Could we build a 20 kilometer high space tower?

Science-fiction novelist Neal Stephenson imagines a 20-kilometer-high steel tower that reaches into the stratosphere.
From that height, planes could save fuel by docking at the tower rather than landing, and space missions could do the same by launching from it.
Stephenson is teaming up with a structural engineer, Keith Hjelmstad at Arizona State University (ASU), to work out how to actually build the tower, New Scientist reports.
Hjelmstad is now analyzing the feasibility of Stephenson’s tower. Preliminary modelling suggests that it could support its own weight, but many questions remain. “The tower pushes well beyond anything anyone has ever done in structural engineering,” says Hjelmstad. “Building [it] would be the biggest project ever undertaken by humans.”
The tower story will be published by Hieroglyph, a joint project between the centre and Stephenson, which will host online forums for further discussion. Other writers can also use Hieroglyph to contribute new stories and collaborate with ASU scientists. Author Cory Doctorow, for example, has a story and accompanying research project based around the idea of sending 3D printers to the moon, where the devices would build a base for humans to inhabit later.
This month will see the launch of the Center for Science and the Imagination, an ASU project that will bring together scientists, engineers, artists and writers and encourage them to think big. The idea is to team artists and authors with ASU researchers to turn science fiction into reality.
For Ed Finn, who will direct the center, success will be measured in part by a change in thinking about “audacious moonshot ideas” that are seen as too ambitious for the incremental process of contemporary science.