Hello! I was wondering how does a mathematical statement come to be an axiom? I understand that an axiom can't be proven using other mathematical statements. But how does one know that a statement can or can not be proven? For example, why isn't Riemann Hypothesis considered an axiom? I also read some stuff about Godel's incompleteness theorem and if I got it right, he proved that using a finite number of axioms there will always be stuff we can't prove, so does't this mean that at a point we can come across a statement that we can't prove using the axioms we have by now? So how can we say that a conjecture can or can't be proved and how can we decide if it can be considered an axiom?