We study the question of when a given countable ordinal α is Σ n 1 - or Π n 1 -reflecting in models which are neither PD models nor the constructible universe, focusing on generic extensions of L . We prove, amongst other things, that adding any number of Cohen or random reals, or forcing with Sacks forcing or any lightface Borel weakly homogeneous ccc forcing notion cannot change such reflection properties. Moreover we show that collapse forcing increases the value of the least reflecting ordinals but, curiously, to ordinals which are still smaller than the ω 1 of L .

Abstract We study a family of variants of Jensen’s subcomplete forcing axiom , upper S upper C upper F upper A comma $\mathsf {SCFA,}$ S C F A , and subproper forcing axiom , upper S u b upper P upper F upper A $\mathsf {SubPFA}$ S u b P F A . Using these, we develop a general technique for proving nonimplications of upper S upper C upper F upper A $\mathsf {SCFA}$ S C F A , upper S u b upper P upper F upper A $\mathsf {SubPFA}$ S u b P F A and their relatives and give several applications. For instance, we show that upper S upper C upper F upper A $\mathsf {SCFA}$ S C F A does not imply upper M upper A Superscript plus Baseline left parenthesis sigma $\mathsf {MA}^+(\sigma $ M A + ( σ -closed) and upper S u b upper P upper F upper A $\mathsf {SubPFA}$

Abstract We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega \omega $ -bounding forcing notions, 2) the class of subproper, T-preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, $[T]$ -preserving forcing notions (where T is an $\omega _1$ -tree) are iterable with revised countable support. In the second part, we adopt Miyamoto’s theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega _1$ , and, in the case of subcompleteness, don’t add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

Abstract Generalizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of $\omega $ . Such sets strengthen maximality, exist under $\mathsf {MA} (\sigma \mathrm {-centered})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak {a}_e$ and $\mathfrak {a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$ , and $\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$ . We also show that there are $\Pi ^1_1$ tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside $\Pi ^1_1$ witnesses for $\mathfrak {a}_e = \mathfrak {a}_p = \aleph _1$ . Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number i is strictly below c, including iterations of Sacks forcing, Miller partition forcing, h-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that i=ℵ1 in the Miller Lite model. An important aspect of the preservation theorem is the notion of “Cohen preservation”, which we discuss in detail.

Abstract We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega ^\omega \to \omega ^\omega $ introduced by the second author in [8]. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg \mathsf {CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, with one exception, for the bounding numbers there are no $\mathsf {ZFC}$ relations between them beyond those in the higher dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak {d}^\lambda _\kappa $ .

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which a=u=i=ℵ1<2ℵ0=ℵ2, each of a, u, i has a Π11 witness and there is a Δ31 well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a Δ31 well-order of the reals is consistent with c=ℵ2 and each of the following: a=u<i, a=i<u, a