Set-theoretic induction principle Example I'm reading Induction, Coinduction, and Fixed Points:Intuitions and Tutorial. At the beginning, page 2, there is a nice example:
$P = \{n ∈ N|2^n > n\}$, the property that each of its members, say $n$, has its exponential $2^n$ strictly greater than $n$ itself.
Then it proves that $P$ is true for any natural number, by proving $F(P) ⊆ P$ where $F$ is the “successor” function (of Peano),
$F(X) = \{0\} ∪ \{x + 1|x ∈ X\}$, where, e.g., $F (\{0, 3, 5\}) = \{0, 1, 4, 6\}$
This means proving that $(0 ∈ P) ∧ (∀x ∈ N.x ∈ P ⇒ (x + 1) ∈ P)$, or, in other words, proving that $2^0 > 0$ and that from $(2^x > x)$ one can conclude that $2^{x+1} > x + 1$
My question / misunderstanding is: doesn't $⊆$ mean $⇒$ (something like: if a set of objects that have a property is included in another set of objects, then the the original set has the more inclusive property also)? Itsn't F(P) = $2^{x+1} > x + 1$ ?
 A: It is not quite accurate to say that $\subseteq$ means $\implies$.
But if you introduce variables to stand for various sets and elements thereof, you can say this:

For any sets $A,B$, the relation "$A \subseteq B$" means "$x \in A \implies x \in B$ (for all $x$)".

Regarding your last question, asking whether $F(P) = 2^{x+1} > x+1$ violates mathematical grammar and so doesn't make any sense.
What you can say is this (and I'm going to go through this very slowly, step by step):

$F(P) \subseteq P$ means that $x \in F(P) \implies x \in P$.

This tells you what you have to prove:

Assume that $x \in F(P)$ and prove that $x \in P$.

And now we slowly translate this.
First translation:

Assume $x=F(n)$ for some $n \in P$ and prove $x \in P$.

Next translation:

Assume $x = F(n)$ and that $2^n > n$, and prove $2^x > x$.

Finally, I presume you know the formula for Peano's successor function, which is simply $F(n)=n+1$. Therefore we get the final translation:

Assume $x=n+1$ and that $2^n > n$, and prove $2^{n+1} > n+1$.

And this last thing can actually (and easily) be proved, using the laws of arithmetic and inequality.
