TITLE: Do we know that we can't define a well-ordering of the reals?
QUESTION [20 upvotes]: Folklore has it that it is impossible to define a well-ordering of the reals explicitly.
There exist pointwise definable models of ZFC where every set is definable without parameters: it is the unique element of the model that satisfies some finite formula $\varphi(x)$. So there is a formula $\varphi(x)$ such that
$$ \tag{*} \forall x(\varphi(x)\rightarrow x\text{ well-orders }\mathbb R) \land \exists ! x\,\varphi(x) $$
is consistent with ZFC.
(And it is not difficult to write down a concrete $\varphi$ which will work in a model with $\mathbf V=\mathbf L$).
Is it known that there is no $\varphi(x)$ such that $\text{(*)}$ is a theorem of ZFC?
Of course, since there are models of ZF that don't have a well-order of $\mathbb R$, in such a model $\varphi(x)$ would either be unsatisfiable or be satisfied by something that doesn't well-order $\mathbb R$. So the Axiom of Choice would have to be used in some essential way in the proof of $\text{(*)}$. But that doesn't in itself seem to preclude the possibility that $\varphi$ might exist.

REPLY [15 votes]: Forcing with sufficiently homogeneous forcing that adds reals is enough to obtain the negation of $(*)$. The point is that if a formula $\phi$ defines a parameter-free well-ordering of $\mathbb R$, then for any ordinal $\alpha$, the statement "$x$ is the $\alpha$-th real in the well-ordering defined by $\phi$" uniquely characterizes $x$ in terms of $\alpha$. If the forcing we used is homogeneous, this implies that $x$ is in the ground model, as you can then proceed to uniquely characterize $x$ (for instance, you can verify whether "its first digit is $0$", "its second digit is $1$", etc). The way that homogeneity is used is in noticing that all these statements I am writing have the form $\psi(\check t_1,\dots,\check t_n)$ when written in the forcing language (for some $\psi$ in the language of set theory, and some $t_1,\dots,t_n$ in the ground model).  But any such statement is either always or never forced (by homogeneity), and so, from the ground model, you can read off every real in the extension, so the forcing cannot add reals.
So you are done once you find a homogeneous forcing that adds reals. Cohen forcing is an example. One can go further and obtain models where there are no well-orderings definable using parameters from the ground model by a simple extension of the sketch above, and even models where there are no well-orderings definable, even if (ordinal definable, or small-sized) parameters from the extension are allowed (though now we need to add more than just one Cohen real to achieve this). 
Since many natural forcing posets are sufficiently homogeneous to carry out the arguments just described, one of the surprising realizations that has come out of the study of forcing axioms is that they tend to imply the existence of definable well-orderings (either from a parameter, or even parameter-free if we force from a definable inner model). I have a few papers explaining this phenomenon, or see this talk. Some questions remain; for instance, it is still open whether if Martin's Axiom holds and $\omega_1=\omega_1^L$, then there is a projective well-ordering of the reals. In the early 2000s, I would have thought asking this was ridiculous. But we do not know.