Explain this “contradiction” of the proof of $x > 0$ iff $x in mathbb{R}^{+}$
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I am working through Apostol's Calculus I and just read about the order axioms . He presents a new undefined concept called positiveness , gives the axioms and then defines symbols $<, >, leq, geq$ in terms of this new concept of positiveness. Apostol then states Thus we have x > 0 if and only if x is positive. I attempted a proof of this before reading other proofs and quickly ran into something I know is incorrect (I read other proofs later) but haven't developed an intuition for, hoping someone can set me straight. I'll provide the definitions and axioms from Apostol for reference, then show my attempt. From Apostol We shall assume that there exists a certain subset $mathbb{R}^{+} subset mathbb{R}$ called the set of positive numbers, which satisfies the followi