Like 6 can never be partitive of 14 in the whole number sense such that axiomatically 14/6 = undefined in some vague sense that I can be damned to elaborate or defend currently?

If I am understanding what you're trying to say, then yes, 14/6 does not have a solution in the sets of natural numbers, whole numbers (natural numbers and zero), and integers (whole numbers and the negative numbers). This is because 14 is not a multiple of 6.

However it does have a solution in the set of rational numbers and by extension, the real numbers. Thus, if we're talking about real numbers, 14/6 does have a solution: 2.3̅3 (two point three three repeating).

I recognize that I'm veering very closely, if not already have gone to explaining this in axiomatic terms. I apologize.

At any rate, if you're wondering if there is any similar extensions to the real numbers that can accommodate 1/0, as far as I understand, such extensions, if they do exist in the way you want them to behave, would still run into the same problems other replies to your post have described. Because of those difficulties (among others), in the two extensions I've looked at (hyperreals and the surreals), division by zero (and thus, 1/0) is still impossible in general, but are allowable in certain circumstances corresponding to the situations where y/x at the limit as x→0 exists or is +infinity or -infinity (but not both). However, as far as I understand it, in the real projective line, where there is a "point at infinity" added, you can define 1/0 as being that point at infinity.

EDIT: added some important clarification.