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
Famous quotes containing the words absolute and/or minimal:
“I am not sure but I should betake myself in extremities to the liberal divinities of Greece, rather than to my country’s God. Jehovah, though with us he has acquired new attributes, is more absolute and unapproachable, but hardly more divine, than Jove. He is not so much of a gentleman, not so gracious and catholic, he does not exert so intimate and genial an influence on nature, as many a god of the Greeks.”
—Henry David Thoreau (1817–1862)
“For those parents from lower-class and minority communities ... [who] have had minimal experience in negotiating dominant, external institutions or have had negative and hostile contact with social service agencies, their initial approaches to the school are often overwhelming and difficult. Not only does the school feel like an alien environment with incomprehensible norms and structures, but the families often do not feel entitled to make demands or force disagreements.”
—Sara Lawrence Lightfoot (20th century)