Is there a universe which can always be forced to, which never can be forced from?



Is there a Hotel California of set-theoretic geology?
I think there's a serious confusion going on here, around what sort of background we assume.


If we are working within a single model $V$ of $ZFC$, and considering the multiverse(-like structure) consisting of all the inner models of $V$, then there are lots of possibilities. For one thing, if we include $V$ here, there is obviously a 'terminus.' However, even if we only include (say) generic extensions within $V$ of $L$, there are still lots of possibilities; for example, Monroe's example of a model with precisely two inner models: $L$ and itself. So in this case, there are lots of possibile answers, depending on exactly how you set things up.


If, however, we adopt a truly multiversal picture of things, then at the very least we want to demand that we can always force - in every presentation of the multivierse or anything similar I've ever seen, this has been one of the axioms. In this case, it's trivially the case that there is no terminus, since by definition we can always force over any world in the multiverse.


Where you're getting confused, I suspect, is in mixing the two pictures - talking about multiversal ideas (such as set-theoretic geology) while holding a picture of an 'ultimate' $V$ in the background, which of course constrains the allowed forcings. I suggest you make very precise exactly what your background theory is here, what sort of things we're assuming, etc., because otherwise this question is just too broad to admit a real answer.




---


One particular nice and specific sub-question lurking here: what are the possibilities for "all the set-generic extensions of $L$?" (Within a fixed $V$.) This could be a singleton, or have a top element, or be just a hideously complicated directed system of things! If what you're really interested in is the sort of variety 'the inner models of $V$' can have, I suggest among other things looking at Sy Friedman's Inner Model Hypothesis (<http://www.sciencedirect.com/science/article/pii/S0168007212000280>).


