Does anyone know an example of a limit ultrapower of a structure that is not isomorphic to an iterated ultrapower of that structure? I scoured Chang-Keisler but without any luck.

Here are some background definitions and comments:

Fix a structure $\mathcal M$. If $(Y,<)$ is a linearly ordered set and $\mathcal U_y$ is an ultrafilter on a set $I_y$ for each $y\in Y$, one can consider, for finitely many $y_1<\cdots<y_n$, the finite iterated ultrapower $\mathcal M^{\mathcal U_{y_1}\times \cdots \times \mathcal U_{y_n}}$. The collection of these finite iterated ultrapowers naturally forms a directed system whose direct limit is the iterated ultrapower of $\mathcal M$. If instead one considers a directed system of arbitrary ultrapowers of $\mathcal M$ (not necessarily finite iterated ultrapowers), then one arrives at a limit ultrapower.

Clearly every iterated ultrapower is a limit ultrapower. It seems that the converse should fail. Limit ultrapowers correspond to complete extensions of $\mathcal M$ and these can exist in any sufficiently large cardinality by the Compactness Theorem. There are some cardinality limitations on an iterated ultrapower as presented in the Exercises of Chang and Keisler. It seems that these restrictions should rule out some limit ultrapowers from being iterated ultrapowers.

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    $\begingroup$ Can you define limit ultrapower and iterated ultrapower? For example, with iterated ultrapower, what kind of index set for the iteration do you allow? $\endgroup$ – Joel David Hamkins Dec 10 '19 at 8:13
  • $\begingroup$ It sounds like you've largely answered your own question with the cardinality observation. That should be the easiest way to prove that they aren't always equivalent, although it will probably only show this with some set theoretic assumptions. $\endgroup$ – James Hanson Dec 13 '19 at 0:04

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