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?
> 
> 
> 



What does the Concordant constructible universe model?
We assume we are working in ZF. We assume that the definition of concordant is $concordant(X) \equiv\_{df} \mathcal {R^c}(X) = \mathcal {R}(X)$+1(without the "+1", no set is concordant).


We will show that union does not hold in ‘[]. For simplicity we will use Quine ordered pairs so that if x and y are in ‘[], and is infinite, then (x,y) is in ‘[]. We will construe


formulas as elements of ω. We can express that a formula holds in ‘[], in (+1)‘[]. Let QX be the one element subsets of X. Let A be the set of elements of (ω+1)‘[], given by


(f,p)∈A<-->("f is a formula with two free variable holds" ,p∈(ω+1)‘[], and "f(p,(f,p)) does not hold in (ω+1)‘[]"). We note that there is an infinte p and an f such that


"f is a formula with 2 free variables" holds, and (f,p)∈A.(For instance let f be the formula corresponding to r=r∧s≠s and let p=ω. Then p=p∧(f,p)≠(f,p) does not hold in (ω+1)‘[].)


Then A∉(ω+2)‘[], but QQA∈(ω+4)‘[]. If QA∈(ω+3)‘[], then QA is an example of a set whose union does not exist in ‘[]. If QA∉(ω+3)‘[], then QQA is an example of a set whose union does not exist in
‘[].


