Example 3 In the set $S_{n}=\{1,2, \cdots, n\}$, a subset that does not contain two consecutive natural numbers is called a "good subset". How many good subsets are there in $S_{n}$?
Let $S_{n}$ have $a_{n}$ good subsets.
When $n=1$, $S_{n}$ has 2 subsets, $\varnothing$ and $\{1\}$, both of which are good subsets, $a_{1}=2$;

When $n=2$, $S_{n}$ has 4 subsets, $\varnothing$, $\{1\}$, $\{2\}$, $\{1,2\}$, except for $\{1,2\}$, all are good subsets, $a_{2}=3$.
For $n \geqslant 3$, let $M$ be a good subset of $S_{n}$.
If $n \notin M$, then $M$ is also a good subset of $S_{n-1}$; conversely, a good subset $M$ of $S_{n-1}$ is also a good subset of $S_{n}$.

If $n \in M$, then by the definition of a good subset, $n-1 \notin M$. Therefore, the subset $M^{\prime}$ obtained by removing $n$ from $M$ (where $M = M^{\prime} \cup \{n\}$) must be a good subset of $S_{n-2}$; conversely, a good subset $M^{\prime}$ of $S_{n-2}$, when combined with $n$ to form $M = M^{\prime} \cup \{n\}$, is also a good subset of $S_{n}$. Hence, we have
$$
\left\{\begin{array}{l}
a_{n}=a_{n-1}+a_{n-2}, \\
a_{1}=2, a_{2}=3 .
\end{array}\right.
$$