Play this scenario out, and decide for yourself:

A businessman dips into his fortune to fund a campaign for term limits. Enough voters sign petitions to put the term limits measure on the ballot, and the measure actually passes. The 51 City Council members, along with officials elected citywide, are allowed only two consecutive four-year terms.

That doesn’t sit well with career council members, some of whom have served for decades. They claim the voters didn’t really want term limits, and were improperly influenced by the billionaire’s spending. To prove their point, they overwhelmingly approve their own measure three years later asking voters to change their minds. Much to their surprise, voters reject that idea and term limits survive.

This ushers in a new era of city government, as the entire city council is replaced over two election cycles with a fresh set of lawmakers who know they have no more than eight years to make their mark. They pass a flurry of legislation, from substantive measures that include extending gay rights to headline grabbing wastes-of-paper that include an unenforced fine for using a cell phone in a movie theater.

But as the eight-year mark approaches, the city council members get nervous as they contemplate looking for work.

The first to blink is the members’ chosen leader, known as the speaker. He wants to run for mayor, but his eight-year limit is up before the mayoral election. In a stroke of political genius, he convinces his colleagues to extend his time by two years. It’s a fluke, and the public doesn’t really notice.

But that subtle change empowers other council members. If they can so easily modify a measure voters enacted, perhaps they could actually overrule the voters and end term limits altogether. At the very least, some of them say, they can add another term. What politician doesn’t like the sound of “Four more years”?

This scenario is playing out right now as council members ponder life after City Hall. The persuasive argument that turnover and new ideas are refreshing is being replaced with the twice-failed argument that institutional memory and experience are paramount.

A legal battle appears inevitable. Council leaders point to court rulings that allow them to modify voter-approved ballot measures. The billionaire businessman, philanthropist, and former ambassador who began the whole term limits movement, Ronald Lauder, argues the will of the people can only be changed by the people.

There may be reasons to change or even eliminate term limits. Perhaps three terms would benefit New Yorkers. Perhaps allowing unlimited term limits would allow dedicated public servants to serve the public good. At the very least, term limits imply voters can’t decide on their own when to end a politician’s time in office. These are issues that New Yorkers can presumably decide for themselves after reflecting on the initial impact of a concept they’ve twice approved.

Another variable is what’s known as self dealing – council members might allow a term limits change to benefit themselves. If city lawmakers do go ahead with overruling their constituents, they will surely face pressure to have the change take effect once they’re out of office. That would lend the concept credibility, as Mayor Bloomberg finally learned while pressing his own ballot measure.

In 2003, Mr. Bloomberg spent millions of his own dollars to promote “non-partisan” elections – which would have replaced various party primaries with a single primary that included all candidates on a single ballot. When I interviewed Mr. Bloomberg in June of that year, he argued the change should take effect before his re-election by asking, “How can you look the public in the eye and say the public should be denied the privilege of democracy for any reason whatsoever?” Under pressure, though, Mr. Bloomberg ultimately changed his mind and agreed that he wouldn’t benefit from a change he initiated. Voters rejected the idea anyway, and the primary tradition lives on.

In a sense, ending term limits by legislation is like the council voting out a member elected at the ballot box. Even if the council had that right, voters would surely revolt.

Over the next month, the City Council will select a new leader who will dominate the term limits debate. If council members remain determined to make a change, they have a few options: Make a change that helps themselves, make a change that affects only future members, or ask the voters to revisit the issue. Going back to voters is the riskiest but most credible path. If the Council members are rejected on term limits at the ballot box a third time, maybe then they’ll finally get the message.

Mr. Goldin is a host of NY1’s “Road to City Hall,” which airs weeknights at 7 and 10:30 p.m.