Nesting the same quantifiers I already asked a similar question on StackExchange, but I have the feeling that I don't really understand the issue yet, so let me ask this question to understand it once and for all.
If I have found after long search three objects a, b and c with the property P(a,b,c) then I can claim that there is an x, a y, and a z such that P(x,y,z). Also, if I proved that Q(x,y,z) holds for every combination of objects x, y and z, then I can claim that for all x,y and z, Q(x,y,z) is true. I am NOT thinking of this claim as asserting "For all x, we have that for each y the following holds: for each z, Q(x,y,z)". Similarly, I am NOT thinking of the above existential claim as asserting "There is an x for which there is a y with the property that there is a z such that P(x,y,z)".
In other words: In my naive understanding as a beginner, I regard the natural language expressions "for all x,y,z, ..." and "there are x,y,z such that" NOT as nested quantifiers in the sense above.
Now my question is: in mathematics, is one always regarding statements having the form "for all x,y,z, ..." or "there are x,y,z such that" as being abbreviations of the statement formulated with nested quantifiers (for example, "There is an x for which there is a y with the property that there is a z such that P(x,y,z)" in the existential case). And how can one intuitively understand why the meaning of the natural language formulation is essentially the same as the statement formulated with nested quantifiers?
 A: You are right - this is a point where natural language and mathematical language differ in form (as to content, see below). They don't need to - we could introduce multi-ary quantifiers so that e.g. "$\forall x, y, z$" is a meaningful expression - but this isn't what's generally done (although you'll often see this in practice, reflecting the fact that this is how natural language works).
Why isn't this done? Well, if you look at them, you'll see that they wind up meaning the same thing. Let's look at the two-variable case since it's a bit easier to think about. When is $$(*)\quad \forall x\forall y P(x, y)$$ true? Well, think about counterexamples: the only way $(*)$ could be false is if for some $x$ (say $x=a$) there is some $y$ (say $y=b$) with $\neg P(a, b)$. But then $\forall x, yP(x, y)$ is also false, via $x=a, y=b$.
So there's no extensional difference between the two expressions, which means the next question is whether there is a meaningful intensional difference. This is of course a subjective question; my instinct, though, is no. I see no additional information that one carries but the other doesn't. In general, if you want to argue that there's an intensional difference between two expressions $A$ and $B$, (I would say) you need to argue that there's some informally-described context where one could hold and the other fail. The various notions of implication constitute a good example of this. But here, no such context occurs to me.
Well, there is one context I can think of, which applies to $\exists \exists$ (but not $\forall \forall$): thinking in terms of Skolem functions, "$\forall x\exists y\exists z(P(x, y, z))$" becomes "$\exists F\exists G\forall x(P(x, F(x), G(x, F(x))))$". This is a weaker expression than "$\exists F\exists G\forall x(P(x, F(x), G(x)))$", which essentially has the two existential variables "disentangled" from each other. I could see thinking of this disentangled version as the right interpretation of "$\exists x, y$". If this sounds interesting, then you want to look at independence-friendly logic and the closely-related (but not totally equivalent) dependence logic.
