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On the axioms of the real number system as stated in Apostol’s textbook

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The typical axiom system for the real numbers states that the real numbers satisfy the axioms of an algebraic field plus a few others.

In the mathematical analysis textbook by Apostol, the axioms are written in a slightly different way. In particular, he states that “given any two real numbers $x$ and $y$, there exists a real number $z$ such that $x+z=y$. This $z$ is denoted by $y-x$.”

I think it should have been stated that there exists a unique such $z$. Even the definition of $y-x$ is not well-defined if the uniqueness assertion is missing. Or can the uniqueness be proved using the axioms? I couldn’t see how.

For example, the cancellation law $x+y=x+z \rightarrow y=z$ would hold if we could show that $0_x:=x-x$ is an additive unit. But if we tried to prove it, we would need the uniqueness assertion in the axiom. That would be a circular reasoning.


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