There are two basic ways that I am familiar with generating different transitive models of ZFC:
- Starting with the empty set, and iterating the power set construction to different heights.
- Playing around with the what the power set construction is to begin with, by changing which first-order-undefinable subsets of sets we consider to "exist."
I have heard Joel David Hamkins say that the first property determines how "tall" a model is, whereas the second property determines how "wide" it is. The main question I have is if these are the only two basic degrees of freedom with which we have to create transitive models of ZFC.
In other words: clearly we can add more large cardinals and make it "taller", and clearly we can remove undefinable subsets and make it "narrower" and closer to something like $L$. Is that all we can do?
Of course there are slight variations on this idea - once the model is "narrow enough," we can also make it "shorter" than it otherwise could be. Likewise, I would guess we can probably have "lumpy" models which are "wider" in some parts than others - maybe they agree with $L$ up to some point and then start admitting undefinable subsets beyond that. But at the end of the day, we are still basically playing around with the same two basic ideas of varying how many undefinable subsets exist when you take the power set, and varying how many times you want to iterate the power set operation. Does that basically summarize all of the things we can do when creating a model of ZFC?
To attempt to put question on rigorous footing: Can we completely characterize a transitive model of ZFC by
- the "length" of what the model thinks is the "cumulative hierarchy," expressed as an ordinal, and
- the extent to which the power set operation admits or leaves out undefinable subsets at each stage?