The axiom of constructibility, $V = L$

The axiom of constructibility, written $V=L$, is the assertion that the universe of all sets is exactly the universe of all constructible sets. It is minimalistic in the sense that any inner model $M$ of ZF must contain all sets from Gödel’s constructible universe $L$. The axiom is compatible with some of the smaller large cardinal notions such as weak compactness but is not compatible with any large cardinal notion implying the existence of $0^{\sharp}$ such as measurability.

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