0 is a Set, Not a Number

My calculator holds 12 digits. In other words, 1 + 10^-13 -1 = 0. In other words, in the context of 1, 10^-13 = 0. Or rather, 10^-13∈ [0]. My calculator also has a limit at 10^-99, meaning x <= 0.9*10^-99 = 0.

You are a finite being and no matter what calculation you do there's a finite accuracy to it. Anything below that accuracy ∈[0]. In other words x/[∞]∈[0]  Mathies are wrong again, it is exactly the case that 1/∞ = 0, except that ∞ and 0 are sets, not numbers. [0] is just [∞] viewed upside-down. Or, equivalently, [∞] is just a way of looking at [0]. When you define a [∞], you also unavoidably define a [0] relative to any member of [∞], and vice-versa.

E.g. for my calculator and x <= 0.1, 10^99 ∈ [∞]. Asking it 0.1/10^99 gives the answer 0. Sure I can handle way more nines than that, 10^9999999999999999999999, but eventually I get bored of writing nines and we've found [∞] in the context of me.

If that's not general enough for you, then generalize it yourself.