Your first sentence is true (modulo the word "inductive"), but not in the way you mean: $ZFC$ proves the existence of many set well-orderings of each $V_\alpha$. Now, under some further assumption - say, $V=L$ - there might be a sequence of somehow canonical well-orderings of the $V_\alpha$, in which case we can indeed "glue them together" to get the global well-ordering you seek. However, if there is no such definable sequence, then we can't define your well-ordering, since we would need to pick some specific well-ordering of each of the $V_\alpha$s simultaneously - that is, find a choice function for a class-sized sequence of sets. And this, of course, is exactly global choice.

EDIT: One positive point. Even though choice doesn't imply global choice, $NBG$ is strongly conservative over $ZFC$, so in some sense $ZFC$ is enough to show that a well-ordering of $V$, but is enough to show that a well-ordering of $V$ isn't too destructive to the universe. :)

Your first sentence is true (modulo the word "inductive"), but not in the way you mean: $ZFC$ proves the existence of many set well-orderings of each $V_\alpha$. Now, under some further assumption - say, $V=L$ - there might be a sequence of somehow canonical well-orderings of the $V_\alpha$, in which case we can indeed "glue them together" to get the global well-ordering you seek. However, if there is no such definable sequence, then we can't define your well-ordering, since we would need to pick some specific well-ordering of each of the $V_\alpha$s simultaneously - that is, find a choice function for a class-sized sequence of sets. And this, of course, is exactly global choice.

EDIT: One positive point. Even though choice doesn't imply global choice, $NBG$ is strongly conservative over $ZFC$, so in some sense $ZFC$ is not enough to show that a well-ordering of $V$ exists, but is enough to show that a well-ordering of $V$ isn't too destructive to the universe. :)

Your first sentence is true (modulo the word "inductive"), but not in the way you mean: $ZFC$ proves the existence of many set well-orderings of each $V_\alpha$. Now, under some further assumption - say, $V=L$ - there might be a sequence of somehow canonical well-orderings of the $V_\alpha$, in which case we can indeed "glue them together" to get the global well-ordering you seek. However, if there is no such definable sequence, then we can't define your well-ordering, since we would need to pick some specific well-ordering of each of the $V_\alpha$s simultaneously - that is, find a choice function for a class-sized sequence of sets. And this, of course, is exactly global choice.

Your first sentence is true (modulo the word "inductive"), but not in the way you mean: $ZFC$ proves the existence of many set well-orderings of each $V_\alpha$. Now, under some further assumption - say, $V=L$ - there might be a sequence of somehow canonical well-orderings of the $V_\alpha$, in which case we can indeed "glue them together" to get the global well-ordering you seek. However, if there is no such definable sequence, then we can't define your well-ordering, since we would need to pick some specific well-ordering of each of the $V_\alpha$s simultaneously - that is, find a choice function for a class-sized sequence of sets. And this, of course, is exactly global choice.

EDIT: One positive point. Even though choice doesn't imply global choice, $NBG$ is strongly conservative over $ZFC$, so in some sense $ZFC$ is enough to show that a well-ordering of $V$, but is enough to show that a well-ordering of $V$ isn't too destructive to the universe. :)