Is there a 'Constructible Universe' that is a submodel of a non-transitive model of $ZF$?

Question

In Barwise's book, Admissible Sets and Structures, the following statement is made (on pg. 8):

"...if $ZF$ is consistent, so is $ZF$ + "There is no transitive model of $ZF$"

He mentions this fact as an example of how $ZF$ is, in some ways, "too strong" (also found on pg. 8):

(1) The most obvious advantage of the axiomatic method is lost since $ZF$ has so few recognizable models in which to interpret its axioms.

Let me note here that I am not mentioning the above to criticize $ZF$, but to ascertain the 'pervasiveness' of the constructible sets in models of $ZF$.

So here are my questions:

(i) Since it is known that for transitive models of $ZF$, $L$ (a transitive model of $ZF$) is the smallest submodel of $ZF$, is ther a non-transitive version of $L$ that is the smallest submodel of a non-transitive model of $ZF$?

(ii) If so, then how is the non-transitive version of $L$ formed (since it seems that the usual methods of forming $L$ produce the transitive version of $L$).

(iii) Is there a generalization of $L$ which is the smallest submodel of both transitive and non-transitive models of $ZF$? If so, what is it, and are there any survey articles regarding this generalization?

Finally, if the mathoverflow community or members of the community think these questions are silly, please extend me the courtesy of allowing me to delete this question while I attempt to make them (at least) less so.

Thanks to all for your consideration in this matter.

Answer 1

What $\sf ZF$ actually proves is that there is a class called $L$, and for every axiom of $\sf ZFC$, the relativization of that axiom holds in $L$.

Therefore, the fact we are talking about non-transitive models does not matter. They have their own version of $L$ anyway.

What is provable is that $L$ is a transitive class. But here a transitive class means "with respect to the universe we are working with". In particular, it does not matter that the universe is actually ill-founded in a larger model. It is still going to have $L$, and it will still "think" it is a transitive class.

So to answer your question, the formation of $L$ of a non-transitive model is exactly the same as the formation of $L$ in the usual sense. It is done by an internal transfinite recursion. Much like the von Neumann hierarchy exists even for an ill-founded model, and it is simply not indexed by the actual ordinals.

(Let me also remark on something which seems like a deep running confusion throughout your post. When we say that $L$ is the smallest transitive model, we mean it is the smallest transitive inner model, whereas this implies a proper class. It is certainly conceivable that there are set models, transitive or otherwise, which will certainly not contain $L$.)