- #1

- 3

- 5

- Each ##M_i## is a well-founded model of ZFC.
- ##(I,\leq_I)## is a partially ordered set, and whenever ##i\leq_I j##, there is an embedding ##\tau^j_i:M_i\rightarrow M_j## such that the image of ##M_i## is a
**transitive**subclass of ##M_j##. - For any ##i,j\in I## there is ##k## such that ##i\leq_I k## and ##j\leq_I k##.
- For any ##i##, there is ##j## such that ##\tau^j_i(M_i)## is a countable
**set**in ##M_j##

I included condition 3 because it ensures at least ##M## will satisfy axiom of pairing, and condition 4 because it's the multiverse axiom I'm most interested in; condition 4 implies every set in ##M## is countable. The fundamental case is when ##I## is just the collection of all countable transitive models (ctm) of ZFC and the partial order is inclusion. In this case the direct limit is simply the union of all ctm, which I believe is exactly ##H(\omega_1)##, the collection of hereditarily countable sets (under mild hypothesis, say the existence of one inaccessible cardinal). Will ##M## satisfy separation, replacement and such? Or is there reasonable hypothesis on ##I## that makes it look like ##H(\omega_1)##?

Motivation: I'm trying to read some part of Hamkins' famous paper "The set-theoretic multiverse". On page 23 he explicitly says "we state the multiverse axioms as unformalized universe existence assertions about what we expect of the genuine full multiverse". Maybe the most naive way to formalize multiverse axioms would be just as above, saying that we have a directed collection of models; I don't (yet) like ill-founded models very much so I required them to be well-founded. Then it seems we can actually take the "union" of them, and in a sense we goes from multiverse back to the picture of a single universe. Some may object that this is not in the spirit of multiverse, so there must be a better to formalize it, say using modal logic.