a finite sets problem

Suppose the numbers (positive integers) are $N_1,N_2,\dots,N_m.$ The pilot displays the number $10^{N_1}+10^{N_1+N_2}+\cdots+10^{N_1+N_2+\cdots+N_m}.$

Suppose that passenger $k$ has number $f(k)$.

Then the pilot can show the number $\prod\limits_{n=1}^N p_n^{f(n)}$. Where $p_n$ is the $n$'th prime and $N$ is the number of passengers.

By the fundamental theorem of arithmetic every passenger can determine every person's number.

Edit: here's a second answer, probably shortest so far and easiest to decode.

Suppose the longest number has $d$ digits. The captain pads each number with zeroes on the left so that it's $d$ digits long, then concatenates them in seat order. Since the passengers know the number of seats they can compute $d$ from the length of the number they're told, then count off to their seat number. If the number of seats is unknown too then the caption uses $d$ as a prefix, padded the same way.

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First solution:

Since the captain knows the numbers that correspond to seats he can tell the passengers the product of the numbers $(p_n)^{c}$ where $p_n$ is the $n$th prime and $c$ is the number held by the passenger in seat $n$.

The passengers will have to be pretty good at arithmetic. Maybe their cell phones can factor big numbers.