Last week, kicking off his Independence Day activities, Donald Trump flew off to North Dakota’s Badlands in his new Air Force One, a personal gift from the government of Qatar that Trump gets to keep. His destination was the new Theodore Roosevelt Presidential Library, in Medora, North Dakota, gateway to the Theodore Roosevelt National Park.

Weirdly, Trump has latched onto TR as a kindred soul. He likes TR’s “New Nationalism,” which included colonial excursions in Cuba, Puerto Rico, and the Philippines, as well as the separation of Panama from Colombia for the Panama Canal. At the library, he engaged in a conversation with an AI version of TR, including an exchange about the Panama Canal, which the AI version of TR described as his “proudest achievement.”

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But apart from Trump’s affinity for Teddy Roosevelt’s imperialist excursions, it is hard to imagine two greater opposites than TR and Donald Trump. Roosevelt, of course, was a domestic progressive, promoting antitrust, progressive income taxation, and regulation of the abuses of capitalism. He was a great conservationist, in contrast to Trump’s role as great despoiler.

Teddy Roosevelt was especially noted for his integrity. Earlier in his career, he had led a band of reformers as a progressive Republican assemblyman in Albany. The reformist integrity carried over when he served as a civil service commissioner in Washington, and as police commissioner and then governor of New York.

But the personal differences are the most striking. A sickly child wracked by asthma, TR resolved to strengthen himself by leading what he called the strenuous life. He went west, and later bought and ran a cattle ranch in the Dakotas. In 1898, he resigned as assistant secretary of the Navy, got himself appointed as a colonel, and organized and led a troop of Rough Riders on horseback in the war with Spain over Cuba.

By contrast, President Bone Spurs avoided all military service. His roughest rides are on golf carts.

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Though TR’s family home was in Oyster Bay, Long Island, his presidential library is in Medora, the town nearest TR’s Elkhorn Ranch, a tribute to his affinity for the West. In contrast to the new TR library, which is in keeping with other such presidential libraries, Trump’s latest fixation is to display his trophy Air Force One inside his planned high-rise presidential library and hotel complex in Miami.

It is a mark of Trump’s staggering ignorance of history that he would pick Teddy Roosevelt, of all presidents, as a supposed soulmate.

Two days later, Trump was back in the Dakotas for a major Independence Day speech, using Mount Rushmore as a backdrop. Trump has intermittently said that he belongs on Mount Rushmore, and his supporters have promoted the idea. “There would be no better addition to the iconic Mount Rushmore than the 45th and 47th President of the United States, Donald Trump,” White House spokesperson Taylor Rogers said in a statement to ABC News in advance of Trump’s visit.

In theory, Trump could direct the National Park Service to add a sculpture of himself. This would be entirely in keeping with Trump’s extreme narcissism and his face on everything from public buildings to currency to passports.

But altering Mount Rushmore will be more difficult than demolishing the White House East Wing. For starters, there is no room for another carving. As early as 1936, sculptor Gutzon Borglum, who carved the four presidents, pointed out that the surrounding granite was too unstable to add a fifth likeness. It would be awkward, even for Trump, to order the addition of his own mug by repurposing one of the sculptures of Washington, Jefferson, Lincoln, or his new chum, Teddy Roosevelt.

Robert Kuttner is co-founder and co-editor of The American Prospect, and professor at Brandeis University’s Heller School. His latest book is Notes for Next Time: Surviving Tyranny, Redeeming America. Follow Bob at his site, robertkuttner.com, and on Twitter.