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In a high-stakes negotiation, the most important moves often come not in the end game but at the very start, when one side or the other prevails in defining what is on the table. If you listened closely, you might have heard President Obama try to do just that in his news conference on Monday, when he suggested that Washington will have tamed the government's debt problems if the two parties can agree on another $1.5 trillion or so in spending cuts and tax increases.

Fiscal hawks and small-government conservatives say the White House is setting the bar for fiscal responsibility way too low and just kicking the can down the road again on hard decisions that will only become more painful as time goes by. But Mr. Obama appeared intent on establishing that he was just one more deal away from putting the government back on sound footing, if only Republicans would go along.

His numbers are relatively straightforward. During his re-election campaign he committed himself to $4 trillion in deficit reduction over 10 years — he referred to that figure on Monday as the "consensus" on what is necessary "to stabilize our debt and deficit" — including savings he and Congress had already agreed to. Altogether, they have enacted roughly $2.5 trillion in budget cuts and tax increases so far.

At around $4 trillion in deficit reduction, the United States would have a good shot at achieving what Mr. Obama and a growing number of Democrats consider to be a politically plausible and economically meaningful outcome: holding the national debt steady for a decade or so at under 75 percent of gross domestic product. (As recently as last summer the Congressional Budget Office was projecting a debt-to-G.D.P. ratio climbing into the 80s by the end of this decade if the government did not act to cut spending further and raise more tax revenue.)

Banking that last $1.5 trillion in budget savings would no doubt be achieved only after considerable partisan warfare, and the task will be that much tougher because it is tied up in the showdown over whether Congress will raise the debt ceiling. But if the president can threaten, cajole and compromise his way to one more big deficit-reduction pact, the job of putting the nation's fiscal house in order, in his telling, would be more or less complete.

Best of all, from a Democratic perspective, reaching that goal would not require immediate deep cuts to Medicare or Social Security or a fundamental rethinking of the social welfare compact.

"If we combine a balanced package of savings from spending on health care and revenues from closing loopholes, we can solve the deficit issue without sacrificing our investments in things like education that are going to help us grow," he said, referring to his existing proposals for modest trims to Medicare and tax-code changes that would generate more revenue from the wealthy.

"Solve" might be too strong a term for the impact on the nation's fiscal challenges, and there are a lot of technical assumptions in Mr. Obama's numbers that might prove overly optimistic. But if the United States does not suffer another deep recession, reaching $4 trillion in deficit reduction could buy Washington a decade or so to experiment with ways to curb the main driver of fiscal pressures — rising health care costs. The two parties might be able to nurture greater consensus during that period on whether or how to reshape the entitlement programs to make them sustainable for decades to come.  And if the country was especially lucky, a burst of economic growth could wipe away some portion of the remaining long-term problem.

But Mr. Obama's effort to define success on his terms is coming up against two primary counterarguments as the White House and Congress hurtle toward the next budget showdown in coming weeks.

Fiscal hawks — those most concerned with the potential economic and financial impact of chronic deficits — say Mr. Obama's approach is merely a short-term patch, sufficient only to create a 10-year illusion that the nation has made the hard decisions necessary to maintain long-term solvency. In their view, bolder action is needed now to assure that the debt does not spiral out of control in the following decade as the full costs of providing for an aging population put vastly greater strains on the government's finances.

"Getting some of the easier reforms is not victory, not even close," said Maya MacGuineas, president of the Committee for a Responsible Federal Budget and the leader of Fix the Debt, advocacy groups dedicated to greater budget austerity. "I'm incredibly worried about the argument that we don't need to do anything more quickly."

Then there are the small-government conservatives. They want lower deficits, but they also want to lead the nation to a philosophical shift away from what they see as a society too dependent on entitlements and an economy being suffocated by the growth of government.

"Government can't solve our problems," said Chris Chocola, president of the Club for Growth, a conservative advocacy group. "We can't tax our way out of this. We need to structurally reform the way we do things."

The limits of Mr. Obama's case are laid out in a recent report  from the Center on Budget and Policy Priorities, a liberal research organization. The center's report calculated that Congress and Mr. Obama could stabilize the debt at 73 percent of G.D.P. by later in this decade by agreeing on another $1.2 trillion in tax increases and/or spending cuts (a change that would also yield an additional $200 billion in savings over a decade by reducing what the government would otherwise have to pay in interest on the debt).

Stabilizing the debt during periods of economic growth, the report said, is the "minimum appropriate budget policy," since any nation needs to maintain the flexibility to borrow when times get tough. At 73 percent, the ratio would still be higher than the level recommended for healthy industrial economies by institutions like the International Monetary Fund; the Bowles Simpson report two years ago recommended policy changes that would have put the debt ratio on a declining path for decades to come. And even achieving stability at 73 percent would be a temporary victory.

"In ensuing decades, the aging of America's population and projected increases in per-capita health care costs — which are likely to rise faster than per-capita G.D.P. — will put considerable pressure on federal health and retirement programs," the center's report said, "returning the budget to an unsustainable path of rising debt as a share of the economy."