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JamesGold
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Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?
Consider the set ##\mathbb R^2=\{(x,y)|x,y\in\mathbb R\}## of ordered pairs of real numbers. If we define the addition of two arbitrary members of this set byEvery resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?
To what extent are they arbitrary?
The best one can hope to prove is that there is a vector space in the branch of mathematics defined by this other set of axioms. If you just supply the missing details from what I said about ##\mathbb R^2## above, you're almost done with such a proof. You could take the definition of the real numbers and a few set theory axioms as your starting point (your axioms need to e.g. guarantee the existence of functions and cartesian products), or you can take the axioms of ZFC set theory as your starting point and start by proving the existence of a Dedekind-complete ordered field (i.e. a set whose members have all the properties listed in the definition of the real numbers).If you want to find a lower level mathematical system that can be used to prove the axioms of vector space, I don't know of any work that has done that, but someone else on the forum probably does.