Martin's Axiom and Maximal Orthogonal Families

Martin公理的片段

Fragments of Martin's axiom

本文证明Martin公理对$ω_1$稠密集等价于每个ccc偏序集具有Knaster性质K$_3$的片段，并指出维度3在某种意义上是极小的。

Adding a random real number and its effect on Martin's axiom

We show that adding a random real number destroys a large fragment of Martin's axiom, namely Martin's axiom for partial orders that have precalibre-$\aleph_1$, thus answering an old question of J. Roitman [9]. We also answer a question of J. Steprans and S. Watson [13] by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph_1$ property of a partial ordering while preserving its ccc-ness.

Generalizations of Martin's Axiom, weak square, weak Chang's Conjecture, and a forcing axiom failure

We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{ω_1, ω_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.

Martin's Axiom and Delta^2_1 well-ordering of the reals

Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.

Gregory Trees, The Continuum, And Martin's Axiom

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.

On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $σ$-linked partial orderings. This answers an old question of the first author about the relative strength of Martin's axiom for $σ$-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprans and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can de…

马丁公理、大连续统和全局$Σ^1_n$-统一化

Martin's Axiom, Large Continuum and Global $Σ^1_n$-Uniformization

本文研究在马丁公理、$2^{\aleph_0}=\aleph_3$、实数的轻语气$Δ^1_3$良序及所有$n\geq 2$的$Σ^1_n$-统一化条件下，构造一个满足这些条件的广义扩展。

Remark on the Failure of Martin's Axiom

Let m be the least cardinal k such that MA(k) fails. The only known model for "m is singular" was constructed by Kunen. In Kunen's model cof(m)=omega_1. It is unknown whether "omega_1 < cof(m) < m" is consistent. The purpose of this paper is to present a proof of Kunen's result and to identify the difficulties of generalizing this result to an arbitrary uncountable cofinality.

Martin公理与弱Kurepa假设

Martin's Axiom and Weak Kurepa Hypothesis

该研究证明，在Mahlo基数相容的前提下，可构造模型使Martin公理在$ω_2$成立而弱Kurepa假设失败，回答了相关开放问题，并利用c.c.c.力迫扩张确保弱Kurepa假设始终不成立。

Thin ultrafilters, P-hierarchu and MArtin Axiom

Under MA we prove that for the ideal $\cal I$ of thin sets on $ω$ and for any ordinal $γ\leq ω_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_γ$ of P-hierarchy of ultrafilters. Since the class of ${\cal P}_2$ ultrafilters coincides with a class of P-points, out result generalize theorem of Flašková, which states that there are ${\cal I}$-ultrafilters which are not P-points. It is also related to theorem which states that under CH for any tall P-ideal $\cal I$ on $ω$ there is an ${\cal I}$-ultrafilter, however the ideal of thin sets is …

Martin's maximum revisited

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. The main results is that the forcing axiom MM^{++} (also known as MM^{+ω_1}) decides the Π_2-theory of H_{ω_2} with respect to stationary set preserving forcings. We argue that this is a close to optimal generalization to H_{ω_2} of Woodin's absoluteness results for L(R).

在Martin公理下构造一个具有Grothendieck性质但不具有Nikodym性质的布尔代数

Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property

该研究在弱Martin公理假设下构造特殊完全不连通紧Hausdorff空间，改进了Talagrand的结果，证明Martin公理蕴含存在具Grothendieck性质但无Nikodym性质的布尔代数。

Some infinitely generated non projective modules over path algebras and their extensions under Martin's Axiom

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective (Theorem 1.10). In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{ω_1}$ such that, when $K$ is a countable field, $\operatorname{\sf MA}_{\aleph_1}$ (Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory…

马丁公理与$ω_1^2 \to (ω_1^2, 3)^2$

Martin's axiom and $ω_1^2 \longrightarrow (ω_1^2, 3)^2$

该研究从CH和Hajnal着色出发，通过σ-中心力迫的有限支持迭代构造出满足$MA_{ω_1}$（σ-中心）等条件且$ω_1^2\nrightarrow(ω_1^2,3)^2$成立的模型，同时提出并证明了相关力迫保持原理。

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).

${\sf MM}^{++}$ implies $(*)$

We show that Martin's Maximum${}^{++}$ implies Woodin's ${\mathbb P}_{\rm max}$ axiom $(*)$. This answers a question from the 1990's and amalgamates two prominent axioms of set theory which were both known to imply that there are $\aleph_2$ many real numbers.

Reflexive subgroups of the Baer-Specker group and Martin's axiom

In two recent papers (math.LO/0003164 and math.LO/0003165) we answered a question raised in the book by Eklof and Mekler (p. 455, Problem 12) under the set theoretical hypothesis of diamondsuit_{aleph_1} which holds in many models of set theory, respectively of the special continuum hypothesis (CH). The objects are reflexive modules over countable principal ideal domains R, which are not fields. Following H.Bass, an R-module G is reflexive if the evaluation map s:G ---> G^{**} is an isomorphism. Here G^*=Hom(G,R) denotes the dual module of G. We proved the existence of reflexive R-modules G of…

Weak diamond and Galvin's property

We prove that the Devlin-Shelah weak diamond implies Galvin's property. On the other hand, Galvin's property is consistent with the negation of the weak diamond, and even with Martin's axiom. We show that the proper forcing axiom implies a relative to the negation of Galvin's property for $\aleph_1$.

路径范畴与公理化马丁-洛夫类型理论的双等价

The biequivalence of path categories and axiomatic Martin-Löf type theories

本文研究了路径范畴与公理化马丁-洛夫类型理论之间的双等价关系，证明了公理化类型理论模型与特定路径范畴的等价性，并引入了模块化的显示映射路径范畴。

带Church论题和选择公理的归纳与余归纳拓扑生成

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

本研究通过将mTTcind+CT+AC的一致性归结为CZF+REA，证明了在最小主义框架中添加余归纳定义不增加一致性强度，并给出了两种解释路径。

非OCA与Fréchet空间的乘积

Not OCA and products of Frechet spaces

本文继续研究两个可数Fréchet空间乘积的M-可分性，重点考察马丁公理下的情形，并指出OCA蕴含肯定答案，而已有独立性结果仅适用于$\mathfrak c\leqω_2$的模型。

Parametrized Measuring and Club Guessing

We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $ω_1$ is measured by some club subset of $ω_1$. The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $σ$-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.

BMM is stronger than BSPFA

It is shown that if BMM (= Bounded Martin's Maximum) holds then each set is contained in an inner model with a strong cardinal. This answers a question that has been asked by various people. It follows that BMM has a much larger consistency strength than BSPFA (= the Bounded Semi-proper Forcing Axiom).

On embedding certain partial orders into the P-points under RK and Tukey reducibility

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin's axiom for $σ$-centered po…

The topology of ultrafilters as subspaces of $2^ω$

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $ω$ up to homeomorphism. Here, we identify ultrafilters with subpaces of $2^ω$ in the obvious way. Using the same methods, still under Martin's Axiom for countable posets, we will construct a non-principal ultrafilter $\UU\subseteq 2^ω$ such that $\UU^ω$ is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial resu…

Cosmic dimensions

Martin's Axiom for $σ$-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.

The Stationary Set Splitting Game

The \emph{stationary set splitting game} is a game of perfect information of length $ω_{1}$ between two players, \unspls and \spl, in which \unspls chooses stationarily many countable ordinals and \spls tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to $Σ^{2}_{2}$ maximality with a predicate for the nonstationary ideal on $ω_{1}$, and an example of a consistently undetermined game of length $ω_{1}$ with payoff definable in the second-order monadic log…

一个传递作用于某些模型集合的置换群

A permutation group acting transitively on certain collections of models

基于 σ-centered 的 Martin 公理，构造可数无限集对称群的真正稠密子群，使其传递作用于足够灵活的关系结构，从而一致性地肯定 Neumann 的相关问题。