Consider the equation x + 1 = x.

(Edited again: this example is wrong, and thanks to Kindly for pointing out why. CronoDAS gives a much better answer.)

Curiously enough, the Peano axioms don't seem to say that S(n)!=n. Lo, a finite model of Peano:

X = {0, 1} Where: 0+0=0; 0+1=1+0=1+1=1 And the usual equality operation.

In this model, x+1=1 has a solution, namely x=1. Not a very interesting model, but it serves to illustrate my point below.

sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to "intriguing new ideas", but sometimes it just indicates that your assumptions are wrong.

Contradiction in conclusions always indicates a contradiction in assumptions. And you can always use different assumptions to get different, and perhaps non contradictory, conclusions. The usefulness and interest of this varies, of course. But proof by contradiction remains valid even if it gives you an idea about other interesting assumptions you could explore.

And that's why I feel it's confusing and counterproductive to use ironic language in one example, and serious proof by contradiction in another, completely analogous example, to indicate that in one case you just said "meh, a contradiction, I was wrong" while in the other you invented a cool new theory with new assumptions. The essence of math is formal language and it doesn't mix well with irony, the best of which is the kind that not all readers notice.