Given a model $M$ of some set theory, and two elementary submodels $N$, $Q$ of $M$, is there an elementary embedding between $N$ and $Q$?
$\begingroup$
$\endgroup$
4
-
4$\begingroup$ Since every two elementarily equivalent models $N,Q$ elementarily embed in a common model $M$, you are in effect asking whether any two elementarily equivalent models of your theory are comparable wrt elementary embedding. This can only hold for very tame theories (not set theory). $\endgroup$Emil Jeřábek– Emil Jeřábek2025-12-18 19:19:46 +00:00Commented 18 hours ago
-
2$\begingroup$ @EmilJeřábek I would encourage you to post an answer to that effect, since Gabe's answer uses large cardinals, but these are not required. $\endgroup$Joel David Hamkins– Joel David Hamkins2025-12-19 02:10:47 +00:00Commented 11 hours ago
-
$\begingroup$ @JoelDavidHamkins There is still a somewhat interesting question of whether the question for transitive models requires large cardinals $\endgroup$Gabe Goldberg– Gabe Goldberg2025-12-19 02:48:42 +00:00Commented 10 hours ago
-
1$\begingroup$ @JoelDavidHamkins There are actual model theorists around who would be better equipped to do it than me. $\endgroup$Emil Jeřábek– Emil Jeřábek2025-12-19 08:44:44 +00:00Commented 4 hours ago
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
No, consider normal measures $U$ and $W$ on the first and second measurables of some countable transitive model $P$. Our model is $M=P_{U\times W}$, the ultrapower of $P$ by the product measure. We have factor embeddings $k_U : P_U \to P_{U\times W}$ and similarly $k_W$. But there is no elementary embedding between $P_U$ and $P_W$; none from $P_U$ to $P_W$ since this would imply the first measurable of $P_U$ is at most that of $P_W$; none the other way by considering the second measurable.
-
$\begingroup$ While your argument is probably the conceptually simplest, it might be worth noting that counterexamples with transitive $M$ exist in any model of ZFC. In fact $M=H_\theta$ always works for any $\theta\geq\omega_2$. $\endgroup$Andreas Lietz– Andreas Lietz2025-12-19 04:44:32 +00:00Commented 8 hours ago