What's the difference between 'any', 'all' and 'some'?

The term "any" is troublesome, because in natural usage it could mean "all" or "at least one", depending on the context. Here are examples to consider.

(1) For any $a > 0$ there is an $x > 0$ such that $x^2 = a$.

(2) Does the equation $x^3 + y^3 + z^3 = 33$ have any integral solution?

(3) Have you solved any of those problems?

(4) Using this new technique, I can solve any of the problems from that list.

In the first example, "any" = "all". In the second one, "have any" is asking about existence. In the third, "any" means "at least one" (existence). In the fourth, "any" means "all".

I have known weak math students who are native English speakers and think (1) is proved by showing it works when $a = 1$, even though that way of interpreting (1) makes it into a trivial statement. In other words, they interpret "For any" in (1) as meaning "For some", and hence turn (1) into an existence claim instead of a universal claim. Such usage of "any" is present in non-mathematical English (see the third example), and I think this is the basis for the student's misunderstanding (comparable to having to learn the different meaning of "or" in mathematical English compared to non-technical English). I don't think any native English speaker would misunderstand the different senses of "any" in (3) and (4).

I would advise someone who is not a native English speaker to avoid using "any" in mathematical statements. You can convey what you need with other choices of words.

I just want to point out the difference between "for all" and "for any":

1) "for all" usually used in the end of the sentence, meaning the condition is always satisfied. For example, "$x=x$ for all $x\in\mathbb{R}$".

2) "for any" usually is placed in the beginning of the sentence, stressing that you are choosing an arbitrary element. For example, "for any $x\in\mathbb{R}$, we have $x=x$".

Your interpretation of "for some" is correct.

There is no logical difference between "for all" and "for any", they both mean $\forall$. For any number n in the integers, there is always a number bigger than it. For all numbers in the integers, there is always a number bigger than it.

The difference language-wise is that your sentence must stick to the plural form of nouns when using "for all", while you may use singular nouns when using "for any".