# Can stratification find a meaning in enabling a concordant stratified constructible hierarchy?

Let's define the $$V$$-rank (also can be named as set \ true - rank) of a set $$X$$ as the minimal $$\alpha$$ such that $$X \in V_{\alpha+1}$$.

On the other hand, define the $$L$$-rank (also can be named as constructible-rank) of a constructible set $$X$$ as the minimal $$\alpha$$ such that $$X \in L_{\alpha+1}$$.

A set in $$L$$ is concordant if and only if its $$V$$ and $$L$$ ranks are equal. Formally: $$\operatorname {conc}(X) \iff \rho^L(X)=\rho^V(X)$$

We know that $$L$$ does contain sets that are not concordant, an example is the diagonal of a constructible bijection between $$L_\omega$$ and $$L_{\omega+1}$$.

Now, I want to motivate the criterion of Quine's stratification in relation to this notion of concordance. So, the working assumption is that if we weaken the definition of $$L$$ to stratified-constructible sets (i.e., require an additional condition in defining stages of $$L$$, that of the defining formulas being also stratified), then the resulting hierarchy (call it stratified-$$L$$) can have all of its elements being concordant! That is, stratification can prevent constructible jumps.

Is there a clear counter-example to the above assumption?

• Why do you think restricting attention to stratified formulas would prevent such jumps? Jumps of this sort are unavoidable as long as you want a reasonably rich universe built in the end. Commented May 26 at 23:02
• @NoahSchweber, it depends on how you qualify "rich", for instance do you think the universe of NF[U] rich? Commented May 26 at 23:10
• Note that NFU proves the existence of non-arithmetic sets. But non-arithmetic sets aren't in $L_{\omega+1}$, so they automatically lead to jumps of the sort you want(?) to avoid. Commented May 26 at 23:11
• @NoahSchweber, I'm not sure of that. Can you give examples of NFU doing that? Commented May 26 at 23:35
• Rank $\alpha$ should mean that $\alpha$ is least such that $x\in V_{\alpha+1}$, not $V_\alpha$, and similarly with $L_{\alpha+1}$, since these hierarchies are both continuous. Commented May 27 at 0:36