In the context of noisy channel coding, a theorem by Shannon says that, by using suitable channel codes, communication with rate up to the channel capacity is possible.
I've seen numerous claims that this capacity (of one way communication) cannot be improved by adding a feedback channel, which tells the sender which sent symbols were corrupted by the noisy channel.
In a lecture by MacKay, he proves this fact for the binary erasure channel (BEC). This channel essentially loses a fraction $p$ of sent symbols and has a capacity $C=1-p$. I have doubts about the proof presented in the lecture, since it seems to assume that any lost symbol can be re-transmitted and will then not be lost a second time. Perhaps asymptotically this assumption can be justified but it's not obvious to me.
On the other hand, I've seen claims in literature saying that the feedback does not help for any channel. I'm not sure how the feedback channel is defined exactly for this claim to hold. Is the feedback channel itself also noisy? I think, with added noiseless feedback, the binary symmetric channel (BSC) and the BEC should have the same capacity, since it no longer matters if the bits are lost of flipped. This seems to contradict the claim, meaning that the feedback channel assumed there is not noiseless. Is it then the same channel added backwards?
Finally, how does one prove this result rigorously (perhaps for BSC as an example, or in general)?