Negative results on precipitous ideals on omega_1

TL;DR

This paper demonstrates that in certain extender models, there are no generic embeddings with critical point ω₁ similar to the stationary tower at the second Woodin cardinal, indicating limitations of these models.

Contribution

It establishes a negative result about the non-existence of specific generic embeddings in extender models, clarifying their structural limitations.

Findings

No generic embeddings with critical point ω₁ in extender models

Limitations on stationary tower embeddings at the second Woodin cardinal

Clarifies the boundaries of extender model capabilities

Abstract

We show that in extender models there are no generic embeddings with critical point $\omega_1$ that resemble the stationary tower at the second Woodin cardinal.