Combinatorial properties of classical forcing notions Original Research Article

We investigate the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum. We show: 1. (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1; 2. (2) Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH; 3. (3) Millerʹs rational perfect set forcing preserves the axiom MA(σ-centered).