Can one build a hierarchy of stratified constructible stages? That is a hierarchy that is built in a manner similar to Godel's [constructible universe](https://en.wikipedia.org/wiki/Constructible_universe) L, but with additionally requiring that the defining formulas must be also [stratified](https://en.wikipedia.org/wiki/Stratification_(mathematics)#In_set_theory)!



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> If that can be built, then my main question is whats the stance of choice in that hierarchy, would choice be satisfied internally in every limit stage of that hierarchy? Can choice fail?
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Can we have stratified L?
Glad to see someone taking an interest in this stuff!
All but one of the Goedel operations are stratified, and the one that isn't is the existence of $\in$ "locally." Replace this operation by one that gives you the local version of $\in$ composed with singleton, so that you get
$A \cap \{\langle \{x\},y \rangle: x \in y \in B\}$. You can then prove an exact analogue of the original result that anything closed under the rudimentary functions and power set is a model of $\Delta\_0$ separation - for the stratified operations. Any set closed under the stratrud functions and power set is a model of stratified $\Delta\_0$ separation. This actually gives a finite axiomatisation of NF. Naturally one wonders about what happens if one tries to construct an analogue of $L$ in the fashion and the (annoying) answer is that what one gets depends very sensitively on how often one "sweeps up." The more often you sweep up the more of $L$ you get. I have some quite extensive notes on this, but they are from about 10 years ago and i would need to do some archaeology. I will if pressed.


