A general tool for consistency results related to I1

TL;DR

This paper introduces a versatile method for establishing the consistency of the I1(λ) axiom with various combinatorial properties at λ, applicable in complex set-theoretic contexts without deep forcing knowledge.

Contribution

It provides a general, accessible tool for proving the consistency of I1(λ) with multiple combinatorial properties at λ, broadening the scope of set-theoretic consistency results.

Findings

Proves the consistency of I1(λ) with the failure of GCH at λ.

Demonstrates I1(λ) can be consistent with the existence of a very good scale.

Shows I1(λ) can be compatible with the negation of the approachability property.

Abstract

In this paper we provide a general tool to prove the consistency of $I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of the forcing notions involved. Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at $\lambda^+$ and $\lambda^{++}$.