A version of $$\kappa $$-Miller forcing

We consider a version of $$\kappa $$ κ -Miller forcing on an uncountable cardinal $$\kappa $$ κ . We show that under $$2^{<\kappa } = \kappa $$ 2 < κ = κ this forcing collapses $$2^\kappa $$ 2 κ to $$\omega $$ ω and adds a $$\kappa $$ κ -Cohen real. The same holds under the weaker assumptions that $${{\,\mathrm{cf}\,}}(\kappa ) > \omega $$ cf ( κ ) > ω , $$2^{2^{<\kappa }}= 2^\kappa $$ 2 2 < κ = 2 κ , and forcing with $$([\kappa ]^\kappa , \subseteq )$$ ( [ κ ] κ , ⊆ ) collapses $$2^\kappa $$ 2 κ to $$\omega $$ ω .

The Bristol model: An abyss called a Cohen real

We construct a model [Formula: see text] of [Formula: see text] which lies between [Formula: see text] and [Formula: see text] for a Cohen real [Formula: see text] and does not have the form [Formula: see text] for any set [Formula: see text]. This is loosely based on the unwritten work done in a Bristol workshop about Woodin’s HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking us one step closer to understanding models of [Formula: see text], and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of [Formula: see text], and we use it to show the failure of Kinna–Wagner Principles in [Formula: see text].

THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL

It is shown that $K|{\omega _1}$ need not be solid in the sense previously introduced by the authors: it is consistent that there is no inner model with a Woodin cardinal yet there is an inner model W and a Cohen real x over W such that $K|{\omega _1}\,\, \in \,\,W[x] \setminus W$. However, if ${0^{\rm{\P}}}$ does not exist and $\kappa \ge {\omega _2}$ is a cardinal, then $K|\kappa$ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of $\neg {0^{\rm{\P}}}$, the core model is contained in the solid core, previously introduced by the authors.It is also shown, assuming ${0^{\rm{\P}}}$ does not exist, that if there is a forcing that preserves ${\omega _1}$, forces that every real has a sharp, and increases $\delta _2^1$, then ${\omega _1}$ is measurable in K.

Forcing Properties of Ideals of Closed Sets

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ -ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal.We also study the 1–1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1–1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P1 adds a Cohen real, or else I has the 1–1 or constant property.

Extending Baire property by uncountably many sets

We show that for an uncountable κ in a suitable Cohen real model for any family {Av}v<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets Av, into the algebra of Baire subsets of 2κ modulo meager sets such that for all Borel B,The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.

Orthogonal families of real sequences

For x, y ϵ ℝω define the inner productwhich may not be finite or even exist. We say that x and y are orthogonal if (x, y) converges and equals 0.Define lp to be the set of all x ϵ ℝω such thatFor Hilbert space, l2, any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4].Theorem 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of lp for every p > 2.It was already known that there exists a family of continuum many pairwise orthogonal elements of ℝω. A family F ⊆ ℝω∖0 of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of ℝω orthogonal to every element of F is 0, the constant 0 sequence.It is somewhat surprising that Kunen's perfect set of orthogonal elements is maximal (a fact first asserted by Abian). MAD families, nonprincipal ultrafilters, and many other such maximal objects cannot be even Borel.Theorem 2. There exists a perfect maximal orthogonal family of elements of ℝω.Abian raised the question of what are the possible cardinalities of maximal orthogonal families.Theorem 3. In the Cohen real model there is a maximal orthogonal set in ℝω of cardinality ω1, but there is no maximal orthogonal set of cardinality κ with ω1 < κ < ϲ.By the Cohen real model we mean any model obtained by forcing with finite partial functions from γ to 2, where the ground model satisfies GCH and γω = γ.

Simple forcing notions and forcing axioms

In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [4] in which several problems were posed. We answer some of those problems here.In the first section we deal with the problem of adding Cohen reals by simple forcing notions. Here we interpret simple as of small size. We try to establish as weak as possible versions of Martin Axiom sufficient to conclude that some forcing notions of size less than the continuum add a Cohen real. For example we show that MA(σ-centered) is enough to cause that every small σ-linked forcing notion adds a Cohen real (see Theorem 1.2) and MA(Cohen) implies that every small forcing notion adding an unbounded real adds a Cohen real (see Theorem 1.6). A new almost ωω-bounding σ-centered forcing notion ℚ⊚ appears naturally here. This forcing notion is responsible for adding unbounded reals in this sense, that MA(ℚ⊚) implies that every small forcing notion adding a new real adds an unbounded real (see Theorem 1.13).In the second section we are interested in Anti-Martin Axioms for simple forcing notions. Here we interpret simple as nicely definable. Our aim is to show the consistency of AMA for as large as possible class of ccc forcing notions with large continuum. It has been known that AMA(ccc) implies CH, but it has been (rightly) expected that restrictions to regular (simple) forcing notions might help.