<b>Main Question.</b> Can there be an embedding $j:V\to L$ of the
set-theoretic universe $V$ to the constructible universe $L$, if
$V\neq L$?

By *embedding* here, I mean merely a proper class isomorphism from
$\langle V,{\in}\rangle$ to its range in $\langle L,{\in}\rangle$, or in other words a
quantifier-free-elementary map $j:V\to L$, a class map $j$ for
which $x\in y\iff j(x)\in j(y)$. 
 
This embedding concept is considerably weaker than usually considered in set theory, where one typically has embeddings that are at least $\Delta_0$-elementary if not much more. Of course, we may easily refute the existence of nontrivial fully elementary or even of $\Delta_0$-elementary embeddings $j:V\to L$. Those arguments, however, simply fail with this much weaker embedding concept. One can begin to see this by observing that $$j(x)=\{\
j(y)\mid y\in x\
\}\cup\{\
\{0,x\}\ \}$$ defines an embedding $j:L\to L$ with $j(x)\neq x$ for every $x$. In particular, the existence of a nontrivial embedding $j:L\to L$ in this weak sense is consistent with $V=L$ and carries no large cardinal strength, and does not prove the existence of $0^\sharp$.

The question arises in connection with my paper,

 - [J. D. Hamkins, "Every countable model of set theory embeds into its own
constructible universe"](http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe/), (see also the [arxiv entry](http://arxiv.org/abs/1207.0963)),

where it appears in the final section with the other questions I
ask here, among others. I have half an expectation, a gnawing
suspicion, however, that this questions may admit an easy answer,
and this is why I am asking it here. But I don't know which way
the answer will go.

The main theorem of the paper shows that every countable model of
set theory $M$ has an embedding $j:M\to L^M$. But the proof
establishes the existence of such embeddings only in an external
way, using the countability of $M$. The main question above
inquires from an internal perspective whether one can ever find
such an embedding as a class inside the model.

The existence of such an embedding as a definable class would of
course imply $V=HOD$, since one could pull back the canonical
order from $L$ to $V$. More generally, if $j$ is merely a class in
G&ouml;del-Bernays set theory, then the existence of an embedding
$j:V\to L$ implies global choice. So we cannot expect every model
of ZFC or of GB to have such embeddings. Can they be added
generically? Do they have some large cardinal strength? Are they
outright refutable?

There are several more concrete versions of the question.

<b>Question.</b>
 Does every set $A$ admit an embedding $j:\langle A,{\in}\rangle \to \langle L,{\in}\rangle$?  If not, which sets do admit such embeddings?

It follows from the main theorem of the paper that every countable
set $A$ embeds into $L$. What about uncountable sets?

<b>Question.</b> Does $\langle V_{\omega+1},{\in}\rangle$ embed into
$\langle L,{\in}\rangle$? How about $\langle P(\omega),{\in}\rangle$ or $\langle \text{HC},{\in}\rangle$?

These latter questions are interesting principally when $V$ has non-constructible reals. I would be very interested in learning the answer.