If our billionaire mayor, Michael Bloomberg, moves to lift term limits so he can serve a third term as mayor, the billionaire businessman who brought term limits to the city, Ronald Lauder, will move to block him. Such is the news brought in today on our front page, illuminating what could shape up as an exceptional struggle.

We had been urging Mr. Bloomberg to run for president, but he has declined to do so, leaving the field to the anti-free-trade Senators Clinton and Obama and the pro-gun Senator McCain. The editorial page editor of the New York Post has been urging Mr. Bloomberg to run for governor, reasoning that it is in Albany where many of Mr. Bloomberg’s key initiatives, from the 2012 Olympics to congestion pricing, have been stymied.

We are going to withhold judgment on the 2009 mayoral race until we know who the other candidates are, but this much is clear: even if Mr. Bloomberg succeeds in changing the city charter to win a third term, he is not going to be mayor forever. So, if the mayor wants to preserve his legacy, he’d do well to devote some time to thinking about a successor and to preparing the public for that transition.

Mayor Giuliani talked, after September 11, 2001, of potentially extending his term to deal with the emergency’s aftermath. In the end, to his credit, he turned the city over to a capable successor in Mr. Bloomberg, a transition made possible by Mr. Giuliani’s endorsement. The talk of a third term for Mr. Bloomberg reminds us of Mr. Giuliani’s emergency plan — an understandable instinct, to be put aside when a worthy successor emerges.