Question: A friend of mine asked me if I could explain this statement: "It's not logically possible to prove that something can't be done". The actual reason is the understanding of this strip: 

![enter image description here](https://i.stack.imgur.com/YtHMp.gif)

Since I'm not an expert on logic, but at least know the basics, I told him that the statement is false. To prove it I gave him the counterexample of the impossibility of trisecting an angle by using only straightedge and compass, which was proved by Pierre Wantzel. This shows clearly that it's possible to prove that something can't be done.

When I told him this, he explained to me that this is sort of naive because Asok, the guy who makes the statement, is a graduate from IIT and also I read that he's always a brilliant character in the strip. 

Can you guys please tell me what you think about this? Also I'd like to know how you would interpret this dialog.

Answer: 
It's a paradox: by saying "it's not logically possible to do X", Asok is, in effect, claiming that there must exist a *proof* that X is impossible. Which is impossible, if Asok is right.

Hence Asok must be wrong!
Technically, you *can* prove that something can't be done, provided that you choose an appropriate "something" to prove impossible. Several examples have been cited in answers.

On the other hand, the illogical scientist says Asok's software will not *work*.
How does he define that claim? Does he mean that there is a pathological sequence of
events that is at least remotely possible and that will cause Asok's software not to
deliver the desired result in that case? He may be right, but there are plenty of
very successful software systems that have that characteristic.

Suppose Asok is writing a program to analyze the company's other programs and tell
which of the functions will halt on what input. We know Asok's program cannot
produce a "yes" or "no" answer in $100\%$ of all possible cases, but even
if Asok's program produces a "yes" or "no" answer for only $90\%$ of the cases
on which it is actually tried, it can be very useful, so to say "it does not work"
in nonsense.

Taken word for word without any additional context,
Asok's statement is overly broad, and you have (I think) a valid counterexample.
In the context of software engineering, however, where something "works" or
"does not work" depending on whether it exhibits suitable behaviors over
finite periods of time and finite amounts of input, at least the well-known
"impossibility" proofs say only that the software will *not always* work.
This is not at all the same thing as proving it *never* can work.