To be honest, this is a question that has been bothering me (provided that I understand this correctly), which probably means I'm not quite sure about what I ask, but I'm looking for some clarification.
I am a graduated mathematician who mostly work on graph theory. I admit I have to learn more about mathematical logic, but when I started to dig deeper, I found (and read) something about statements that are independent of some systems of axioms (not just ZFC: it's just what I heard most, more than PA or others). I read that formally some statements cannot be deduced from a selected system.
I found some independent statements (of ZFC) listed in Wikipedia page, and questions similar to this in Math SE too, but I seem to not get the answer I am wondering about. Some said that when you are trying to prove/deduce a statement, you don't necessarily need to care about its independency or something.
I mean, within my simple, curious mind, which currently might not get the idea of independency at all, isn't it natural to ask:
"Does this mean that if we face a math statement and want to know it is TRUE or FALSE, instead of (usually) only conjecturing that this statement is TRUE or FALSE and providing the reasoning (by this I mean I can try to prove it without any knowledge about axiomatic systems that I use at all), we actually need to also consider the possibility that this statement is independent (of something?) or anything like that?"
I hope anyone gets what I am saying and what my struggles are. This is because when I found some statements that are independent of ZFC or something, I was like "Wait, so does this mean that mathematicians every time must realize that when facing a statement, they must either: a) show it is true, b) show it is false, OR c) show it is axiomatic (can't be proven true or false, relative to a system)?"
I am pretty sure I've never considered option c when doing math (or maybe I have, but I did not care and simply accept that I need AC to show each vector space has a basis). Is that such a normal, and preferable approach? Should we just accept that my above concern will rarely happen? Is mine even a valid concern (since if I read it right, I (generally) should always consider doing additional work when trying to verify a statement)???
Maybe future (and more) researches will lead me to have to consider option c one day, though.
*feel free to criticize this question, if any *some of my tags possibly are wrong ones