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

There are lots of expressions like, for all x, for any x, for some x, etc.

I think 'for some x in R s.t ~' means that there exists at least one point in R s.t ~~. right?

However, I can't know the difference between 'all' and 'any'.

(It may be because of the fact that I am not a native english speaker.)

Could you explain it? thank you~!

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".