115. In how many non-empty subsets of the set $\{1,2,3, \ldots, 10\}$ will there be no two consecutive numbers?
115. 143. 

Let $A_{n}$ denote the set of non-empty subsets of the set $\{1,2,3, \ldots, n\}$ that do not contain two consecutive numbers, and let $a_{n}$ be the number of such subsets. Clearly, the set $A_{1}$ consists of the subset $\{1\}$, and the set $A_{2}$ consists of the subsets $\{1\}$ and $\{2\}$. Thus, $a_{1}=1$ and $a_{2}=2$. For $n \geq 3$, we can partition the sets in $A_{n}$ into two classes. The first class consists of sets that do not contain the number $n$, which coincides with $A_{n-1}$ and contains $a_{n-1}$ sets. The second class consists of sets that contain $n$. Among these sets, there is the set $\{n\}$, and the rest have two or more elements. By removing the element $n$ from these sets, we obtain the set $A_{n-2}$. Thus, we get the formula $a_{n}=a_{n-1}+a_{n-2}+1$ for finding the answer. Therefore, we sequentially find the values of $a_{n}: 4,7,12$, $20,33,54,88$ and 143.