Uri Abraham, Saharon Shelah
Assuming the continuum hypothesis there is an inseparable sequence of length omega_1 that contains no Lusin subsequence, while if Martin's Axiom and the negation of CH is assumed then every inseparable sequence (of length omega_1) is a union of countably many Lusin subsequences.
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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…
Martin's Axiom does not imply that all automorphisms of P(N)/[N]^{<aleph_0} are somewhere trivial.
We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $ω_1$ that don't seem to have been considered in the literature before.
It is shown that Martin's Axiom for sigma-centred partial orders implies that every maximal orthogonal family in R^N is of size 2^{aleph_0}
The axioms of ZFC provide a foundation for mathematics, however, there are statements independent of ZFC, such as the Continuum Hypothesis (CH). We discuss Martin's axiom, which is an alternative to CH that roughly states that if there is a cardinal strictly between $ω$ and $2^ω$ it "behaves" like $ω$.
Distinguishing Martin's axiom from its restrictions
该研究提出一种对强着色破坏极小的可数链条件力迫迭代方法，证明了Martin公理严格强于两类受限形式的Martin公理，给出了更精细的公理强度区分。
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。
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.
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 $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.
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.
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.
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…
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。
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'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。
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 …
I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of targets independently and simultaneously. I primarily investigate jointness in the case of various kinds of Laver diamonds. In the case of measurable cardinals I show that, while the assertions that there are joint families of Laver diamonds of a given length get strictly stronger with increasing length, they are all equiconsistent. This is contrasted with …
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).
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。
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…
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。
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).
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.
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…
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。
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。
If $A_n$ is a sequence of C*-algebras, then the C*-algebra $\prod A_n / \bigoplus A_n$ is called a reduced product. We prove, assuming Todorcevic's Axiom and Martin's Axiom, that every isomorphism between two reduced products of separable, unital UHF algebras must be definable in a strong sense. As a corollary we deduce that two such reduced products $\prod A_n / \bigoplus A_n$ and $\prod B_n / \bigoplus B_n$ are isomorphic if and only if, up to an almost-permutation of $\mathbb{N}$, $A_n$ is isomorphic to $B_n$.
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.
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…
Martin's Axiom for $σ$-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.
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…
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。
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。
We prove several rigidity results for corona $C^*$-algebras and Čech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todorčević's $\OCA$ and Martin's Axiom at level $\aleph_1$ imply: (i) that if $X$ and $Y$ are locally compact second countable topological spaces, then all homeomorphisms between $βX\setminus X$ and $βY\setminus Y$ are induced by homeomorphisms between cocompact subspaces of $X$ and $Y$; (ii) that all automorphisms of the corona algebra of a separable $C^*$-algebra are trivial in a topological sense; (iii) that if $A$ is …
The notion of a $δ$-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis that for each $δ< ω_{2}$, any $δ$-generic sequence of P-points can be extended to an $ω_{2}$-generic sequence. This shows that the Continuum Hypothesis implies that there is a chain of P-points of length ${\mathfrak{c}}^{+}$ with respect to both Rudin-Keisler and Tukey reducibility. The proofs can be easily adapted to get such a chain of length ${\mathfrak{c}}^{+}$ under a more general hypothesis like Martin's Axiom. These results answer an old question of Andreas Blass.
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。
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