2. Any general proof that all expressions which are indeterminate forms (in the...

[Secret]

...context of limits) are undefined in ALL CONCEIVABLE MATHEMATICAL STRUCTURES (e.g. number systems, sets etc.)?

Example of indeterminate forms:
http://en.wikipedia.org/wiki/Indeterminate_forms

They are indeterminate because what they are equal to depends on which functions you choose

P.S. From wikipedia and Dr Maths, all 7 of them are one and the same, thus the question boils down to: How to prove 0/0 or infinity/infinity is undefined in ALL CONCEIVABLE MATHEMATICAL STRUCTURES?

P.S.2.

As an aside, I think there is a basic misconception about the possibility or impossibility of dividing by zero. If someone says "we can't divide by zero", someone else will invariably hear "nobody has yet managed to divide by zero". And since it's so extremely easy to come up with a way of dividing by zero, they think they have seen something original. (I'm not saying this is what Robo37 does here, but it certainly is the way the BBC story on Anderson's nullity was worded.) The fact, as you point out and as the article maybe needs to be clearer on, is that 0/0 can be defined to be something (and that it already has been done, several times!), it just can't be defined to be something more useful than "undefined". A useful division by zero should let us, for instance, completely solve the equation x2 = x by dividing both sides by x. As far as I know, not even wheel theory accomplishes that, because it's just not accomplishable. Now, there's a gauntlet for all presumptive zero divisors (pun intended) to pick up. —JAO • T • C 05:14, 14 August 2009 (UTC)

How to formally prove the bolded statement for all conceivable mathematical structures (i.e. all possible cases)?

 

[Fednis48]

I'm not a mathematician, so I don't know if this will be completely rigorous. Rigor in formal math is nightmarishly difficult to pull off. But here goes:

1. Zero is that which, when multiplied by anything finite, returns itself. This is the definition of the zero element and should be true independent of number systems.

2. $$y*0=2*0$$ reduces to $$0=0$$. If we could divide by zero, we should be able to start with $$0=0$$ and divide both sides by zero, leaving $$y=2$$.

3. $$y*0=3*0$$ reduces to $$0=0$$. If we could divide by zero, we should be able to start with $$0=0$$ and divide both sides by zero, leaving $$y=3$$.

4. Points 2 and 3 contradict each other, so we cannot divide by zero.

 

[rpenner]

The number system where $$1=0$$ has division by zero -- it's just not useful.

Example: $$2 = 1+1 = 0+0 = 0\\ 2x = x \leftrightarrow 2 = 1 \leftrightarrow 1 = 0 \quad \textrm{(which is our assumption)}$$

So all numbers x are solutions to "2 x = x" because all numbers are equal to zero.