I Direct limit of multiverse models of ZFC

[omega]

Let $(M_i)_{i\in I}$ be a multiverse of models of ZFC. By that I mean:

1. Each $M_i$ is a well-founded model of ZFC.
2. $(I,\leq_I)$ 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 $M_j$.
3. For any $i,j\in I$ there is $k$ such that $i\leq_I k$ and $j\leq_I k$.
4. For any $i$, there is $j$ such that $\tau^j_i(M_i)$ is a countable set in $M_j$

Now we can take the direct limit of these models, similar to direct limit of algebraic structures. More precisely, define two elements $x\in M_i,y\in M_j$ to be equivalent iff they become the same in some $M_k$, in other words $\tau^k_i(x)=\tau^k_j(y)$, and define $x\in y$ similarly. My main question is whether the direct limit $M$ is well-founded, and how much set theory does it satisfy? I thought it should be straightforward that $M$ is well-founded and the canonical map embeds $M_i$ as a transitive subset of $M$, but then I got stuck (assume for contradiction that there is an decreasing sequence $x_1\ni x_2\ni\cdots$ in the direct limit; identify the same element in different models for brevity; for each $n$ there is a model $M_n\ni x_1\ni x_2\ni\cdots\ni x_n$; where's the contradiction?).

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.

 

[andrewkirk]

Doesn't the fact that experts like Hamkins write about a multiverse rather than an all-encompassing model suggest they can see no way to create any kind of amalgamation (direct limit / disjoint union) that satisfies the requirements for a single model, as the above attempts to do?

As well as considering well-foundedness of M we need to identify whether it aligns with all the other ZFC axioms, and hence qualifies as a model, and I am not sure that we could do that.

On the question of transitivity, one can imagine an infinite descending sequence whose elements come from an infinite collection of $M_i$ with no global upper bound in the order relation. That is, given a descending sequence $x_1\ni x_2\ni ...$, for any $n\in\mathbb Z^+$, there exists $M_k$ such that all of $x_1,..., x_n$ can be traced back to elements of $M_k$. But there also exists $m>n$ such that $x_m$ cannot be traced back to an element of $M_k$. So I see doubt as to whether transitivity of sequence components would carry through to the limit.

Also, I think you need to include additional conditions on your $\tau_i^j$ embedding functions, in order to make a direct limit possible, per items 1 and 2 here.

 

[omega]

Doesn't the fact that experts like Hamkins write about a multiverse rather than an all-encompassing model suggest they can see no way to create any kind of amalgamation (direct limit / disjoint union) that satisfies the requirements for a single model, as the above attempts to do?

As well as considering well-foundedness of M we need to identify whether it aligns with all the other ZFC axioms, and hence qualifies as a model, and I am not sure that we could do that.

On the question of transitivity, one can imagine an infinite descending sequence whose elements come from an infinite collection of $M_i$ with no global upper bound in the order relation. That is, given a descending sequence $x_1\ni x_2\ni ...$, for any $n\in\mathbb Z^+$, there exists $M_k$ such that all of $x_1,..., x_n$ can be traced back to elements of $M_k$. But there also exists $m>n$ such that $x_m$ cannot be traced back to an element of $M_k$. So I see doubt as to whether transitivity of sequence components would carry through to the limit.

Also, I think you need to include additional conditions on your $\tau_i^j$ embedding functions, in order to make a direct limit possible, per items 1 and 2 here.

You are right, I should have said it is a "commutative diagram" of models.

Hamkins considers ill-founded models, which I guess is not super popular, plus he considers not just "M is a transitive subclass of N" but all kinds of embeddings, like those coming from ultrapowers, so it is indeed more difficult to find or even formulate the appropriate "direct limit" for his multiverse (note however that he does say "The multiverse view is one of higher-order realism", so it seems he treats the multiverse of models as the "ultimate" structure, which I personally find not that different from treating the universe of sets as the ultimate structure).

One of my motivation is to see if we only consider well-founded models, do we end up with an all-encompassing model like you say. I think that is actually the case. The point is that I require $\tau_i^j(M_i)$ to be transitive in $M_j$. For each $i$ there is the "absolute rank function" $\rho_i:M_i\rightarrow Ord$, which is the composition of the internal rank function $M_i\rightarrow {Ord}^{M_i}$ with the map that idenfities an $M_i$-ordinal with the isomorphic von-Neumann ordinal in the background universe. It follows from absoluteness between transitive models that $\rho_j\circ\tau_i^j=\rho_i$. If there were an infinite decreasing sequence $x_1\ni x_2\ni\cdots$ in the direct limit $M$, then the absolute rank of $x_{i+1}$ is smaller than that of $x_i$, a contradiction in the background universe. I assume we work in a background universe in order to formalize multiverse, just like Hamkins uses the collection of computably saturated models as a toy model for his multiverse axioms. I think the argument does suggest that if we only allow well-founded models, then multiverse is in a sense just a different perspective of universe.