L Is Absolute and Minimal
If W is any standard model of ZF sharing the same ordinals as V, then the L defined in W is the same as the L defined in V. In particular, Lα is the same in W and V, for any ordinal α. And the same formulas and parameters in Def (Lα) produce the same constructible sets in Lα+1.
Furthermore, since L is a subclass of V and, similarly, L is a subclass of W, L is the smallest class containing all the ordinals which is a standard model of ZF. Indeed, L is the intersection of all such classes.
If there is a set W in V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the L of W. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.
Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets which are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.
Because both the L of L and the V of L are the real L and both the L of Lκ and the V of Lκ are the real Lκ, we get that V=L is true in L and in any Lκ which is a model of ZF. However, V=L does not hold in any other standard model of ZF.
Read more about this topic: Constructible Universe
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