Graph-like Distributions and Types in Ultrapowers

关于Solovay模型的Spector超幂

On a Spector ultrapower of the Solovay model

证明了可数Solovay模型$\gM$（其中所有实数集都是勒贝格可测的）的Spector-like超幂扩张$\gN$等于泛型扩张$\gM[\al]$中所有实数集的可构造集合，其中$\al$是$\gM.$上的随机实数。证明涉及Solovay模型中的几乎处处均匀化定理。

Saturating the random graph with an independent family of small range

Motivated by Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters $D$ on $I$, $|I| = λ> \aleph_0$, the fact that $P(I)/\de$ has little freedom (as measured by the fact that any maximal antichain is of size $<λ$, or even countable) does not prevent extending $D$ to an ultrafilter $D_1$ on $I$ which saturates ultrapowers of the random graph. "Saturates" means that $M^I/\de_1$ is $λ^+$-saturated whenever M is a model of the theory of the random graph. This was known to be true…

The Seed Order

This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that order, the seed order is consistently a linear ordering even under strong large cardinal assumptions. In fact, the linearity of the seed order is a feature of all known canonical inner models for large cardinal axioms. We develop the theory of the seed order under the assumption that it is linear. Augmented by large cardinal hypotheses currently out of reach of inner model theory (namely supercompactness), this linearity…