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@Enrico you have not provided a good reason to reject an axiomatic formulation. In fact, you accepted it for vectors but rejected it for spacetime. Your justification was to capitalize the word THE. That is not a valid justification. Nor is bold or italics or underlining.
Since the professional scientific literature uses axioms then those axioms are acceptable here on PF. Material has been provided using those acceptable axioms as a starting point and reasoning from there. As usual, there are multiple such equivalent starting points and approaches that have been provided.
If you find that unsuitable for your preferences then it is up to you to show that your preferences are also suitable by finding a professional scientific reference that embodies your preferred starting point and construction. If your preferences are incompatible with the literature then your preferences should change.
When you have found such a reference please feel free to open a new thread on the topic. Alternatively, while reading the existing material that has been provided please feel free to open a new thread on any point in any of those references that you find confusing.
As this thread has become unproductive and inconsistent with the professional scientific literature it is now closed. The question in the OP has been answered, and the restrictions that you wish to apply are inappropriate, unjustified, and unclear.
Since the professional scientific literature uses axioms then those axioms are acceptable here on PF. Material has been provided using those acceptable axioms as a starting point and reasoning from there. As usual, there are multiple such equivalent starting points and approaches that have been provided.
If you find that unsuitable for your preferences then it is up to you to show that your preferences are also suitable by finding a professional scientific reference that embodies your preferred starting point and construction. If your preferences are incompatible with the literature then your preferences should change.
When you have found such a reference please feel free to open a new thread on the topic. Alternatively, while reading the existing material that has been provided please feel free to open a new thread on any point in any of those references that you find confusing.
As this thread has become unproductive and inconsistent with the professional scientific literature it is now closed. The question in the OP has been answered, and the restrictions that you wish to apply are inappropriate, unjustified, and unclear.