At least one mathematician (I forget his name) considers V=L to be a reasonable axiom to add. Informally put, it says that nothing exists except the things that are required to exist by the axioms. ZF + V=L implies choice, the generalised continuum hypothesis, and many other things. His argument is that just as we consider the natural numbers to be the numbers intended to be generated by the Peano axioms, i.e. the smallest model, so we should consider the constructible universe L to be the sets intended to be generated by the ZF axioms. The axioms amount to an inductive definition, and the least fixed point is the thing they are intended to define. One can think about larger models of ZF, just as one can think about non-standard natural numbers, but L and N are respectively the natural models. I don’t know how popular this view is.
In Peano arithmetic, the induction axiom (not axiom schema) basically says ”… and nothing else is a natural number”. It can only be properly formulated in second-order logic, and the result is that Peano arithmetic becomes “categorical”, which means it has only one (the intended) model up to isomorphism. The real or complex number systems and geometry also have categorical axiomatizations. Standard (first-order) ZFC is not categorical, since it allows both for models that are larger than intended (like first-order Peano arithmetic) and smaller than intended (unlike first-order Peano arithmetic). However, second-order ZFC is also not categorical, although I think it rules out some part of the non-standard models. But your description of the theory ZF+V=L sounds like this theory (i.e. a second-order version) would indeed be categorical. Though presumably this would be somewhat of a big deal but is nowhere mentioned in the Wikipedia article. So I guess the theory probably is still not categorical.
At least one mathematician (I forget his name) considers V=L to be a reasonable axiom to add. Informally put, it says that nothing exists except the things that are required to exist by the axioms. ZF + V=L implies choice, the generalised continuum hypothesis, and many other things. His argument is that just as we consider the natural numbers to be the numbers intended to be generated by the Peano axioms, i.e. the smallest model, so we should consider the constructible universe L to be the sets intended to be generated by the ZF axioms. The axioms amount to an inductive definition, and the least fixed point is the thing they are intended to define. One can think about larger models of ZF, just as one can think about non-standard natural numbers, but L and N are respectively the natural models. I don’t know how popular this view is.
In Peano arithmetic, the induction axiom (not axiom schema) basically says ”… and nothing else is a natural number”. It can only be properly formulated in second-order logic, and the result is that Peano arithmetic becomes “categorical”, which means it has only one (the intended) model up to isomorphism. The real or complex number systems and geometry also have categorical axiomatizations. Standard (first-order) ZFC is not categorical, since it allows both for models that are larger than intended (like first-order Peano arithmetic) and smaller than intended (unlike first-order Peano arithmetic). However, second-order ZFC is also not categorical, although I think it rules out some part of the non-standard models. But your description of the theory ZF+V=L sounds like this theory (i.e. a second-order version) would indeed be categorical. Though presumably this would be somewhat of a big deal but is nowhere mentioned in the Wikipedia article. So I guess the theory probably is still not categorical.