[purveyor of knowledge]

Ok thanks Neutralino, I appreciate for your question which is outstandingly merited and which I will now answer.

Here is all the justification for division by zero of one value.

One value that is divided by nothing is simply a value which is fully connected, integrated, and complete. But in order for it to be this, it must also be a consistent contradiction. This is what allows time to be the way that it is and means that all of the most important scientific laws are wrong and now have to be rewritten.

I am very glad you asked this question in fact, because now I can explain that 1/0 is not infinity or zero, it is just one thing that is so amazing that it is not divided by anything at all, in fact, it is fully unified and completely paramount. It is everything. It is the unified field. It is the nearest point of energy. It is the opposite of nothing. It is the reciprocal of 0.

By saying that multiplicative inverse of a number, defined in this case as 0/n=0 is not unique, I assume you mean that it can be shown that one number can equal any other number. And yes in fact this is true, because it signifies that one moment constantly changes into the next moment, and that energy is constantly being created and can be destroyed or reversed. But zero is not a number and zero is not a moment, and so one moment divided by 0 is all moments undivided. And yes it is very good for maths and for humanity because it means that everything is possible and we can get ourselves out of this mess, which we got ourselves into and have chosen.

And here are all your axioms you seek for

algebraic options concerning division by zero often represent a counterpart to their option for multiplication by zero

For example, any number multiplied by nothing is nothing
Any number divided by nothing is everything
that is because everything is the unified field

Also, nothing times everything is anything, or the number 0/0

Nothing added or subtracted to anything ever changes it's value

And anything added or subtracted to anything never changes it's value

Everything is equal and paramount and the absolute greatest value. Everything is the combination of positive and negative infinity. Positive infinity is our time and negative infinity is there time.