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: 
I think the confusion is caused by mixing a technical meaning of "proof" (deriving a result from some axioms and rules) with an informal meaning (showing that something can/cannot 'exist' in the 'real world').

One of the assumptions in Wantzel's proofs is the use of Euclidean geometry. However, general relativity says that space is non-Euclidean, so that proof doesn't apply to the 'real world'.

Every proof can be defeated in this way, using "what if" questions. These kinds of questions, about what is 'real' or 'true', caused a lot of debate before the 20th century. For example, Hilbert sought a single, consistent set of rules which could prove every true statement and disprove every false statement. Many Mathematicians didn't believe Cantor's proofs regarding infinities were 'real'. Kronecker claimed that "God created the Integers, all else is the work of man". There are many more such examples.

These questions are still unresolved, but now fall into the realm of philosophy. Mathematics avoids these issues by accepting that some assumptions are always necessary, that "truth" is an ill-defined concept and we should instead focus on *provability*.
I agree with your initial response. There exist some tasks T for which it is possible to prove that T cannot be achieved.