That is very reductive view of what constitutes most axioms.
Merriam-Websters has:[1]
>2: an established rule or principle or a self-evident truth
Etymonline has:[2]
>"statement of self-evident truth," late 15c., from French axiome, from Latin axioma, from Greek axioma "authority," literally "that which is thought worthy or fit,"...
So, no, not arbitrary at all. Which particular axioms do you find arbitrary? Is game theory arbitrary?
Nothing you say conflicts with being arbitrary. Axioms are arbitrarily self-evident—they are "arbitrary" because they are the result of some person's judgement, not somehow produced with some other processes, and they are self-evident because they provide the basis for reasoning and cannot be contradicted within reasoning parameterized by the axioms.
https://math.stackexchange.com/questions/327201/are-there-in...
Change axiom and you end with -another- math.
In Western cultures, the belief in the intrinsic value and sanctity of human life is often considered a foundational and self-evident moral principle. It underlies many ethical and legal frameworks, including principles related to human rights, the value of individual lives, and the importance of protecting life.
An axiom, in this context, is a fundamental and self-evident belief or principle that serves as a basis for other beliefs and actions within a particular culture or belief system.