That's a pretty silly statement. Surely you can prove that 1+1!=3, which is the negative of 1+1=3. You can also prove that pi is not a rational number, 1/2 is not an integer, etc. There's a post on the front page about attempting to prove P!=NP.

The typical formulations of all of those proofs are by contradiction, though—so it may be worth distinguishing “prove” from “demonstrate by proof”.

Is there a distinction between proving and demonstrating by proof? I did a quick search (I'm hardly a mathematician), to quote Wikipedia:

> a proof must demonstrate that a statement is always true

It would seem that, if you can demonstrate a statement is true for a given logic, that is proof. Contradiction and exhaustion (which is a bit more controversial, I think), while less tidy than one might expect, are valid techniques for showing that a statement is always true, and therefore are valid ways to construct a proof.

Note: I encourage any and all maths professors to come and tell me off if I'm way off base.

I am wearing only two socks. Both socks are black.

Is it impossible for me to prove that I'm not wearing red socks?

They can prove we don't live in the specific type of simulation they are testing for.

But they have no reason to believe we are in that type of simulation, they just figured out a way to falsify that type of simulation.

1 down, an infinite number to go!