There is never a nanosecond when there is not a current president you idiot.
So if Trump never takes the oath, for the whole four years, who is President? Either the President elect has to take the oath, or he doesn't. I think he has to. Until he does, whether the interval is 30 seconds, or 48 months, we do have an acting President. And for 30 second or so this Friday, it will be John Kerry.