Those treasures include a marvellous library, right out of Harry Potter, and a dizzying spiral staircase that was actually used as a location for one of the Harry Potter films. There is also a large room devoted solely to a vast model, made by the architect of St Paul's, Sir Christopher Wren, of his original design. It is very different, in many ways, from the building we see today. Wren was not just a great architect. He was a revolutionary architect, whose ideas did not all find favour with the authorities of St Paul's. Some of his notions were rejected by them, but he sneaked a number back when the construction took place and it was too late to get rid of them.
Today this marvellous late 17th-century building sits in the heart of the old city, occupying a vast space and looking, in its way, like a stately battleship topped by a vast dome. It is arguable that, with St Peter's in Rome, this building is one of the two most famous churches in the world. It has the Wow factor.
Why, today, with all the ingenuity of modern architecture, do we have so few new Wow buildings erected, not just in London but in the whole of Britain?
Last month, in Los Angeles, I visited two such edifices. One, if it can be described simply as a building, is the Getty Museum. Designed by Richard Meier, it sits atop a mountain, has fabulous views, consists of pavilions, imaginative open spaces and dazzling gardens (including one covering 134,000 ft2), occupies 750 acres, and cost $1bn. It amazes visitors who may never, in a full day, even have time to set foot in the pavilions and inspect the works of art contained therein.
It could be contended that there is no possibility of such a complex in London, since neither the space nor the funds would be available. No such argument can be put forward as a reason why we in Britain cannot come up with much, if anything, to compare with the other Los Angeles building I saw, due to open in October: the Disney Concert Hall.
Why are so many trumpets blown for a drab and boring art gallery like the Tate Modern?
The budget for this is a mere £150m, about half what London's new (and unnecessary – but that's another column) Olympic stadium is scheduled to cost, with about one-fifth of the money contributed by the Disneys themselves: hence the name of the hall. Although planned before the Guggenheim Museum in Bilbao, Spain, its swirling, intoxicating lines bear a family resemblance to the Guggenheim.
This is not surprising since the Disney hall, like the Guggenheim, was designed by the planet's hottest architect, Frank Gehry. It occupies a modest space in dingy, downtown Los Angeles and, both its cost and area would be more than feasible either in London or any other British city. And, yes, it is a Wow building.
It made me both excited and ashamed. Britain is a rich country, teeming with distinguished designers. Why do neither government nor corporate wealth have the daring or imagination to take chances?
Why is the most exciting recent addition to London's skyline in recent years the London Eye, which would not be claimed by its clever designers as anything other than a wonderful piece of fun, and was not even intended by them as a permanent structure? Why are so many trumpets blown for a drab and boring art gallery like the Tate Modern, which would not excite the milk off a cooling cup of coffee? And do not tell me about the Great Court of the British Museum, which is an interior. And do not cite the British Library, which, whatever might be thought of it, was planned in the third quarter of the 20th century.
Gerald Kaufman is MP for Manchester Gorton and chairman of the culture, media and sport select committee.