A "missing" axiom, in my experience, is not truly a missing axiom that otherwise has no impact on other axioms. A "missing" axiom is one that exposes a bad assumption in another axiom currently being relied upon.
For instance. Socrates is a man, all men are mortal, therefore Socrates is mortal.
But then you discover that a couple of eons have passed and Socrates is still alive. Clearly there must be a "missing" axiom. And after some investigation you realize that Socrates is a Venusian man, and Venusians are immortal.
"Socrates is Venusian" is a missing axiom, but really the problem is that "All men are mortal" is actually false, since it had implicit assumptions that "All men are human" (false) and "All humans are mortal" (true).
Again, I am sorry for being direct, but this does not make sense. If a Venusian man is immortal, then the "axiom" (preposition) that all men are mortal is false. In other words, the issue is not that the preposition "Socrates is Venusian" was missing but that the preposition "all men are mortal" is false.
It is possible to develop significant mathematical theory without using some axioms. For example, mathematicians sometimes choose not to use the "axiom of choice" when working with Zermelo–Fraenkel set theory. That does not mean that mathematical theorems proven without using the axiom of choice are invalid, even if you later assume that this axiom is true (or false).
The point is that the axiom that all men are mortal was thought to be true, and then was later discovered to be false. My comment was actually in agreement with your previous comment.
For instance. Socrates is a man, all men are mortal, therefore Socrates is mortal.
But then you discover that a couple of eons have passed and Socrates is still alive. Clearly there must be a "missing" axiom. And after some investigation you realize that Socrates is a Venusian man, and Venusians are immortal.
"Socrates is Venusian" is a missing axiom, but really the problem is that "All men are mortal" is actually false, since it had implicit assumptions that "All men are human" (false) and "All humans are mortal" (true).