(CNN) President Donald Trump blinked this week, abandoning a pledge to proudly shut down the government if Democrats wouldn't fund his border wall and potentially signaling the project could never come to be.

But Trump won't stop talking about it, taking to Twitter to regurgitate some old claims -- that Mexico will pay for it indirectly and that the military will build it -- and add some new explanations, like that his wall will actually be "artistically designed steel slats."

Let's pick through each of those, starting with the most important immediate claim, that Mexico will pay for it.

Trump's claim that trade deal will pay for wall

Twice as it became clear in recent days that Congress would not write him a check for the wall, Trump said Mexico would ultimately pay for it, albeit indirectly.

"Mexico is paying (indirectly) for the Wall through the new USMCA, the replacement for NAFTA! Far more money coming to the U.S.," he said in a tweet Wednesday morning.

Mexico is paying (indirectly) for the Wall through the new USMCA, the replacement for NAFTA! Far more money coming to the U.S. Because of the tremendous dangers at the Border, including large scale criminal and drug inflow, the United States Military will build the Wall! — Donald J. Trump (@realDonaldTrump) December 19, 2018

"I often stated, 'One way or the other, Mexico is going to pay for the Wall,'" Trump said on Twitter last Thursday. "This has never changed. Our new deal with Mexico (and Canada), the USMCA, is so much better than the old, very costly & anti-USA NAFTA deal, that just by the money we save, MEXICO IS PAYING FOR THE WALL!"

There are multiple important problems with this claim.

First, the USMCA is not yet official. While Trump signed it last month with the leaders of Mexico and Canada, it has not yet been ratified by the House of Representatives or the Senate, both of which will have to ratify the trade deal.

It's not at all clear the USMCA has the votes it needs, as leaders from both parties have raised concerns. It will have to be brought up for a vote in the Democratic-controlled House first, and soon-to-be Speaker Nancy Pelosi has said she would like to see stronger enforcement measures for the labor and environment provisions.

"Democratic leadership has never been super enthusiastic about trade deals. It's hard to see the incentive for them to vote for one that will have Trump's signature," said Geoffrey Gertz, a Brookings fellow who focuses on the politics of trade.

Trump has said he plans to unilaterally end the original North American Free Trade Agreement, essentially forcing the Democrats to either adopt his new trade agreement within six months or have no deal in place at all. But he has yet to act on that threat.

The second problem with his USMCA claim is that there's no indication the new deal will bring enough or all that much new money into the US Treasury.

Is the US going to be able to use the NAFTA replacement -- the United States-Mexico-Canada Agreement -- to pay for the wall? "We're not," said Phil Levy, a senior fellow at the Chicago Council on Global Affairs. He ticked through possible revenue from a trade deal -- either directly through higher tariffs or indirectly via tax revenue from increased economic activity. There's nothing to suggest the USMCA will land the government a windfall on either count.

Even if Mexico did start pumping money into the USMCA after it passes, Levy said, that doesn't mean it would go to the wall.

"There's nothing that earmarks this type of money for a wall," he said. "Your income taxes or corporate taxes -- they go into general revenue. But it's still the budget process that determines where those funds go."

And it's that budget process that has Trump stymied. Republicans in the Senate are trying to find Trump a way back from the ledge of his shutdown boast and are poised to suggest a bill that would fund the government through February 8, so it's possible this show will be in reruns then.

The military will build it

In addition to his claim that Mexico will pay for it, Trump added Wednesday that the military would build it. On some level, it makes sense for the President to believe this. Congress has not authorized the building of a wall, and without that authorization, he can't seek bids and carry forward with construction. But he doesn't need congressional approval to deploy the military, as he did by sending thousands of US service members to help Border Patrol agents along the border.

However, there is an exponential leap from deploying the military to help secure the border and deploying the military, the country's fighting force, to actually construct a wall.

Pentagon spokesman on December 11 did not deny the military could technically be called upon for wall construction under certain circumstances.

"To date, there is no plan to build sections of the wall. However, Congress has provided options under Title 10 US Code that could permit the Department of Defense to fund border barrier projects, such as in support of counter-drug operations or national emergencies," Lt. Col. Jamie Davis told CNN.

But Democrats have made clear they would fight any such move by Trump and have pointed out that type of national security authority is not usually invoked inside the US.

The wall is 'artistically designed steel slats'

U.S. Department of Homeland Security Secretary Kirstjen Nielsen, center, speaks in front of a newly fortified border wall structure Friday, Oct. 26, 2018, in Calexico, Calif.

Trump once pledged his barrier on the southern border would absolutely not be a fence, but rather a "big beautiful wall."

It's worth noting a change in his language that occurred this week when the President said it would still be beautiful, but is now set to be "artistically designed steel slats, so you can easily see through it...."

Back in 2015, during the Republican primary, Trump was clear when he shot back at Jeb Bush, who was then running for president, that it would be a wall:

"Jeb Bush just talked about my border proposal to build a 'fence.' It's not a fence, Jeb, it's a WALL, and there's a BIG difference!"

After winning the election, during his first news conference as President-elect, Trump again dismissed the idea of a fence.

"On the fence -- it's not a fence. It's a wall," he said

But the type of structure Trump now seems to be referring to, with his talk of steel slats, appears very similar to the "bollard-style" barriers referred to as fences under previous administrations. Congress has authorized construction of similar types of fencing.

But at an event at the border in October, unveiling a barrier to replace existing fence, Trump's homeland security secretary, Kirstjen Nielsen, said it was actually wall.

"It's different than a fence in that it also has technology. It's a full wall system," she said in Calexico, California, according to NBC . "It's a wall. This is what the President has asked us to do. It's part of a system."

It will be interesting and important to note if this system of what was previously known as fence is accepted as Trump's wall. Fence or wall, the construction Nielsen unveiled is certainly a barrier. But not one for which Trump has funding to continue.