Originally Posted by

**Speakpigeon**
I voted "not valid".

It's a standard proof. So, you wanted to know about the impact of having what you believe to be the "wrong" logic on mathematics? Well, as I pointed out, it's pervasive. This proof is not weird, or suspect, or anything like that. This is (pretty much, not every detail; I don't remember every detail, plus I simplified it a little because the only prime I consider is 2) a proof that was given in a basic algebra course I took (when talking about the applications of prime factorization, iirc). That was before I took a course in mathematical logic. I understood the proof. All of the other students understood the proof. Very probably (almost certainly), none of them had taken a course in mathematical logic, either (that's generally for considerably more advanced students). We all reckoned the proof was correct. That is our intuitive sense of logic.

If you want evidence that this is a standard proof, then look it up: look for proofs of the irrationality of the square root of two, and you'll see that, other than details, they're essentially of the same sort. I don't think you will find any that is valid in your logic. For example: just google "square root 2 irrational" without quotation marks to see the arguments.