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How would a space tether be built if they had the materials to build it?

There have been many different attempts by scientists and engineers to devise a way to build a space tether or ‘elevator’ and yes most of them involve constructing a really long cable of some kind – 38,000km or more to be exact!
The idea is that the cable would be in a geostationary orbit around Earth, one end of the cable would be attached to either a fixed or mobile platform on the Earth and the other end would connect with some kind of counterweight, high above the Earth’s atmosphere.
The elevator would prove highly useful in being able to get things into space without the need for launching rockets, however such a project is not without its share of difficulties.
The main technical issue to overcome in building such a cable is to keep it from collapsing under its own weight. One idea is to vary the thickness of the cable to allow for the tension to stay constant throughout, as the gravitational force on the cable increases the closer it is to the Earth. It would need to be built out of a material which is incredibly strong, yet very light. In fact the strength required from such a material would need to be at least twice that of diamond!
Currently there are no such materials which could handle this strain, however there is a lot of exciting research happening at the moment in the field of carbon nanotubes which may prove useful. Carbon nanotubes are extremely strong carbon structures which have been synthesised in laboratories and it has been theorised that these nanotubes may have what it takes to withstand the strain while still being a relatively light material.