Main Question. 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 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 $x\in y\iff j(x)\in j(y)$.
Set theorists quite commonly consider embeddings with at least a small degree of elementarity, and for such kind of embeddings, the question has a strong negative answer. There can be no
elementary embedding $j:V\to L$ when $V\neq L$, and indeed, there
can be no nontrivial $\Delta_0$-elementary embedding $j:V\to L$. So the embeddings of the question will necessarily exhibit very little elementarity. Meanwhile, it is easy to see that if $V=L$, then $$j(x)=\{\
j(y)\mid y\in x\
\}\cup\{\
\{0,x\}\ \}$$ is a nontrivial embedding $j:L\to L$, and there are
many other similar such embeddings from $L$ to $L$, and these carry no large cardinal strength and do not imply the existence of $0^\sharp$.
The question arises in connection with my paper,
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
Gödel-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.
Question.
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?
Question. 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.
