electromagnetism - Does rest exist?

I initially thought that the concept of rest depended on an inertial frame of reference. So for example, if the Earth and everything on it were the only things in the universe, and the Earth was floating with constant velocity, we would assume anything with the same velocity from an outsider's point of view as being at rest.

However, today, while taking a quick skim at my physics book's chapter on electromagnetic induction, I saw a diagram showing how the movement of a magnet could induce a current on a coil of wire. I quickly concluded that this was just a consequence of how the electrons are moving from the frame of reference of the magnet.

Then I thought about two electrons moving at the same nonzero velocity, assuming that these two electrons, a person "at rest" (basically some person that can measure this velocity and get a nonzero value), and a person moving at this velocity are the only objects/people in the universe. From the frame of reference of the person at rest, the electrons have a net magnetic force that they apply on each other, and they must have this force, else the definition of the ampere makes no sense. However, from the frame of reference of the person who is moving at the same velocity of the electrons, the calculations show that there is no magnetic force.

So does absolute rest actually exist? Or have I made a mistake in my reasoning.

Answer

All your reasoning is correct, except for the conclusion!

You take your thought experiment to mean that absolute rest exists. But what it really says it that absolute electric and magnetic fields don't exist: observers can disagree on what they are, just as they disagree on space and time. The real, invariant object is the electromagnetic field tensor $F$. Its components $F^{\mu\nu}$ in a particular frame tell you the electric and magnetic fields in that frame.

This is still consistent with your observation that there's a magnetic force in one frame and no magnetic force in the other, because we never measure magnetic force. We can only measure the total electromagnetic force $f$, given by the Lorentz force law $f^\mu = qv_\nu F^{\mu\nu}$.