The P-ideal dichotomy, Martin’s axiom and entangled sets

Abstract

We build a model of the P-ideal ichotomy (PID) and Martin’s axiom for ω1 ($${\rm MA}_{{\omega}_1}$$ MA ω 1 ) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for ω1-dense sets are not consequences of $${\rm PID} + {\rm MA}_{{\omega}_1}$$ PID + MA ω 1 . We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.