I asked this earlier today and received lots of confusion and misunderstanding. Here are some clarifications:
I am not asking for intuitive reasoning that division by zero is impossible or nonsensical. I am asking for a mathematical proof that $0/0$ is specifically not equal to zero. Also, the argument that $a/b$ is the unique solution to $a=bc$ does not apply in this scenario because $b=0$ therefore $b/b \ne 1 $*$ $ and you can't move it to the other side of the equation by multiplying the equation by $b$.
Finally, please don't mark this as a duplicate of my previous question. Said question was marked as a duplicate of something that didn't answer the question. Nobody responded to the question post with an answer to the question.
Sorry for previous confusion. Once again, please prove that $0/0 \ne 0$. And once again, please try to avoid an exclusively intuitive proof.
*There exist multiple proofs for this, and one (of which I know) for the more broad statement: $0/0 \ne c \ :\ c/c=1$. In order to avoid confusion, I won't attempt to transcribe said proof for the broad statement.