Question about mathematical logic with set and inclusion relation Let $L = \{ \subseteq \}$ and let $M$ be the L-structure whose universe is $P^\mathbb{N}$, the set of all subsets of $\mathbb{N}$, and where $\subseteq$ is interpreted in the usual way.
1) Show that for all integer $n$, there is a formula $\psi_n[x]$ such that $M \vDash \psi_n[a]$ holds if and only if $a$ has at least $n$ elements.
I get stuck so badly on this question and really need some hints
2) Show thatt for any automorphism $\sigma$ of $M$, we have $\sigma(\emptyset)=\emptyset$.
Attempt: This is just a routine check of the definitions of automorphism. But I doubt if there is some hidden corners, since it's labeled as the 2nd question, so it should be easier than the 1st question, but I couldn't do the 1st.
3) Let $\sigma$ bean automorphism of $M$ such that $\sigma( \{ n \})$ = $\{ n \}$ for all $n \in \mathbb{N}$. Show that $\sigma$ is the identity.
Attempt: I did a very straight forward induction. If $n = 0$, we are done as proved in (2). If $n=1$, we're done, as it's the hypothesis. So suppose that for a given given set of $K$ of $k$ elements, we have $\sigma(K)=K$. We try to prove that for a set of $K'$ of $k+1$ elements, we have $\sigma(K')=K'$. This will be done by the standard technique, i.e., one is a subset of the other. Am I on the right track?
4) Find all automorphism of $M$.
Attempt: The most educated guess I can think of is that the automorphism on $M$ is precisely the identity, i.e., for an $a \in P^\mathbb{N}$, then if $f$ is an automorphism on $M$, then $f(a)=a$. Could you please give me some hints on this question?
Thanks!

Regarding the first question:
$$\psi_n[x] \equiv\exists y_1 \dots \exists y_n \ (\bigwedge_{i=1}^n y_i \subseteq x) \wedge (\bigwedge_{1 \le i < j \le n} y_i \neq y_j)$$ 
