Perhaps a nitpick, but: axioms are defined to be true. Mathematics, roughly speaking, is the business of exploring the consequences of chosen sets of axioms. Sometimes published mathematical theorems turn out to be wrong.
That distinction is important, and it seemed to me you'd skimmed over it.
Axioms are directly defined to be true for the purposes of mathematical inquiry, and as such can't really be wrong. At worst, a set of axioms can be inconsistent.
Emergent properties are different; they're not directly defined to be true. This is of practical consequence, as mathematicians sometimes get them wrong.
The baffling thing is how few and simple axioms we need for them to be useful.
Color happens in your head.
That is why trichromacy, tetrachromacy, pentachromacy, dodecachromacy etc. is a thing.
If I look outside right now, the sky is probably better described as cyan; quite obviously a different colour from the blue exercise mat that happens to be on my floor.
The sky can also be described by many of the other colours depending on weather or the time of day.
Only the person who responded to the person who made such a statement were more polite about it, hence this "discussion".
I think that in the definition you gave "accepted" should be replaced by "assumed".
The first sentence of that problem is an axiom. It is absolutely 100% true, within the context of that story problem. It could contradict another axiom, but it can't turn out to be false, because it's not a statement about the real world.
There are not "truths" in mathematics, it's literally a system of logic with definitions at the bottom (axioms) and then a following of the results based upon the agreed upon logic.