Definability of the direct limit

Question

Suppose $\kappa$ is a measurable cardinal and $U$ is a normal measure on $\kappa$. $M_1$ is the ultrapower of the universe $V$ constructed form $U$,$j_{01}$ is the induced elementary embedding, $M_2$ is the ultrapower of $M_1$ constructed from $j_{01}(U)$...We can iterate this process. We know $M_0,M_1,M_2,...$ and $j_{01},j_{02},j_{12},..$ are all classes of the universe, i.e. they are all definable. Also, they form a direct system, so we can define the direct limit $(M_{\omega},E_{\omega})$. $M_{\omega}$ and $E_{\omega}$ are both subset of the universe $V$, but my question is why they are definable subset of the universe?

Answer 1

There are in fact many different definable ways to represent the direct limit. One particularly concrete representation of the direct limit is to use maximal threads, that is, maximal sequences $\langle x_i \mid j\leq i\lt\omega\rangle$, where $x_{i+1}=j_{i,i+1}(x_i)$, where $j_{i,i+1}:M_i\to M_{i+1}$ is the ultrapower of $M_i$ by $U_i=j_{0,i}(U)$, a normal measure on $\kappa_i$, where $j_{i,j}:M_i\to M_j$ is the corresponding composition of these embeddings. One then defines the $\in$ relation in $M_\omega$ by consulting the common coordinates, and this is well-defined. Alternatively, one may take the disjoint union of the $M_i$, and declare that any two such points are equivalent when they map to the same point in some later $M_i$. And there are many other representations of the direct limit, and these may all be defined by defining the constituent elements of the system used to define the direct limit, often taking the quotient by an equivalence relation, which is definable from the system and the measure.

The point is that any of these concrete representations of the direct limit can be defined from the measure $U$. With $U$ as a parameter, one may define any of the classes $M_i$ and also the embeddings $j_{i,j}:M_i\to M_j$, and furthermore, we may do so uniformly in $i$ and $j$. Thus, the point is that not only are the various $M_i$ and embeddings $j_{i,j}$ definable individually, but uniformly in the parameters $i$, $j$, and this is what it takes to define the direct limit using any of the usual representations of the direct limit. In short, we use the fact that we can define the entire system of embeddings $j_{i,j}:M_i\to M_j$ uniformly in order to know that we can build a representation of the direct limit of that system.

But what is more, since the direct limit is well-founded, there is in fact a canonical way to represent the direct limit: the Mostowski collapse of your favorite representation. This provides $\langle M_\omega,{\in}\rangle$ as a transitive model of set theory, using the actual $\in$ relation. And the result will be a proper transitive class $M_\omega\subset V$, definable from $U$ as a parameter, with definable class embeddings $j_{i,\omega}:M_i\to M_\omega$.