I think I see what you're asking. I think the answer is: Before we start talking, you have to tell me what all the rules are. So when you say "start with 0/0 = 'undefined'", what is the ambient number system? The proof I gave earlier showed the number system can't be a ring - so it's not the real numbers, the integers, the rationals ... at least not by the standard definitions of those number systems. When you do math, you don't make up the rules as you go.
As far as the example you gave goes, you start by assuming 0/0 is defined. But if you're trying to show 0/0 is defined you're assuming what you want to prove. The logic isn't correct.
Note that giving 0/0 the name "x" doesn't do anything.
Simply naming something doesn't establish any fact. It just makes "x" shorthand for "0/0".
Anyway, observing that 0/0 * 0 = 0 but also 1 * 0 = 0, 2 * 0 = 0, and so on doesn't establish any necessary connection between "0/0" and 1, 2, ... You wouldn't conclude from "1 * 0 = 0" and "2 * 0 = 0" that "1 = 2", or that "1 could be 2", for instance. So nothing has happened. But what could happen? Remember that you started by assuming that "0/0" was defined. "Assume" in math means you've assumed it's true. In that case, you're done, right? Its "definedness" isn't probabilistic. And if starting with that assumption you did find out something true, it doesn't follow that the assumption is "independently" true. (The truth of "if P, then Q" and the truth of "Q" do not together imply the truth of "P".)
You might want to look at the post on wheels higher up this thread. It shows what you could do - namely, use a different set of rules.
As far as the example you gave goes, you start by assuming 0/0 is defined. But if you're trying to show 0/0 is defined you're assuming what you want to prove. The logic isn't correct.
Note that giving 0/0 the name "x" doesn't do anything. Simply naming something doesn't establish any fact. It just makes "x" shorthand for "0/0".
Anyway, observing that 0/0 * 0 = 0 but also 1 * 0 = 0, 2 * 0 = 0, and so on doesn't establish any necessary connection between "0/0" and 1, 2, ... You wouldn't conclude from "1 * 0 = 0" and "2 * 0 = 0" that "1 = 2", or that "1 could be 2", for instance. So nothing has happened. But what could happen? Remember that you started by assuming that "0/0" was defined. "Assume" in math means you've assumed it's true. In that case, you're done, right? Its "definedness" isn't probabilistic. And if starting with that assumption you did find out something true, it doesn't follow that the assumption is "independently" true. (The truth of "if P, then Q" and the truth of "Q" do not together imply the truth of "P".)
You might want to look at the post on wheels higher up this thread. It shows what you could do - namely, use a different set of rules.
This may be more than you wanted to know ...