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 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", (see also the arxiv entry),
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.
