"Any" is ambiguous and it depends on the context. It can refer to "there exists", "for all", or to a third case which I will talk about in the end.

https://en.oxforddictionaries.com/definition/any

Oxford Dictionary:

- [usually with negative or in questions] Used to refer to one or some of a thing or number of things, no matter how much or how many.

[as determiner] ‘I don't have any choice’

**which means there does not exist a choice**
- Whichever of a specified class might be chosen.

[as determiner] ‘these constellations are visible at any hour of the night’

**which means for all hours**

Also in Oracle database I remember a query to return the employees where employee.salary > any(10,20,30,40) which means the salary of the returned employee must be bigger than 10 or 20 or 30 or 40 which means "**there exist" one salary** in that tuple such that the employee.salary is bigger than it.

Same ambiguity is in math, since math did not come from nothing, rather it a notation system for the language.

**However:**

Some times "any" is used for the meaning of "any" and not "exist" or "all". For example, in the definition of the little o asymptotic notation we have:

$o(g(n))$ = { $f(n)$: for any positive constant c>0, there exists a constant $n_0>0$ such that $0 \leq f(n) < c g(n)$ for all $n \geq n_0$ }

Here "any" means "any" which is two things "there exists" and "for all" how??

If you take any as "for all c" then the meaning is wrong because $n_0$>0 is attached to some choice of c and for each c there may be a different $n_0$>0.
And you can not find a fixed $n_0$ for all c that satisfies the remaining because c can go very close to zero like c=0.000....00001

If you take any as "there exist c" then the meaning is wrong also because for some c the remaining may apply but for another c the remaining may not apply.

Example: let $f(n)=n$ and $g(n)=2n$:

If $c=1$ then $n < 1 \times (2n)$ for $n \geq n0>0$

But if $c=0.1$ then $n > 0.1 \times (2n)$ for all $n\geq n0>0$

**So** here "any" means "any" which is for all but one at a time , so in the little o asymptotic notation "any" means for all c>0 pick one at a time and the remaining should be satisfied.

**Conclusion:** Either do not use "any" in Math or explain to the reader what it means in your context.