Martin's Axiom is Consistent with the Existence of Nowhere Trivial Automorphisms

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).

梯级饱和与力迫公理

Echeloned saturation and forcing axioms

针对Paul Larson的问题，研究结合不同力迫公理与集合论假设，证明特定条件下ℵ₁上无弱Laver理想，以及Baumgartner公理下ℵ₂上无Laver理想。

${\sf MM}^{++}$ 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…

分离颠覆力迫公理

Separating Subversion Forcing Axioms

本研究通过引入子完备和子恰当力迫公理的变体，开发了证明非蕴含关系的通用技术，并展示了SCFA不蕴含MA^+(σ-closed)及SubPFA不蕴含Martin's Maximum等应用。

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。

关于对称群的度量约化积的自同构群

On automorphism groups of metric reduced products of symmetric groups

在OCA和马丁公理下，研究对称群度量约化积的同构，刻画其自同构群。

梯子系统染色的均匀化与平稳预 caliber 力迫

Uniformization of ladder system colorings and stationary precaliber forcings

本文研究了梯子系统染色均匀化变体与 Martin 公理片段的关系，提出了平稳预 caliber $\aleph_1$ 性质的力迫公理，并证明了其与 $σ$-链接偏序集力迫公理的分离。

Parametricity, automorphisms of the universe, and excluded middle

It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming specific instances of non-parametricity. We also address the interaction between classical axioms and the existence of automorphisms of a type universe. We work over intensional Martin-Löf dependent type theory, and in some results assume further principles including function extensionality, propositional extensionality, propositional truncation, and the univalence axiom.

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.

Somewhere trivial automorphisms

It is shown to be consistent that there is a non-trivial autohomeomorphism of beta N while all such autohomeomorphisms are trivial on some open set. The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.

Comparisons of polychromatic and monochromatic Ramsey theory

We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erd{ö}s and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various specia…

针对 $\aleph_1$ 规模偏序集的真力迫公理、$ω_1$-链式对称真力迫与连续统的大小

The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum

本文证明针对 $\aleph_1$ 规模偏序集的真力迫公理与任意大的连续统相容，并进一步构造模型同时满足 Martin's Maximum 及多种组合原则。

反射与递归

Reflection and Recurrence

本文研究结构性反射原则与递归公理对 ZFC 的扩展，提出其极大一致组合 Laver-generic Maximum（LGM），将连续统确定为 aleph_2 并整合多种已知独立陈述。

可分离C*-代数无需可数选择公理

Set-theoretic absoluteness for analysts and separable \cstar-algebras without Choice

本文在不使用选择公理的情况下，研究了可分离C*-代数的理论发展，并展示了某些结果在ZF中的可证明性。

A 1-separably injective space that does not contain $\ell_\infty$

We show that the problem whether every $1$-separably injective Banach space contains an isomorphic copy of $\ell_\infty$ is undecidable. Namely, unlike under the continuum hypothesis, assuming Martin's axiom and the negation of the continuum hypothesis, there is an $1$-separably injective Banach space of the form $C(K)$ (which means that $K$ is an $F$-space) without an isomorphic copy of $\ell_\infty$. This result is a consequence of our study of $ω_2$-subsets of tightly $σ$-filtered Boolean algebras introduced by Koppelberg for which we obtain some general principles useful when transferring …

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。

Indestructible colourings and rainbow Ramsey theorems

We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{ω_2}]^2\to 2$ establishing $ω_2 \not\to [(ω_1;ω)]^2_2$ such that some colouring $g:[ω_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{ω_1}$ is arbitrarily large, and a function $g$ establishes $2^{ω_1} \not\to [(ω_1,ω_2)]^2_{ω_1}$ such that there is no uncountable $g$-rainbow subset of $2^{ω_1}$. We also show that for each $k\in ω$ it is consistent with Martin's Axiom that the negative partition relation $ω_1 \not\to^* [(ω_1;ω_1)]_{k-bdd}$ holds.

Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

We answer two questions of Hindman, Steprāns and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\mathrm{cov}(\mathcal M)=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.