Shooting a club with finite conditions

Abstract

We study cohabitation of the poset $P_S$ shooting a club through a given stationary subset $S$ of $ω_1$ with finite conditions with other forcings. Sample results: (1) $P_S$ "sometimes" preserves presaturatedness of $NS_{ω_1}$ (2) $P_S=P_T$ if $S=T mod NS_{ω_1}$ (in Boolean algebra sense) (3) Cons ( one can embed $Q,$ the poset adding $\aleph _1$ Cohen reals, to $P_S$ such that reals in $V^Q$ are the same as reals in $V^{P_S}.$ )