Example 8 Choose $k$ numbers from $1,2, \cdots, 2004$, such that among the chosen $k$ numbers, there are definitely three numbers that can form the side lengths of a triangle (here it is required that the three side lengths of the triangle are all different). Then the minimum value of $k$ that satisfies the condition is $\qquad$
The original problem is equivalent to "Select $k-1$ numbers from $1,2, \cdots, 2004$ such that no three of these numbers can form the sides of a triangle with unequal sides. Find the maximum value of $k$." To maximize $k$, the selected numbers should be equal to the sum of the two largest numbers already in the set, leading to:
$$
\begin{array}{l}
1,2,3,5,8,13,21,34,55,89,144,233, \\
377,610,987,1597,
\end{array}
$$

a total of 16 numbers. For any array $a_{1}$, $a_{2}, \cdots, a_{n}$ that meets the above conditions, it is clear that $a_{i}$ is always no less than the $i$-th number in (1). Therefore,
$$
n \leqslant 16 \leqslant k-1, k-1 \geqslant 16 \text {. }
$$

Solving for $k$ gives $k \geqslant 17$.
Thus, the minimum value of $k$ is 17.