The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to "see" events occurring at the event horizon?
Similar questions have cropped up on this site many times, and the debate surrounding them is usually fractious because people misunderstand each other's use of words like exist.
One of the lessons of General Relativity is that any observer has to choose a locally convenient coordinate system that may not be globally convenient. We on Earth (quite sensibly) choose time as measured on our clocks and distance as measured by our rulers, and these coordinates are known as the Schwarzschild coordinates (strictly speaking they are shell coordinates, but the difference at the orbital distance of the Earth from the Sun is negligable). Locally our coordinates work very well, but when the central body is a black hole the coordinates become increasingly curved as you approach the event horizon and at the event horizon they fail completely resulting in a coordinate singularity.
I've made this sound like a mathematical nicety, but it's quite real. Remember that by the time coordinate I mean the time we measure on our clocks, and that means there is a singularity in our measurements of time at the event horizion. This is why it takes an infinite time for anything to reach the event horizon, let alone cross it.
The question is whether it is therefore correct to say that: the event horizon never forms. It is quite true that you and I and everyone outside the black hole will never measure the time the event horizon forms, because it would take an infinite time. However there are lots of coordinate systems that have no singularity at the horizon, such as Gullstrand-Painlevé, Eddington-Finkelstein and Kruskal-Szekeres coordinates. The trouble is that these coordinates are somewhat abstract and do not coincide with the experience of any human observer. However since such coordinates exist, physicists tend to be quite comfortable stating that black holes form even if human experimenters could never observe it.
It's a matter of what you mean by "see". Even for a distant observer, it will take a small amount of time for the gravitational redshift effect to become essentially infinite. If your collapsing gas star redshifts to the point where it won't emit a single photon in the age of the universe, it may not have yet technically "redshifted to zero", but it has functionally redshifted to zero as far as experiment is concerned.
The actual picture seen by an external observer is consistent with the general picture described by a naïve interpretation -- a black object that is not emitting anything other than (what is, for macroscopic holes, a vanishingly small amount of) Hawking radiation, that will absorb objects that enters it. I should also add that the apparent horizon of a collapsing star will move outward at a speed faster than the speed of light, which will solve the infinite redshift issue, as well.