in statistics... what do you do if there are 2 different modes?

There can't be two modes. If there are two most abundant figure I usually settle for the value between them.

Then the mode becomes a set, for example if my modes are 4 and 5, the mode is {4, 5}.

Likewise, a list where each number appears once, would have many modes.

For example the mode of the list:

1,2,3,4,5

can be defined as {1,2,3,4,5}

the data set is Bi-modal

use them both.

if there's more than two, there's no mode.