New Playstation Controller player 1?

I have 2 PS4 controllers that are registered for Player 1 and Player 2, respectively. With my PS4, I always have to use the first one (player 1) to play with the other one (player 2).

My first controller is slowly giving up and I wanted to get a new one anyway.

Now my question. If I should get a new controller, it is registered as player 1 or as player 3. Can I start and use the ps4 normally with this without using the original controller for player 1?

If you turn on the console, no controllers should turn on.

If you then switch on one (e.g. The 2) then this should also be controller 1.

The controllers are not registered by player, but in the order in which the controllers are switched on. Whereby register is the wrong word for this. That changes with every start. If you switch on a controller (no matter which one), it will automatically be recognized as player 1, the second one that will be switched on afterwards, as player 2, etc.

After each restart of the PS4, the controller sequence is reset, so to speak.