If you have different rules, you have a different object and the meaning of the axiom is different. Even if it is written using the same symbols.
To continue the compiler analogy..
Just because the ASCII sequence "int c=0;" means different things in C and Java, doesn't imply "int c=0;" is meaningless when specifically talking about only C.
You can't have the same axioms with different rules of inference. The rules of inference are axioms.
By the axioms being the same, you mean the ink-shapes making up the symbols on a piece of paper being the same.
Not the actual meaning behind them.
Edit: Since you added more content to this reply later on, let me respond. In this case I was talking about specifically a formal system where a=0 makes sense and is expressible, but to keep it short and concise I didn't choose to write out all the details. So the rebuttal is moot.
Omega is an amazing number. We know it has a digit in its binary expansion that is 0 or 1, yet it is impossible to formulate why. It is impossible to come up with a train of thought that explains it. There is no explanation that can be written down on a piece of paper.
So we are left with the weird conclusion:
1. It has the value it has for no reason.
2. It has the value it has for a reason that is impossible to formulate.
It's a faulty analogy. In fact, a paperclip maximizer goes against every single principle of capitalism out there.
The paperclip maximizer is bad exactly because it isn't based on capitalism; it will maximize paperclips regardless of whether someone values them or not! At _any cost_, it will maximize paperclips.
Markets, on the other hand, have the goal of providing products/services that people are willing to pay for. It adapts. The more people value it, the more capital it will attract, and the more resources will be allocated to it.
Isn't that an absolute truth?