, it follows that $B\in\mathcal P(A)$, and therefore $B\subseteq A$.      $\square$

Now, you might ask yourself, what are some examples of transitive sets? Well, the empty set obviously satisfies this. But are there any others?

Exercise 1.1: If $A$ is transitive, then $A\cup\{A\}$ is transitive.

Exercise 1.2: If $A$ is transitive, then for all $B\subseteq A$, $A\cup\{B\}$ is also transitive.

Exercise 2: If $A$ is transitive, then $\mathcal P(A)$ is transitive.

And more importantly, we have the following theorem, at least in the context where every object of the universe is a set (which is the usual setting for set theory).

Theorem. If $A$ is any set, then there is a $\subseteq$-minimum transitive set $X$ such that $A\subseteq X$.

Proof. Define by induction a sequence $X_0=A$; and $X_{n+1}=X_n\cup\bigcup X_n$, where $\bigcup X_n=\{x\mid\exists Y\in X_n: x\in Y\}$ is the union over all the sets in $X_n$. Finally, define $X=\bigcup_n X_n$.

First, $X$ is transitive. If $Y\in X$, then there is some $n$ such that $Y\in X_n$, and therefore $Y\subseteq X_{n+1}$ by definition, so $Y\subseteq X_n$. And of course $A=X_0$, so $A\subseteq X$.

Secondly, if $Y$ is a transitive set and $A\subseteq Y$, then by induction it holds that $X_n\subseteq Y$ for all $n$, and therefore $X\subseteq Y$ as the union of the $X_n$'s.    $\square$

So, what does all these things tell us? They tell us that there is a proper class of transitive sets. All the ordinals, as well as the von Neumann hierarchy. And in fact, much more.

Finally, for the case where $\mathcal P(A)\subseteq A$, this contradicts Cantor's theorem that there is no surjection from $A$ onto $\mathcal P(A)$ (consider it as an exercise to see why exactly). In set theories with a universal set, however, this can still hold (there Cantor's theorem has other reasons to fail, or the power set axiom does not hold, or various separation assumptions prevent $\mathcal P(A)\subseteq A$ from being a falsehood).

However, in the common naive and $\sf ZFC$-derived systems, it is impossible for $\mathcal P(A)\subseteq A$ to holds for any set $A$. This is due to Zermelo, and it is a reformulation of Russell's paradox: For any set $A$, $\{a\in A\mid a\notin a\}\notin A$. I will leave you this as a final exercise.