Intereting Posts

Every collection of disjoint non-empty open subsets of $\mathbb{R}$ is countable?
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find area of the region $x=a\cos^3\theta$ $y=a\sin^3\theta$
How rigorous are pictorial proofs?
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Divergent products.
Reformulation of Goldbach's Conjecture as optimization problem correct?
Any Implications of Fermat's Last Theorem?
Weird limit $\lim \limits_{n\mathop\to\infty}\frac{1}{e^n}\sum \limits_{k\mathop=0}^n\frac{n^k}{k!} $
Combinatorics-discrete mathematics
Uniform convergence of sequence of convex functions
angle inside a chordal quadrilateral
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Deduction Theorem Intuition

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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