TITLE: Explanation of method for showing that $\frac{0}{0}$ is undefined
QUESTION [31 upvotes]: (This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes)
Ignoring limits, I would like to know if this is a valid explanation for why $\frac00$ is undefined:

$x = \frac00$
   $x \cdot 0 = 0$  
Hence There are an infinite number of values for $x$ as anything multiplied by $0$ is $0$.

However, it seems to have got comments, with two general themes.
Once is that you lose the values of $x$ by multiplying by $0$.
The other is that the last line is: 

$x \cdot 0 = \frac00 \cdot 0$

as it involves a division by $0$.
Is there any merit to either argument? More to the point, are there any major flaws in my explanation and is there a better way of showing why $\frac00$ is undefined?

REPLY [3 votes]: Suppose that division by $0$ is possible. Then consider the following equation,
$$x=0\implies1\cdot x=0\cdot x\implies \large{\large{\color{blue}{1=0}}}$$
This implies that the successor of $0$ is equal to it and that contradicts the Peano Axioms. The above was possible because we assumed that division by $0$ is possible.