I have the infinite list of numbers $$[2, 1, 3, 4, 5, 6, 7, 8, \ldots]$$
It is the set of the natural numbers with the one difference that the two numbers $2$ and $1$ are flipped. The aim of the puzzle is to have every number in its correct position (so in the end we want to have exactly the set of natural numbers). There is one change that is allowed to made: It is allowed to rotate 3 numbers that are next to each other in this form: $$\ldots, a, b, c, \ldots \quad\to\quad \ldots, b, c, a, \ldots$$
so a example after the first rotation could be: $$[1, 3, 2, 4, 5, 6, 7, 8, \ldots]$$
Is it possible to solve this?
Look up "parity" of a permutation. Your initial state is a single transposition away from the goal, making it an odd permutation, but the moves you allow are products of two transpositions, so are even. The fact that you have an infinite set here means you have to be a little careful, but since you are only allowed finitely many moves, they will all take place in some finite subset of the naturals anyway, so that's not a problem.