Proving Set is Not Complete

Method

1. Use Proof by Structural Induction to define a set $\mathcal{H}$ of formulas that uoc $C$
2. Define a Predicate $P(F)$ that holds for every $F\in \mathcal{H}$, but not in general
3. Use Proof by Structural Induction to prove $P(F)$ holds for every formula $F\in \mathcal{H}$
4. Give a specific formula $F$ and show $P(F)$ does not hold, then we show $C$ is not complete

Intuition

Create a property that uoc $C$ has, but one that not all formulas have, then show that a formula does not satisfy the property, thereby showing incompleteness.