Consider a metric without the positivity axiom. Then this set of axioms is inconsistent.

Consider a metric on V such that $$i)\quad d(x,y)=0\iff x=y,$$ $$ii)\quad d(x,y)=d(y,x)\forall x,y\in V,$$ $$iii)\quad d(x,y)+d(y,z)\geq d(x,z).$$
So we do not take $d(x,y)\geq 0$ as an axiom. With this we will try to show that this set of axioms is inconsistent i.e. there exists a statement for which both itself and its denial can be derived from the axioms.

Consider $d(a,b)<0 $. Then by axiom $i$ we have that $$d(a,a)=0.$$
Now axiom $iii$ implies that $$d(a,b)+d(b,a)\geq 0\iff d(a,b)\geq -d(b,a)\implies d(a,b)>0.$$
This contradicts our initial assumption that $d(a,b)<0.$
Now is it correct to be stated that without the positivity axiom these statements are inconsistent or can the four axioms be considered redundand i.e. that positivity can be implied by the other 3 axioms?

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