A combinatorial forcing for coding the universe by a real when there are no sharps

Potential theory and forcing

We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.

Combinatorial properties of Hechler forcing

In this work we use a notion of rank first introduced by James Baumgartner and Peter Dordal and later developed independently by the third author to show that adding a Hechler real has strong combinatorial consequences. We prove: 1) assuming omega_1^V = omega_1^L, there is no real in V[d] which is eventually different from the reals in L[d], where d is Hechler over V; 2) adding one Hechler real makes the invariants on the left-hand side of Cicho'n's diagram equal omega_1 and those on the right-hand side equal 2^omega and produces a maximal almost disjoint family of subsets of omega of size ome…

Coding and reshaping when there are no sharps

Assuming 0^sharp does not exist, kappa is an uncountable cardinal and for all cardinals lambda with kappa <= lambda < kappa^{+ omega}, 2^lambda = lambda^+, we present a ``mini-coding'' between kappa and kappa^{+ omega}. This allows us to prove that any subset of kappa^{+ omega} can be coded into a subset, W of kappa^+ which, further, ``reshapes'' the interval [kappa, kappa^+), i.e., for all kappa < delta < kappa^+, kappa = (card delta)^{L[W cap delta]}. We sketch two applications of this result, assuming 0^sharp does not exist. First, we point out that this shows that any set can be coded by a…

Controlling iterated jumps of solutions to combinatorial problems

Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their it…

Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > ω_{1}$ is equivalent to the statement that 1, $ω$, $ω_{1}$, $ω\times ω_{1}$, and ${\left[ω_{1}\right]}^{< ω}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation $ω_{1} \rightarrow (ω_{1}, α)^2$ for all $α< ω_{1}$. To this effect, we i…

More notions of forcing add a Souslin tree

An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $λ^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.

通用Sacks不可毁的实数组合族

Universally Sacks-indestructible combinatorial families of reals

本研究引入实数组合族的算术型概念，证明对Sacks力迫可数乘积不可毁的算术型组合族必为通用Sacks不可毁，并在连续统假设下给出这类族的统一构造及极大共有限群的存在性结果。

Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters

We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $ω$-hitting and $ω$-s…

通过ssp力迫添加穿过多个正则基数的共尾可数序列

Adding cofinal countable sequences through multiple regular cardinals by ssp forcing

本研究提出一种基于ZFC的初等平稳集保持力迫构造方法，可使指定的大于ω₁的正则基数集元素都具有ω共尾性，同时保证基模其他不可数正则基数不会在力迫扩张中获得可数共尾性。

Sufficient conditions for the forcing theorem, and turning proper classes into sets

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself), including the three properties presented in this paper, imply yet another regularity property for class forcing notions, namely that proper classes of the ground model cannot become sets in a generic extension, that is they do not have set-sized names in the ground model. We then show that over certain models of Gödel-Bernays set theory without the power set axiom, …

Higher random indestructibility of MAD families

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $κ$ is indestructible by the higher random forcing $\mathbb Q_κ$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_κ) = \mathfrak b_κ= \mathfrak c_κ$ implies the existence $\mathbb Q_κ$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.

Laver forcing的性质与其所关联的co-ideal表达形式的Katetov序相关

Properties of Laver forcing associated with a co-ideal expressed via the Katetov order

本文研究了与co-ideal相关的Laver forcing的性质，分析其在Katetov序下的组合特性，并探讨了添加半Cohen实数而不添加Cohen实数的问题。

A version of the Jensen-Johnsbråten coding at arbitrary level $n\geq 3$

Theorem: Let $n\ge 2.$ There is a CCC in $L$ forcing notion $P=P_n\in L$ such that $P$-generic extensions of $L$ are of the form $L[a],$ where $a\subseteqω$ and 1) $a$ is $Δ^1_{n+1}$ in $L[a]$; and 2) if $b\in L[a],$ $b\subseteqω$ is $Σ^1_n$ in $L[a]$ then $b\in L$ and $b$ is $Σ^1_n$ in $L$. In addition, if a model $M$ extends $L$ and contains two different $P$-generic sets $a,\,a'\subseteqω,$ then $ω^M_1 > ω^L_1$. Comment: For $n=2,$ this is a result of Jensen and Johnsbråten, 1974. In this case, 2) is a corollary of the Shoenfield absoluteness theorem.

与更高巴乌斯特公理相关的组合性质

Combinatorial Properties Related to the Higher Baumgartner's Axiom

本文研究两个可由特定结构表达的组合性质，每个性质与CH相容，其结合与2^ω≤ω_2及2^{ω_1}=2^{ω_2}=ω_3相容时，可推出存在一个c.c.c.强制使更高巴乌斯特公理成立。

Generic Coding with Help and Amalgamation Failure

We show that if $M$ is a countable transitive model of ZF and if $a,b$ are reals not in $M$, then there is a $G$ generic over $M$ such that $b \in L[a,G]$. We then present several applications such as the following: if $J$ is any countable transitive model of ZFC and $M \not\subseteq J$ is another countable transitive model of ZFC of the same ordinal height $α$, then there is a forcing extension $N$ of $J$ such that $M \cup N$ is not included in any transitive model of ZFC of height $α$. Also, assuming $0^\#$ exists, letting $S$ be the set of reals generic over $L$, although $S$ is disjoint fr…

Not collapsing cardinals <= kappa in (< kappa) --support iterations II

We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues [Sh:587]. We also deal with complimentary combinatorial results.

Ramsey Theory on Infinite Structures and the Method of Strong Coding Trees

This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the author's recent method of trees with coding nodes are presented, showing how they can be useful both for coding structures with forbidden configurations as well as those with none. Using forcing as a tool for finite searches has allowed the development of Ramsey theory on such trees, leading to solutions for finite big Ramsey degrees of Henson graphs as well as infinite dimensional Ramsey theory of copies of the Rado…

Large Cardinals and Definable Well-Orderings of the Universe

We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at a proper class of successor cardinals.

The grounded Martin's axiom

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, which asserts that the universe is a ccc forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of Martin's axiom. The new axiom is shown to be consistent with the failure of Martin's axiom and a singular continuum. We prove that the grounded Martin's axiom is preserved in a strong way when adding a Cohen real and that adding a random real to a model of Martin's axiom preserves the grounded version (even though it destro…

Finite powers and products of Menger sets

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.

Products of Hurewicz spaces in the Laver model

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel's conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.