But what he said was closer to
I don’t dispute that [formula] has no solution. … But what in my set of math axioms makes that so? I know the answer to the formula. I don’t know how/why I know.
Please explain that to me simply."
It’s the meta question that’s interesting, not the brute algebra.
But the other thing he said in the OP — the thing we’d maybe need to explain to someone else, but which he’s already spelled out — is that: “If we are asked to solve the equation for x we end up with the equation 0 = 5 which is false.”
So he doesn’t merely tell us that he doesn’t dispute that it has no solution; he tells us that 0=5 is false, and he tells us that solving this would mean ending up with the conclusion that 0=5; that’s why he says he doesn’t dispute it.
When is it okay for me to (a) be at a loss for words because I (b) note that those are already all of my words?
I suppose my motives for the OP might have been a little cryptic. I actually do know the basics of Proof by Contradiction, Axioms of the Real Numbers, The Law of the Excluded Middle, what an equal sign means, etc. I didn’t really need an explanation for me. By including “explain this to me like I’m five years old” I was hoping to get some ideas on how to better explain this to others who have very little experience with math but need to learn this for work. Nonetheless it was fun reading all of the responses. Thank you.
While looking for something else I stumbled on this old thread that might be tangentially useful to the OP.