Downward transference of mice and universality of local core models

TL;DR

This paper proves that under certain conditions, inner models of ZFC are universal for all countable mice in the universe, demonstrating a form of downward transference and universality of local core models.

Contribution

It establishes conditions under which local core models are universal for all mice in the universe, extending the understanding of inner model theory.

Findings

K^M  omega_2 is universal for all countable mice in V.

Under certain assumptions, K^M  delta is universal for mice of size less than delta.

Shows downward transference of mice to inner models under specific set-theoretic conditions.

Abstract

If M is a proper class inner model of ZFC and omega_2^M=omega_2, then every sound mouse projecting to omega and not past 0-pistol belongs to M. In fact, under the assumption that 0-pistol does not belong to M, K^M \| omega_2 is universal for all countable mice in V. Similarly, if M is a proper class inner model of ZFC, delta>omega_1 is regular, (delta^+)^M = delta^+, and in V there is no proper class inner model with a Woodin cardinal, then K^M \| delta is universal for all mice in V of cardinality less than delta.