President Donald Trump is making broad generalizations about people who arrive illegally in the United States, casting them all as violent criminals when statistics say otherwise. He’s also suggesting that a newly signed executive order would solve the problem of family separations at the border, even while continuing to blame Democrats for separations that result from his own policy of criminally prosecuting every adult entering the U.S. illegally.

His comments capped a dizzying week of questionable claims, non-sequiturs and outright misstatements amid fallout over his treatment of refugees crossing the U.S.-Mexico border.

Meanwhile, on North Korea, Trump contradicted his own Defense Department by saying that leader Kim Jong Un had already begun to fully rid the country of nuclear weapons and wavered back and forth on statements regarding the level of threat the North posed to the U.S.

He also left out important context about the creation of new U.S. jobs and rising wages, overstated the impact of a new health insurance option for small businesses that appeared aimed at undercutting an Obama-era law, and repeated misleading claims about a special counsel’s probe into Russian meddling in the 2016 election.

A look at the statements:

TRUMP: “They can be killers, they can be thieves, they can be horrible people.” — weekly address Saturday.

TRUMP: Democrats “think immigration is being weak on the border, which is therefore allowing tremendous crime to come into our country.” — remarks Saturday to Nevada GOP convention.

TRUMP: “I always hear that, ‘Oh, no, the population’s safer than the people that live in the country.’ You’ve heard that, fellas, right? You’ve heard that. I hear it so much, and I say, ‘Is that possible?’ The answer is it’s not true. You hear it’s like they’re better people than what we have, than our citizens. It’s not true.” — remarks Friday.

THE FACTS: His generalization about people in the U.S. illegally is overdrawn, dismissing without evidence several studies showing immigration does not lead to increased crime.

The studies from social scientists and the libertarian think tank Cato Institute have shown that people here illegally are less likely to commit crime than U.S. citizens, and legal immigrants are even less likely to do so.

A March study by the journal Criminology found “undocumented immigration does not increase violence.” The study, which looked at the years 1990 through 2014, argues that states with bigger shares of such people have lower crime rates.

A study last year by Robert Adelman, a sociology professor at University of Buffalo, analyzed 40 years of crime data in 200 metropolitan areas and found that immigrants helped lower crime. New York City, for example, has the nation’s largest population of immigrants living in the country illegally — about 500,000 — and last year had only 292 murders among a total population of 8.5 million people. A city murder rate is often used as a benchmark for overall crime because it’s difficult to fudge murder statistics.

And Ruben Rumbaut, a University of California, Irvine sociology professor, co-authored a recent study that noted crime rates fell sharply from 1990 to 2015 at a time when illegal immigration spiked.

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TRUMP: “Open Borders Democrats support the loopholes that prevent families from being detained and removed together — they just want everyone to be released into our country no matter how dangerous they are.” — weekly address Saturday.

TRUMP: “Democrats, fix the laws…Strong Borders, No Crime!” — tweet Sunday.

THE FACTS: His assertion that Democrats are fully to blame for continued family separations is misplaced.

The separations are a consequence of a Trump administration policy to maximize criminal prosecutions of people caught trying to enter the U.S. illegally. That means more adults are jailed, pending trial, so their children are removed from them. Before the policy, many people who were accused of illegal entry and did not have a criminal record were merely referred for civil deportation proceedings, which generally did not break up families.

While Republicans hold a 51-49 majority in the Senate, there are enough Democrats to block legislation sought by Trump on a procedural vote that would also provide money for a border wall. House Republicans, meanwhile, are internally divided over a suitable plan.

Trump has blamed “bad legislation passed by the Democrats” for driving families apart. But the 2008 law he appears to be referring to passed unanimously in Congress and was signed by a Republican president. It was focused on freeing and otherwise helping children who come to the border without a parent or guardian. It does not call for family separation.

While blaming Democrats, Trump in the past week has also told congressional Republicans to “stop wasting their time” on immigration legislation until after November.

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TRUMP on Kim Jong Un: “We get along very well…He’s a great negotiator. I think he sees a tremendous future for North Korea. But we want to have it denuclearized, and that’s what’s happening.” — remarks Saturday at roundtable on tax reform in Las Vegas.

TRUMP: “The big thing is, it will be a total denuclearization, which has already started taking place.” — remarks Thursday at Cabinet meeting.

THE FACTS: That’s not what his Pentagon chief, Jim Mattis, says. When asked by a reporter on Wednesday whether he had seen any sign that North Korea had begun steps toward denuclearization, Mattis replied, “I’m not aware of any. Obviously, we’re at the very front end of the process. Detailed negotiations have not begun.”

At a Singapore summit with Trump earlier this month, North Korea’s leader committed to “complete denuclearization of the Korean Peninsula,” but no details were worked out.

In May, prior to the summit, North Korea demolished tunnels at its sole underground nuclear test site, although outsiders have not inspected the result. Its nuclear program has many other elements, including nuclear materials production facilities, nuclear warheads, ballistic missiles and missile launchers.

Trump had pointed to the summit’s discussions on denuclearization in asserting that North Korea was no longer a nuclear threat, but on Friday he appeared to reverse course by declaring that the North still poses an “extraordinary threat” to the U.S. In an executive order, the president extended for one year the so-called “national emergency” with respect to the nuclear-armed nation, re-authorizing economic restrictions against it.

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TRUMP: “And ultimately, we have to have a real border — not judges. Thousands and thousands of judges they want to hire. Who are these people? When we vet a single federal judge, it goes through a big process. Now we’re hiring thousands and thousands. … And it got so crazy that all of these thousands — we now have thousands of judges — border judges — thousands and thousands.” — remarks Tuesday to the National Federation of Independent Business.

TRUMP: “We shouldn’t be hiring judges by the thousands, as our ridiculous immigration laws demand.” — tweet Thursday.

THE FACTS: He’s incorrect about the U.S. having “thousands and thousands” of immigration judges and about thousands more judges being hired. The Justice Department’s immigration courts division has about 335 judges currently on staff nationwide, with the budget for 150 additional judges.

Dana Leigh Marks, past president of the National Association of Immigration Judges who also works in the Justice Department’s executive office for immigration review, said funding for immigration courts has increased modestly amid a growing backlog of cases. With a backlog of 700,000, each judge would have to take on more than 2,000 cases to clear the docket.

The figures also don’t take into account a wave of expected retirements that would shrink the ranks of judges. A June 2017 Government Accountability Office report determined that 39 percent of immigration judges are now eligible for retirement. Congressional investigators blamed the mounting caseload in part on the slow hiring of immigration judges.

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TRUMP: “We’re keeping families together, and this will solve that problem.” — remarks Wednesday at signing of order to halt his administration’s policy of separating children from their parents when they are detained illegally crossing the U.S. border.

THE FACTS: It doesn’t solve the problem.

Trump’s executive order will continue his “zero tolerance policy” of criminally prosecuting all adults caught crossing the border illegally, and will now seek to keep families together in detention instead of separating them while their legal cases are heard by the courts.

But a 1997 landmark settlement known as the Flores agreement that generally bars the government from keeping children in immigration detention for more than 20 days remains in place. Trump is seeking to have the settlement amended, but his Justice Department says the 20-day policy remains in effect until Congress or the courts take action to change that.

That means without further action from Congress or the courts, the Trump administration could be forced to again separate the immigrant children from their parents in three weeks.

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TRUMP: “We have created more than 3.4 million new jobs since Election Day. 3.4 million. Think of what that means.” — remarks Tuesday to the National Federation of Independent Business.

THE FACTS: Well, one thing it means is that job creation has slowed a bit compared to its previous pace. Trump is right that U.S. companies have added 3.4 million jobs in the 18 months since his election, a healthy total. But in the previous 18 months, 3.7 million jobs were added. That’s not entirely surprising or a sign of a weakening economy — job gains typically decline as the unemployment rate falls and there are fewer people to hire. The unemployment rate is currently at an 18-year low of 3.8 percent. But Trump’s remarks suggest there has been a turnaround in job creation, when there hasn’t.

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TRUMP: “Wages for working people are finally, after 22 years, rising again in our country.” — remarks Tuesday.

THE FACTS: Wages, before adjusting for inflation, have ticked up in recent months. But so has inflation, which is offsetting those gains. In May, average hourly pay rose 2.7 percent from a year earlier. Yet inflation rose slightly more during that time: 2.8 percent. Household incomes rose at a solid pace in 2015 and 2016, according to the Census Bureau, partly because inflation was much lower during that time.

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TRUMP, on a health insurance option for small businesses and self-employed people: “You’re going to save massive amounts of money and have much better health care. It’s going to cost you much less. It’s going to be, I think, fantastic. And it’s very comprehensive. I will tell you, a lot of people — big, big percentages of this country — are going to be doing that.” — remarks Tuesday.

THE FACTS: Trump paints an overly rosy picture. It’s not projected that “big, big percentages” of people will benefit.

The nonpartisan Congressional Budget Office estimates a more modest impact of 4 million people who will be covered by the plans within five years but only some 400,000 — just 10 percent of those — who would have been uninsured. That’s compared to about 160 million who are covered by job-based insurance.

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TRUMP: “‘I can’t think of something more concerning than a law enforcement officer suggesting that their (sic) going to use their powers to affect an election!’ Inspector General Horowitz on what was going on with numerous people regarding my election. A Rigged Witch Hunt!” — tweet Tuesday.

THE FACTS: Trump’s statement is misleading. He suggests that findings of a Justice Department watchdog report by inspector general Michael Horowitz on the 2016 Hillary Clinton email investigation means that special counsel Robert Mueller’s Russia probe — which he often criticizes as a “witch hunt” — is “rigged.” The IG report determined that the FBI had made mistakes in the handling of the Clinton probe, which Horowitz described as “extremely serious.” But Horowitz dismissed the notion the probe had been rigged, saying that investigators did not uncover evidence that political bias had influenced the FBI’s conclusion that Clinton should not be prosecuted.

The IG report released earlier this month also did not address questions of collusion between the Trump campaign and Russia, the subject of the investigation led by Mueller, who was appointed last year to take over the FBI’s Russia probe after Trump fired former FBI Director James Comey.

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TRUMP: “We’ve eliminated record numbers of job-killing regulations … we’ve cut more regulation than any other president in the history of our country whether it’s four years, eight years or in one case 16 years, we’ve cut more regulations in 500 days than any president. Even our 16-year president.” — remarks Wednesday at Duluth, Minneosta, rally.

THE FACTS: Trump gets his history wrong, repeating a claim that a U.S. president once served 16 years. Franklin D. Roosevelt, the country’s longest-serving president, died after serving 12 years in office from 1933 to 1945.

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TRUMP, on the return of remains from the Korean War: “We got back our great fallen heroes, the remains sent back today, already 200 have been sent back.” — remarks Wednesday.

THE FACTS: No remains have been returned, although Pentagon officials say they are prepared to receive them. Although the Singapore declaration said this would happen immediately, U.S. officials have given no indication that North Korea has committed to any specific timetable for the return.

On Thursday, in remarks at a Cabinet meeting, Trump modified his claim, saying, “They’ve already sent back or are in the process of sending back the remains of our great heroes who died in North Korea during the war.”

Aside from uncertainty over when North Korea will return the remains it has collected over the years, it’s unclear whether all will be in a condition to permit their positive identification, or whether they all are even Americans. A number of allied soldiers who fought alongside the U.S. during the war also are missing.

Nearly 7,700 American service members are listed as unaccounted for from the Korean War, of which an estimated 5,300 were lost in North Korea.

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TRUMP: “There was a story two days ago, in a major newspaper, talking about people living in Canada, coming into the United States, and smuggling things back into Canada because the tariffs are so massive. The tariffs to get common items back into Canada are so high that they have to smuggle them in. They buy shoes, then they wear them. They scuff them up. … We’re treated horribly.” —remarks Tuesday.

THE FACTS: The president appears to be referring to a column in the New York Post about Canadians who buy shoes, jeans and outdoor goods in the United States and take them into Canada without paying duties or taxes. In some cases, the items aren’t available in Canada or were more expensive.

But most items made in the United States, including shoes, can be imported into Canada duty-free under NAFTA, an agreement that Trump has strongly criticized and is renegotiating with Canada and Mexico.

For its part, the United States imposes some of its highest tariffs on shoe imports. Duties on footwear average nearly 11 percent but for some products can reach nearly 68 percent, according to the U.S. International Trade Commission.

Yet even with the higher costs imposed by tariffs, roughly 95 percent of shoes in the United States are imported from countries like China, Vietnam and Italy, the ITC says.

(AP)