A Version of $\kappa$-Miller Forcing

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, a specific form of $ ext{-}$-Miller forcing collapses the continuum to countable and adds a $$-Cohen real, revealing new interactions between forcing and cardinal characteristics.

Contribution

It establishes that $$-Miller forcing with club many splitting nodes collapses $2^$ to $$ and adds a $$-Cohen real under specific assumptions.

Findings

$$-Miller forcing collapses $2^$ to $$

Forcing adds a $$-Cohen real

Results depend on set-theoretic assumptions about $$ and $$-Miller forcing

Abstract

Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions the $\kappa$-Miller forcing with club many splitting nodes collapses $2^\kappa$ to $\omega$ and adds a $\kappa$-Cohen real.