My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further.
Generally, the Real Number System is said to exist and then introduced through axioms, like the ones in the Wikipedia's Construction of the real numbers. The $=$ sign is always used in all the field axioms and in 1 of the order axioms. Why is an equivalence relation, that is the $=$ sign, never defined for reals? Why is always taken for granted?
My understanding is that for a field, an $=$ sign must be defined separately, while for an ordered field, it may be enough to say that
$$ \forall a, b \left[ a, b \in \Re \Rightarrow \left( a \leq b \wedge b \leq a \Rightarrow a = b \right) \right] $$
(antisimmetry) that is that this property may define the = sign, in the sense that every $=$ or $\neq$ sign in the field axioms may be replaced by
$$ a=b \Leftrightarrow a \leq b \wedge b \leq a \\ a \neq b \Leftrightarrow \neg (a \leq b \wedge b \leq a) $$
where the direction $a=b \Rightarrow a \leq b \wedge b \leq a$ is added to the theory exclusively for brevity, but not needed. For example the axiom "existance of multiplicative identity"
$$ \exists 1 \left[ 1 \in R \wedge 1 \neq 0 \wedge \forall a \right(a \in \Re \Rightarrow a \cdot 1=a \left) \right] \\ $$
may be replaced by
$$ \exists 1 \left[ 1 \in R \wedge \neg (1 \leq 0 \wedge 0 \leq 1) \wedge \forall a \right(a \in \Re \Rightarrow (a \cdot 1 \leq a \wedge a \leq a \cdot 1) \left) \right] \\ $$
The differences between the Wikipedia's article and the book I'm reviewing now is that transitivity is not included in the book as an axiom (but proved as a theorem), and reflexivity is not mentioned at all in the book. So reflexivity should also be redundant in the axioms listed in the Wikipedia's article and one should be able to prove it from the others (even though right now I'm now managing to do that).
Is my understanding correct?