5. A non-empty set $S$ satisfies:
(1) $S \subseteq\{1,2, \cdots, 2 n+1\}, n \in \mathbf{N}_{+}$;
(2) If $a \in S$, then $(2 n+2-a) \in S$.

Then, the number of non-empty sets $S$ that satisfy (1) and (2) is $\qquad$
5. $2^{n+1}-1$.

Pair the natural numbers $1,2, \cdots, 2 n+1$ into $n+1$ pairs:
$$
\{1,2 n+1\},\{2,2 n\}, \cdots,\{n, n+2\},\{n+1\} \text {. }
$$
The elements of $S$ are chosen from the above $n+1$ pairs, and there are
$$
C_{n+1}^{1}+C_{n+1}^{2}+\cdots+C_{n+1}^{n+1}=2^{n+1}-1
$$

ways to choose.