127 Elements are non-empty sets $S$ of natural numbers that satisfy “if $x \in S$, then $14-x \in S$”. The number of such sets $S$ is $\qquad$ .
127 127. From $x, 14-x \in \mathbf{N}$, we know $1 \leqslant x \leqslant 13$. Pairing $1,2, \cdots, 13$ into $\{1,13\}$, $\{2,12\},\{3,11\},\{4,10\},\{5,9\},\{6,8\},\{7\}$. $S$ is the union of some of these 7 sets. Therefore, the number of $S$ is $2^{7}-1=127$.