Mr. Romney said Mr. Obama had doubled the deficit. That is not true. When Mr. Obama took office in January 2009, the Congressional Budget Office had already projected that the deficit for fiscal year 2009, which ended Sept. 30 of that year, would be $1.2 trillion. (It ended up as $1.4 trillion.) For fiscal year 2012, which ended last week, the deficit is expected to be $1.1 trillion — just under the level in the year he was inaugurated. Measured as a share of the economy, as economists prefer, the deficit has declined more significantly — from 10.1 percent of the economy’s total output in 2009 to 7.3 percent for 2012.

Mr. Obama and Mr. Romney repeatedly sparred over whether Mr. Romney has proposed a $5 trillion tax cut.

It is true that Mr. Romney has proposed “revenue neutral” tax reform, meaning that he would not expand the deficit. However, he has proposed cutting all marginal tax rates by 20 percent — which would in and of itself cut tax revenue by $5 trillion.

To make up that revenue, Mr. Romney has said he wants to clear out the underbrush of deductions and loopholes in the tax code. But he has not yet specified how he would do so.

This week, in a television interview, Mr. Romney did shed some light — floating the idea of capping each household’s deductions at $17,000.

“As an option, you could say everybody’s going to get up to a $17,000 deduction. And you could use your charitable deduction, your home mortgage deduction, or others, your health care deduction, and you can fill that bucket, if you will, that $17,000 bucket that way,” he said. “Higher-income people might have a lower number.”

The deduction cap has the virtue of avoiding the tough negotiations over which tax expenditures to unwind. Many tax expenditures are highly popular, like the deduction for charitable giving. Moreover, many are important to the stability of the economy. Suddenly ending the home mortgage interest deduction, for instance, would threaten to destabilize the housing market.

But a number of unanswered questions about Mr. Romney’s tax plan remain.

For instance, Mr. Romney did not address how his proposed cap on deductions would affect tax credits. (Generally, deductions lower a family’s level of taxable income and credits erase part of their overall tax bill.)

It is also unclear whether his proposal to cap deductions would raise enough revenue to pay for his income tax rate cuts — at least not without increasing the tax burden on families making less than $200,000 a year, which Mr. Romney has vowed that he will not do.

Mr. Romney said that Mr. Obama’s health care overhaul would allow the federal government to “take over health care,” an assertion rejected by the president.

The 2010 health care law clearly expands the role of the federal government. But it also builds on the foundation of private health insurance, providing subsidies for millions of low- and moderate-income people to buy private insurance.

Under the law, close to 30 million Americans are expected to gain health coverage, according to the Congressional Budget Office. Many of them would receive insurance through the expansion of Medicaid. The federal government will initially pay the entire cost of Medicaid coverage for newly eligible beneficiaries and would never pay less than 90 percent.

In addition, the federal government would subsidize the purchase of private insurance for millions of people with incomes up to four times the poverty level (up to $92,200 for a family of four). Private insurers would thus have many new customers.

When Mr. Romney and other Republicans complain of a federal takeover, they are referring to more than spending and enrollment in government health programs. They say the new health care law will require most Americans to purchase “government-approved insurance” or pay a new tax. The tax issue was at the heart of the Supreme Court’s much-debated 5-to-4 decision in June to uphold the president’s health care overhaul law, the Affordable Care Act.

Mr. Obama first brought up Mr. Romney’s frequent criticism that the president cut $716 billion from Medicare, by saying the cost savings were from reduced payments to insurance companies and other health care providers. But Mr. Romney repeated the claim, suggesting that the $716 billion in Medicare reductions would indeed come from current beneficiaries.

While fact-checkers have repeatedly debunked this claim, it remains a standard attack line for Mr. Romney.

The charge that Mr. Obama took $716 billion from Medicare recipients to pay for “Obamacare” has several problems — not least the fact that Mr. Romney’s running mate, Representative Paul D. Ryan, included the identical savings in his budget plans that House Republicans voted for in the past two years.

Mr. Obama did not cut benefits by $716 billion over 10 years as part of his 2010 health care law; rather, he reduced Medicare reimbursements to health care providers, chiefly insurance companies and drug manufacturers. And the law gave Medicare recipients more generous benefits for prescription drugs and free preventive care like mammograms.

According to nonpartisan analysts, it is Mr. Romney who would both cut benefits and add costs for beneficiaries if he restored the $716 billion in reductions. Restoring higher payments to insurers and other companies would in turn increase Medicare premiums because beneficiaries share in Medicare’s total cost. Marilyn Moon, a vice president at the American Institutes for Research, has calculated that a Medicare recipient’s out-of-pocket expenses would increase $577 a year on average by 2022.

Also, the Obama reductions added eight years to the life of Medicare’s financially troubled trust fund, to 2024, according to Medicare trustees. If the cuts were restored, the insolvency date would revert to 2016.

But the cuts to providers could cause private Medicare plans to raise their premiums, which is expected to reduce enrollment in them. Those changes have not materialized yet.