I had read that the space elevator wouldn't work beacause the tensional stress required in the cable was unendurable by any known material. Has that changed in the recent past?
Kevlar was introduced about the time Arthur Clarke wrote "Fountains of Paradise", with a popular science fiction plan for the elevator. It was about an order of magnitude away from the strength/weight necessary.
Today nanotube technology is approaching the point of necessary properties, but it will be decades before the massive production scale and uniform quality make the elevator possible.
The project is ten times the size of Apollo or Manhattan or the space station, so the biggest questions will be about economics and strength of leadership.
So, they don't have a material that's strong enough, yet. But given the rate of progress in the nanotech field and the number of people working on it, it's only a matter of time.
How do we go up and increase lateral speed without bending the cable into a big "C" shape?
Lateral speed is everything. Taking an object up the elevator and pushing it out the door at 100 miles in altitude (shuttle orbiting height) will do no good. It doesn't have orbital speed to stay there. At any elevator height.
True. Science fiction it is and science fiction it will remain.
Has that changed in the recent past?
They are looking for the ability to buy unobtainium with the investors bank accounts.
Carbon Nanotubes can handle the stress.. at this point there this can become reality.. there are obviously technical issues to overcome, but none of them are major.
Basically the only factor that is going to determine if this happens is whether or not it will continue to be funded.