Use of “any”

It is better not to use “any” in mathematics, beacuse its meaning depends on the context. In English, “for any \(x\)” can mean both “for all \(x\)”, i.e., \(\forall x\), and “for some \(x\)”, i.e., \(\exists x\). The choice depends on the context: this is not admissible in mathematics.

Here are two examples from wordreference.com:

  1. I’ll watch any film.
    (I will not watch all movies, but I will just not choose)
  2. Any teenager can tell you how useful the internet can be.
    (All teenagers can answer the question, because you don’t need to choose a specific one)