Definition. We say a set $Y$ is inductive if $0 \in Y$ and, $\forall y \in Y, y^+ \in Y$, where $y^+$ is the sucessor of $y$, i.e., $y^+ = y \cup \{y\}$.
Define the set $\cal{A}$$ = \{B \in PA : 0 \in B \text{ is inductive}\}$. Then $\cal{A}$ is non-empty and from this set we form the natural number set, which is given by: \begin{equation*} \omega = \bigcap \cal{A} \end{equation*}
After this, we've been given some properties and this is where my trouble begins.Take the following Lemma as an example:
Lemma. For all $n \in \omega$ e $m \in n$: $m \subseteq n$.
And my problems here are: What's the difference between $m \in n$ and $m \subseteq n$ ?
Thanks for any help in advance.
