# V=L and a Well-Ordering of the Reals

A fairly simple question: I've read in multiple sources that Godel proved that if we accept the axiom of constructibility in ZFC, then we can create an explicit formula that well-orders the real numbers. I tried searching for a paper or some other source that explains what this formula is, but I came up empty-handed. Can someone explain what this formula is, or perhaps point me to a resource that does?

The order is very easy. Under $$V=L$$, the set-theoretic universe is built according the hierarchy $$(L_\alpha \mid \alpha \in \mathrm{Ord})$$, where $$L_0$$ is empty, $$L_{\alpha+1}$$ consists of all definable subsets of $$L_\alpha$$, and $$L_\lambda$$ is the union of all earlier $$L_\alpha$$ when $$\lambda$$ is a limit ordinal.
Since we can order the definitions used to go from $$L_\alpha$$ to $$L_{\alpha+1}$$, we obtain a definable well-ordering of the entire universe. Namely, $$x$$ is less than $$y$$ iff
1. $$x$$ appears before $$y$$ in the hierarchy or
2. they appear at the same stage, but $$x$$ appears with an earlier definition than $$y$$.
If one analyzes the complexity of the resulting definition for real numbers, it has complexity $$\Delta^1_2$$ in the descriptive set theoretic hierarchy.