Games with creatures
英文摘要：
Many forcing notions obtained using the creature technology are naturally connected with certain integer games.
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Creature forcing and topological Ramsey spaces
This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples.
math.LO
Creature forcing and five cardinal characteristics in Cichoń's diagram
We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
math.LO
Creature forcing and large continuum: The joy of halving
For $f,g\inω^ω$ let $c^\forall_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e., for every branch $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. Let $c^\exists_{f,g}$ be the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. We show that it is consistent that $c^\exists_{f_ε,g_ε}=c^\forall_{f_ε,g_ε}=κ_ε$ for continuum many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we introduce a new mixed-limit creature forcing construction.
math.LO
Norms on possibilities I: forcing with trees and creatures
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the solutions we develop various ideas interesting per se.These include a new iterable condition for ``not adding Cohen reals'' (which has a flavour of preserving special properties of p-points), new intriguing properties of ultrafilters (weaker than being Ramsey but stronger than p-point) and some new applications of variants of the PP--property.
math.LO
Sheva-Sheva-Sheva: Large Creatures
We develop the theory of the forcing with trees and creatures for an inaccessible lambda continuing Rosłanowski and Shelah math.LO/9807172, math.LO/9909115. To make a real use of these forcing notions (that is to iterate them without collapsing cardinals) we need suitable iteration theorems, and those are proved as well. (In this aspect we continue Rosłanowski and Shelah math.LO/9906024.)
math.LO
Partition theorems from creatures and idempotent ultrafilters
We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.
math.COmath.LO
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Towards Martin's Minimum
We show that it is consistent with MA + the negation of CH, that the Forcing Axiom fails for all forcing notions in the class of omega^omega-bounding forcing notions with norms of "Norms on possibilities I: forcing with trees and creatures"(by Roslanowski and Shelah - math.LO/9807172).
math.LO
Nonproper Products
We show that there exist two proper creature forcings having a simple (Borel) definition, whose product is not proper. We also give a new condition ensuring properness of some forcings with norms.
math.LO
Measured creatures
Using forcing with measured creatures we build a universe of set theory in which: (a) every sup-measurable function f:RxR-->R is measurable, and (b) every function f:R-->R is continuous on a non-measurable set. This answers a question of Balcerzak, Ciesielski and Kharazishvili and von Weizsacker's problem (see Fremlin's list of problems).
math.CAmath.GNmath.LO
The splitting number can be smaller than the matrix chaos number
Let chi be the minimum cardinal of a subset of 2^omega that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of creature forcing we show that s<chi is consistent. We thus answer a question by Vojtas. We give two kinds of models for the strict inequality. The first is the combination of an aleph_2-iteration of some proper forcing with adding aleph_1 random reals. The second kind of models is got by adding delta random reals to a model of MA_{< kappa} for some delta in [aleph_1,kappa). It was a conjecture of Blass that s=aleph_1<chi=kappa holds in such…
math.LO
Specializing Aronszajn trees by countable approximations
We show that there are proper forcings based upon countable trees of creatures that specialize a given Aronszajn tree.
math.LO
Many Different Uniformity Numbers of Yorioka Ideals
Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.
math.LO
具有ccc的糖性
Saccharinity with ccc
本文利用creatures技术构造了若干Suslin ccc非甜forcing概念，探讨了ZFC与ZF+实数集等于Borel集模null理想闭包的等价性，以及ω₁序列的构造。
math.LO
Cichoń's Diagram and Localisation Cardinals
We reimplement the creature forcing construction used by Fischer et al. (arXiv:1402.0367) to separate Cichoń's diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals.
math.LO
Forcing, games and families of closed sets
We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ($\mathbfΠ^1_1$ on $\mathbfΣ^1_1) $σ$-ideals we show that if a $σ$-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the $σ$-ideal $\E$ generated by closed null sets and $σ$-ideals connected with not piecewise continuous functions.
math.LO
在forcing下某些拓扑性质的保持
Preservation of some topological properties under forcing
Czech Academy of Sciences（捷克科学院）
本文研究了在forcing下某些拓扑性质的保持问题，证明了在特定游戏中玩家II的获胜策略在集合forcing扩展中保持不变。
math.LO
Continuum Many Different Things: Localisation, Anti-Localisation and Yorioka Ideals
Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $ω^ω$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from arXiv:1805.11005.
math.LO
*变化的Banach-Mazur游戏与forcing公理
The *-variation of the Banach-Mazur game and forcing axioms
Nagoya University（名古屋大学）
本文提出一种加强(ω_1+1)-strategic closedness的poset性质，并证明PFA在特定forcing下的保持性，同时应用该性质重新证明了PFA与弱square原理的一致性。
math.LO
通过强 ssp 力迫增加第二一致不可辨元
Increasing the second uniform indiscernible by strongly ssp forcing
本文提出一种新的平稳集保持力迫 $\mathbb P^{c-c}(λ,μ)$，在特定大基数假设下将第二一致不可辨元 $\mathbf{u}_2$ 增加至超越给定序数 $λ$，并利用双人捕获博弈控制力迫行为。
math.LO
Many countable support iterations of proper forcings preserve Souslin trees
We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
math.LO
Forcing Iterated Admissibility in Strategic Belief Models
Iterated admissibility (IA) can be seen as exhibiting a minimal criterion of rationality in games. In order to make this intuition more precise, the epistemic characterization of this game-theoretic solution has been actively investigated in recent times: it has been shown that strategies surviving m+1 rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality (RmAR) in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, $R\infty AR$, might not be satisfied…
math.LO
Combinatorics on Ideals and Axiom A
Throughout this abstract let U be a fixed p-point ultrafilter and let I be the dual ideal. Grigorieff forcing is P(U)={p:omega to 2|dom(p) is an element of I} ordered by reverse inclusion. It is well known that Grigorieff forcing is proper. The main result of this paper is the following: THEOREM: Gregorieff forcing does not satisfy Axiom A. To prove this we use the following game, denoted G(U), for two players playing alternatively: Player I plays a partition of omega, {J_n| n<omega}, such that for all n<omega, J_n is an element of I; At the nth turn Player II plays a finite subset F_n of J_n.…
math.LO
Uniform Unfolding and Analytic Measurability
We regard Forcing Notions P adding real numbers and the algebras of P-measurable sets. As for Cohen- and Random-Forcing we can show that each analytic set is P-measurable using Solovay's Unfolding Trick for infinite games. To show this we develop the notion of strategic fusion and give a proof of a uniform unfolding theorem which allows us to reduce the proof of analytic measurability to the definition of a certain function.
math.LO
Sacks模型中的Menger集与consonant集
Menger and consonant sets in the Sacks model
本文通过迭代Sacks力迫和拓扑博弈，证明Cantor立方体中基数为连续统的全不完美Menger集的存在性独立于ZFC，并分析了Sacks模型中Hurewicz与consonant子集的结构。
math.GNmath.LO
