An Iteration Theorem for ω₁-preserving Forcings

2024/03/14 by Lietz, Andreas
#03E35 03E55 03E57 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2403.09018

Abstract

We prove an iteration theorem which guarantees for a wide class of nice iterations of ω₁-preserving forcings that ω₁ is not collapse, at the price of needing large cardinals to burn as fuel. More precisely, we show that a nice iteration of ω₁-preserving forcings which force SRP at successor steps and preserves old stationary sets does not collapse ω₁.

Related

• More on forcing iteration
• Property B: A Baumgartner-style Property that Applies to Preservation of ℵ₁ and ℵ₂ under Iterations with Supports of Size ℵ₁
• Not collapsing cardinals <= kappa in (< kappa) --support iterations
• Understanding Preservation Theorems, Chapter VI of Proper and Improper Forcing, I
• Long iterations for the continuum