7
$\begingroup$

It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set like $\omega_7$ or $\mathcal P(\mathbb R)$, how many distinct equivalence classes does the ultrapower of a countable model by $U$ have? Keisler proved that a certain kind of regularity property implies that the answer is the maximum possible. The failure of this property was connected with large cardinals by Magidor, Kanamori, and others.

It seems that there were still many possibilities left open by the work of these set theorists, but it also seems like there hasn't been that much written on this in recent years. Is there still interest from contemporary model theorists in being able to come up with various consistent answers for the size of ultrapowers on small cardinals? Are there possible applications?

Improve this question
$\endgroup$
2
  • 3
    $\begingroup$ Is your question really a sociological question about model theorists? Here's my take: I think most model theorists would tell you that such questions are questions of set theory, not model theory. If you can say something interesting about cardinalities of ultrapowers, many model theorists will certainly be interested (e.g., those model theorists who are also interested in set theory). But this is just not the kind of question most model theorists ask these days. $\endgroup$
    – Alex Kruckman
    Commented Oct 10, 2023 at 16:49
  • $\begingroup$ @AlexKruckman I wouldn’t call it a sociological question, because it’s not about their behavior in general. It’s just a question about the direction of current research. $\endgroup$
    – Monroe Eskew
    Commented Oct 10, 2023 at 20:42

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .