The Puzzling Paradox of Broad’s Basement

"If 90% of your work is in storage you need to begin lending it to other institutions. Get art out of the basements."
—Eli Broad, speaking at the annual meeting of the American Association of Museums, New York

The Hilbert Hotel is always full, yet it always has room for one more guest. It appears that Eli Broad has taken a leaf from this famous paradox (or maybe he saw Inception). Broad has an equally mind-bending plan for getting all the nation's art out of museum basements. Just lend it to other museums.
Museum #1 wants to show the great art it's got in storage. To do that, it must find a place to put the art it's already displaying in its exhibition galleries. It lends that art to Museum #2. This means that Museum #2 needs to free up space, so it lends the art it was exhibiting to Museum #3. And so on, and so on.
This isn't Broad's idea, it's David Hilbert's. His hotelkeeper can always free up a room by moving every guest from room N to room N+1. Needless to say, Hilbert was a mathematician, not a hotelkeeper. There's a fairly serious catch. Hilbert's scheme works only when the hotel has an infinite number of rooms.
Broad has long struck the quasi-populist note that evil, elitist museum directors are scheming to put art (Broad's art!) in storage. He's used this versatile talking point to justify yanking his collection from BCAM and building yet another museum to house it. (The new museum will reportedly be smaller than BCAM: Broad is a man of many paradoxes.) Broad talks as if everything in his 2000-piece collection can and must eventually be on permanent view. The art that's not in his planned museum will be lent out, notwithstanding the fact that this would require the equivalent of about ten Whitney Museums, sitting empty out in the hinterlands.
The bottom line is that there is more art than museum space to show it. Thus museum installations, particularly of contemporary art, are ever-changing and (to use the fashionable term) "curated." What's so bad about that?