TITLE: a question about subset of R
QUESTION [0 upvotes]: Suppose we have a set $S$ of real numbers,such that 
$ n \neq \Pi_{m \in S-{n}}    m^{r_m}$.......  $\forall n\in S$ where  $r_{m}$ is a rational number....
 The set of prime number is an example of such a ser...now my question is that...Is such a set necessarily countable?

REPLY [3 votes]: Since we are talking about rational exponents, I assume it makes sense to restrict to positive real numbers. If I got it right the question is whether a set $S$ where no element is a product of some others to rational powers is necessarily countable. 
One can show non-constructively that the answer is no, that is, that there exist uncountable sets with this property. This is because the condition on $S$ is equivalent to $\{\log x: x\in S\}$ being linearly independent over the rational numbers. 
Since there is a basis for $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and such a basis is necessarily uncountable, there is an uncountable set $S$ with the given property.
{"set_name": "stack_exchange", "score": 0, "question_id": 65254}