Define a ranking function $\cal R$ as:

$\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y) $

Now the constructible rank $\mathcal R^c$ of a set $X$ in $L$ is the ordinal index of the first constructible stage $X$ appears as an element of.

Accordingly for some set $X$ we may have $\mathcal {R^c}(X) > \mathcal {R} (X) +1$

Now define: $concordant(X) \equiv_{df} \mathcal {R^c}(X) = \mathcal {R}(X) + 1$; otherwise, $X$ is discordant.

Let $ [Cc]$ be the class of all concordant sets. Let $ L`[Cc]$ be defined as $ L`[Cc] = \bigcup L_\alpha`[Cc]$

Where each $L_{\alpha+1} `[Cc] $ is the set of all concordant subsets of $L_\alpha `[Cc]$ that are definable after formulas with parameters and quantifiers restricted $L_\alpha `[Cc]$; with limit stages being unions of all prior stages.

Now $L`[Cc]$ is a proper subclass of $L$ so it cannot be a model of $\sf ZFC$.

Now what sentences $L`[Cc]$ satisfy? In particualr does $L`[Cc]$ satisfy set union?

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