We construct a generic extension of $L$ satisfying Martin's Axiom, $2^{\aleph_0}=\aleph_3$, a lightface $Δ^1_3$ wellorder of the reals, and $Σ^1_n$-uniformization for every $n\geq 2$ simultaneously.
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…
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.
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 $ω$.
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.
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…
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.
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 …
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).
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…
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…
We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Lof's type theory (hence acceptable from a constructive and generalised-predicative standpoint). In addition, it is strong enough to prove the Set Compactness Theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more gene…
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…
A permutation group acting transitively on certain collections of models
E. T. S. I. I. Universidad Politécnica de Madrid（马德里理工大学）；Einstein Institute of Mathematics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）；Department of Mathematics, Rutgers University（罗格斯大学数学系）
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。
