# On independency of ZFC of statements in math problem solving

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

• See Examples of unprovable statements : "fter the appearance of Gödel's Theorem a variety of statements more or less directly related to provability were shown to be unprovable in Peano arithmetic and certain other axiom systems. Starting in the 1960s the so-called method of forcing allowed certain kinds of statements in strong axiom systems—like the Continuum Hypothesis in set theory (see page 1155)—to be shown to be unprovable. 1/2 – Mauro ALLEGRANZA Nov 30 '18 at 15:23
• Then in 1977 Jeffrey Paris and Leo Harrington showed that a variant of Ramsey's Theorem (see page 1068)—a statement that is much more directly mathematical—is also unprovable in Peano arithmetic." 2/2 – Mauro ALLEGRANZA Nov 30 '18 at 15:23
• When you say “true, false, or independent,” you are mixing metaphors. It is “provable, unprovable, or independent” in a given system. – spaceisdarkgreen Nov 30 '18 at 15:45
• Note that even that third possibility isn't fully satisfactory: there will be statements which are "unprovably unprovable" in ZFC. Basically, there is a sense in which we as mathematicians can never be fully satisfied, or even confident that we will be fully satisfied with respect to a particular problem. Whether that's a positive or a negative depends on one's outlook - personally, I find it quite exciting! But as Carl says, it's a phenomenon one runs into very rarely (and indeed one very rarely needs the full strength of ZFC, or anything close to that), so it's mostly ignorable if one wants. – Noah Schweber Nov 30 '18 at 17:54
• When you are trying to prove something in ZFC, you can always use the fact that any particular statement is true or false. That is, ZFC proves $\theta\lor\neg\theta$ for every sentence $\theta$ (formulated in the language of ZFC). That $\theta$ and $\neg\theta$ might both be unprovable is an entirely different matter. – Andreas Blass Nov 30 '18 at 18:52