How can axioms be not always true? I'm reading this book: https://www.amazon.com/Learning-Reason-Introduction-Logic-Relations/dp/047137122X/ref=sr_1_1?dchild=1&keywords=learning+to+reason&qid=1584305939&sr=8-1 and at page 193 it says:
"The axioms serve as the truth foundation of that particular system. If that system does not suit our needs for a particular purpose, then we can use another slate of axioms and build another axiomatic system. Thus, the axioms are relative truths. An axiom in one system is not necessarily true in another system."
giving an example, a little vague, saying:
"Until the 19th century it was believed that Euclid's postulates were absolute truths [..] From this unquestioning faith in the five axioms of Euclid, a geometric picture of our universe emerged with straight lines traveling across cosmic distances, straight lines that behaved as they were perceived to behave here on Planet Earth. Current scientific evidence, though, suggests that our view of the universe as Euclidean may be outdated.."
I still dont understand how can some axioms be false in another system, can you guys please give some simple examples on how an axiom can be true in one system and false in another? (please simple, i'm just starting out learning math)
 A: It may be helpful to think of mathematics as a game. Axioms need not have any real-world meaning (though they often do), they're simply a set of rules that lets us play our game. We manipulate the axioms in order to prove theorems.
The example given of geometry is specific and relatively easy to understand. The parallel postulate in Euclid's axioms for geometry states that for any point and any line not passing through that point, there is exactly one line passing through the point that is parallel to the other line. This works for geometry where points and lines in a plane have their usual meaning.
However, let's say we change the definition of what a point and a line are. Let's say instead of a plane we're working on the surface of a sphere. What we call a "point" is now a pair of points that are exactly opposite each other on the sphere, and a "line" is a great circle on the sphere (like an equator). Then we still say that two "lines" are parallel if they don't intersect. It turns out, though, that there are no parallel "lines" when we use this definition. Thus the parallel postulate becomes: "Given a point and a line not passing through that point, there does not exist any line passing through the point that is parallel to the original line." So Euclid's parallel postulate is no longer true. It directly contradicts the new axiom. But the new axiom would not be true in the Euclidean plane.
We have much more freedom than simply changing the interpretation, though. To an extent, you have the freedom to begin with any axioms you want, though some are more useful than others. Then in particular the negation of those axioms are not "true," or at least not provable from the axioms (unless the theory is inconsistent). "True" and "provable" turn out to be distinct concepts, even if your axioms do have a real-world interpretation.
