Yes, the aperture value corresponds to the diameter of the aperture (i.e, twice the radius). The radius is a linear measure. The amount of light admitted through the lens is proportional to the area of the aperture, which is proportional to the square of the radius.

The reason why f-stops are given in increments of 1.4 is that 1.4 is approximately the square root of 2. 2 is a doubling of the area of the aperture => a 1.4x change in the diameter of the aperture. That's why you have common aperture values of 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32. Each stop is approximately 1.4x the last one and corresponds to twice the light admitted through the lens.

An f/1.0 lens would have an effective aperture of 1/1.0 times the focal length. On a 50mm lens, a 50mm aperture. An f/2.0 lens would have an effective aperture of 1/2 the focal length, i.e. a 25mm aperture on a 50mm lens. An f/4.0 lens would have an effective aperture of 1/4 the focal length, i.e. a 12.5mm aperture on a 50mm lens.