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I apologize for not reading the entire post, but I don't have the patience tonight. Couldn't this problem be solved by saying that 0 is not a normal number with normal properties? Therefore, it might not fit into all of the properties of equality.

1/0, 0/0, (5)x(0) = (88)x(0), 5^0 = 77^0

are a few example of really weird math things. Accepting 0 can not be used as a number but as a placeholder for an ideal would be better because then you won't make logical fallacies.

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In general, this is sort of how things are done. In abstract algebra you often define 0 as the additive identity (and then prove there can be only one).

But, you don't want to give it too much special treatment. You can prove that x * 0 = 0. Why is this nice? Because you can apply abstract algebra to systems that are not just arithmetic. It can, for example be applied to set theory where you do this:

Addition --> Set intersection Multiplication --> Set union

If you do that, and all the axioms are met, then you get a bunch of theorems about set theory for free. And its not just set theory. I've heard about this being applied to material science as well.

The thing to remember is that the operators such as +, *, -, / are very specific with arithmetic but in the more abstract fields, they are place holders for operators that share some generic properties.

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