This question might be characterized as a soft-question, and I am just curious to know if some of the notions below make sense or are consistent.
So to kick things off, we may note that the existence of most large cardinals is linked to the existence of certain elementary embeddings which have powerful properties. And in some cases(in the cases where the embeddings are class-sized) our embeddings are of the form $:V \rightarrow M$ for some inner model $M$, which by Kunen's theorem we have $M \neq V$. I was wondering about embeddings in the reverse direction. To be precise, embeddings of the form $j:M \rightarrow V$ for some inner model $M\neq V$.
So to me, it seems plausible to define dual notions of large cardinals as follows:(a * behind an object refers to it's dual version)
- We may define *measurable cardinals as those which appear as the image of the critical point of a non-trivial embedding $j:M\rightarrow V$.
- We may also define *$\gamma$-supercomapct as those $\kappa$ who appear as the image of the critical point of some non-trivial embedding $j:M\rightarrow V$ with $\gamma < \kappa$ and $M^\gamma \subset M$.
So I have a few questions:
($0$) In a previous edit of the question I thought such embeddings $j: M \rightarrow V$ could exist in the case of $V=L[A]$ and some collapse of some Skolem hull $M$, but I don't see why. Are there examples of such embeddings $j$ such that $V$ can "see" them/ be definable in $V$?
(I) Can we somehow define the above types of embeddings and large cardinals in ZFC?(Because for example to define measurability we form ultrapowers by ultrafilters. Is there an analogue version for the dual notion *measurability for example?)
(II) Are the above notions consistent?(relative to known large cardinals of course)
(III) Are the above dual elementary embeddings and large cardinals studied before?
(IV) If the answer to both (II) and (III) is positive, do they give distinct large cardinals to those which are standard today, and do they form their own hierarchy?
EDIT I:
As noted by Farmer S and Jason Zesheng Chen below in the comments, we can't have a definable class $M$ such that $j:M\rightarrow V$ is elementary. And again Farmer S and I noted that we need some form of amenability condition on $j$, since otherwise embeddings of the type $j:L\rightarrow L$ trivialize the * notions.
So at this stage I think the main question would be:
$(*)$ Can we have a universe $V$, an inner model $M$ which isn't a definable subclass of $V$ and an elementary embedding $j:M\rightarrow V$ such that $j$ is amenable relative to $V$, i.e. for every $X\in V$, $X\cap j \in V$?
EDIT II:
Over on twitter, Toby Meadows mentioned that in the paper Vickers-Welch "On the elementary embeddings from an inner model to the universe" found here, there is an example, where by assuming that $\text{On}$ is Ramsey one may find a definable $(j, M)$ over an expansion of the structure $(V, \in)$ such that $j:(M, \in)\rightarrow (V, \in)$ is elementary by using Skolem hulls and indiscernibles (Theorem $2.3$). But what makes this different than the $L$ case, is that here we may have $M\neq V$ which makes it at least non-trivial.
But there is one big catch; in theorem $2.4$ a result attributed to Foreman, it is noted that for $j:M\rightarrow V$, it is always the case that $M^\omega \not\subset M$. So we may forget about *supercompactness, since no embedding can witness it. But now since we do have some examples at hand, I wonder if one can have up to some degree of large cardinals from these dual notions.
So my questions now are (III) and (IV) mentioned above.
Sorry for the long post.