set theory – Notions of “completeness” and “sufficiency” of a mathematical model


I’m modelling a real-world problem as having instances $i$ in a set $P$. The structure of the problem and the model itself are irrelevant to my question so I’ll omit them.

I define certain restrictions $A$ on $P$ using logical formulae over structure of $i$.
$A$ is a necessary condition, i.e. any problem in the problem domain, if represented using $i in P$ would satisfy $A$. We can say that $P$ under $A$, i.e. $P’ = {p : p in P land p~text{satisfies}~A}$.

But it’s possible that $exists i’ in P’$ such that $i’$ is a valid mathematical structure but doesn’t actually correspond to valid any real-world problem.

What is the standard terminology to say

  1. All valid problem instances are a part of $P’$ (I’m informally calling this “sufficiency”)
  2. All problem instances which are a part of $P’$ are valid (I’m informally calling this “completeness”)

I may then use this terminology to say that in my example, $P’$ is sufficient but incomplete (replacing these two words with the actual terminology)