A torsion-free abelian group B of arbitrary rank is called a B_1-group if Bext^1(B,T)=0 for every torsion abelian group T, where Bext^1 denotes the group of equivalence classes of all balanced exact extensions of T by B. It is a long-standing problem whether or not the class of B_1-groups coincides with the class of B_2-groups. A torsion-free abelian group B is called a B_2-group if there exists a continuous well-ordered ascending chain of pure subgroups,
0=B_0 subset B_1 subset ... subset B_alpha subset ... subset B_lambda =B=bigcup_{alpha in lambda} B_alpha such that B_{alpha+1}=B_alpha+G_alpha for every alpha in lambda for some finite rank Butler group G_alpha. Both, B_1-groups and B_2-groups are natural generalizations of finite rank Butler groups to the infinite rank case and it is known that every B_2-group is a B_1-group. Moreover, assuming V=L it was proven that the two classes coincide. Here we demonstrate that it is undecidable in ZFC whether or not all B_1-groups are B_2-groups. Using Cohen forcing we prove that there is a model of ZFC in which there exists a B_1-group that is not a B_2-group.
We show that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext_Z(G, Z) is as large as possible for every prime p and any torsion-free abelian group G . Moreover, given an uncountable strong limit cardinal mu of countable cofinality and a partition of P (the set of primes) into two disjoint subsets P_0 and P_1, we show that in some model which is very close to ZFC there is an almost-free abelian group G of size 2^{mu}= mu^+ such that the p-rank of Ext_Z(G,Z) equals 2^{mu}= mu^+ for every p in P_0 and 0 otherwise, i.e. for p…
The Wallace problem and countably compact torsion-free Abelian groups in ZFC
Independent researcher（独立研究者）；Department of Mathematics, Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学数学、统计与计算机科学学院）
$\aleph_1$-free groups, abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. In this paper, we give a complete proof that the property of being $\aleph_1$-free is absolute; that is, if an abelian group $G$ is $\aleph_1$-free in some transitive model $\textbf{M}$ of ZFC, then it is $\aleph_1$-free in any transitive model of ZFC containing $G$. The absoluteness of $\aleph_1$-freeness has the following remarkable consequence: an abelian group $G$ is $\aleph_1$-free in some transitive model of ZFC if and only if it is (c…
We prove in ZFC that an abelian group $C$ is cotorsion if and only if $\operatorname{Ext}(F,C) = 0$ for every $\aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the $\barλ$-Black Box for $\aleph_k$-free constructions in ZFC.
If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms introduced by Hamkins.
A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set the…
A Borel Concept Class of VC Dimension One with a Non-PAC Consistent Learner in ZFC
Instituto de Matemática, Estatística e Ciência da Computação, Universidade de São Paulo（圣保罗大学数学与统计及计算机科学学院）；Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算机科学学院）
If ZFC is consistent, then each of the following are consistent with ZFC + 2^{aleph_0}= aleph_2 : 1.) X subseteq R is of strong measure zero iff |X| <= aleph_1 + there is a generalized Sierpinski set. 2.) The union of aleph_1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size aleph_2.
Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + κ(ω)$ exists, then $\mbox{TFAG}$ is $a Δ^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $α$-ary Schröder- Bernstein property, and show that $\mbox{TFAG}$ fails the $α$-ary Schröder-Bernstein property for every $α< κ(ω)$. We leave open whether or not $\mbox{TFAG}$ can have the $κ(ω)$-ary Schröder-Bernstein property; if it did, then it would not be $a Δ^1_2$-complete, and hence not Borel complete.
We prove that if ZF is consistent then ZFC+GCH is consistent with the following statement: There is for every k<omega a model of cardinality aleph_1 which is L_{infty,omega_1}-equivalent to exactly k non-isomorphic models of cardinality aleph_1. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC+GCH that there is a ``good'' ladder system having exactly p^m pairwise nonequivalent colourings.
We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line).
A graph G on omega_1 is called <omega-smooth if for each uncountable subset W of omega_1, G is isomorphic to G[W-W'] for some finite W'. We show that in various models of ZFC if a graph G is <omega-smooth then G is necessarily trivial, i.e, either complete or empty. On the other hand, we prove that the existence of a non-trivial, <omega-smooth graph is also consistent with ZFC.
We prove in ZFC that the Baer-Specker group Z^omega has 2^{aleph_1} non-free pure subgroups of cardinality aleph_1 which are almost disjoint: there is no non-free subgroup embeddable in any pair.
Define z to be the smallest cardinality of a function f:X->Y with X and Y sets of reals such that there is no Borel function g extending f. In this paper we prove that it is relatively consistent with ZFC to have b<z where b is, as usual, smallest cardinality of an unbounded family in w^w. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists a set of reals X such that the Borel order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.
We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra $\mathcal{Q}(H)$ is unitarily equivalent to an endomorphism of $\mathcal{Q}(H)$ which is liftable to a unital endomorphism of $\mathcal{B}(H)$. We use this result to classify all unital endomorphisms of $\mathcal{Q}(H)$ up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of $\mathrm{C}^\ast$-algebras that embed into $\mathcal{Q}(H)$ is not closed under tensor product nor countable inductive…
We answer a question of Darji and Keleti by proving in $ZFC$ that there exists a compact nullset $C_0\subset\RR$ such that for every perfect set $P\subset\RR$ there exists $x\in\RR$ such that $(C_0+x)\cap P$ is uncountable. Using this $C_0$ we answer a question of Gruenhage by showing that it is consistent with $ZFC$ that less than $2^ω$ many translates of a compact nullset cover $\RR$.
We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two separable abelian groups of cardinality aleph_1 which are filtration equivalent and one is a Whitehead group but the other is not. This solves one of the open problems of Eklof and Mekler.
We show that (1) If ZF is consistent then the following theory is consistent "ZF + DC(omega_{1}) + Every set of reals has Baire property" and (2) If ZF is consistent then the following theory is consistent "ZFC + `every projective set of reals has Baire property' + `any union of omega_{1} meager sets is meager' ".
We show that the existence of an almost Souslin Kurepa tree is consistent with $ZFC$. We also prove their existence in $L$. These results answer two questions from Zakrzewski.
In [arXiv:1605.02261] Rosłanowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely whether it is consistent with $ZFC$ that in every locally compact group there are no meager but non-null subgroups. The answer is affirmative for both questions [arXiv:1605.02261], however, in this paper we provide a much simpler proof for the special case of Abelian Polish groups.
We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph_1-separable abelian groups of cardinality aleph_1. In fact, there is an aleph_1-separable abelian group M such that M is isomorphic to M oplus M oplus M but not to M oplus M .
In light of the celebrated theorem of Vopěnka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amena…
We discuss the existence of (injectively) universal C*-algebras and prove that all C*-algebras of density character $\aleph_1$ embed into the Calkin algebra, $Q(H)$. Together with other results, this shows that each of the following assertions is relatively consistent with ZFC: (i) $Q(H)$ is a $2^{\aleph_0}$-universal C*-algebra. (ii) There exists a $2^{\aleph_0}$-universal C*-algebra, but $Q(H)$ is not $2^{\aleph_0}$-universal. (iii) A $2^{\aleph_0}$-universal C*-algebra does not exist. We also prove that it is relatively consistent with ZFC that (iv) there is no $\aleph_1$-universal nuclear …
We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).
We show that it is relatively consistent with ZFC that 2^omega is arbitrarily large and every sequence s=(s_i:i<omega_2) of infinite cardinals with s_i<=2^omega is the cardinal sequence of some locally compact scattered space.
Given a group G, we let Z(G) denote its center, G' its commutator subgroup, and Phi (G) its Frattini subgroup (the intersection of all maximal proper subgroups of G). Given U leq G, we let N_G (U) stand for the normalizer of U in G. A group G is FC iff every element g in G has finitely many conjugates. A p-group E is called extraspecial iff Phi (E) = E' = Z(E) cong Z_p, the cyclic group with p elements. When generalizing a characterization of centre-by-finite groups due to B. H. Neumann, M. J. Tomkinson asked the following question. Is there an FC-group G with vert G / Z(G) vert = kappa but [G…
We show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property.
In [arXiv:1605.02261] Rosłanowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with $ZFC$ that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper we give affirmative answers for the general case.
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<ł<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $ł$ and its existence is not a result of …
We show that it is relatively consistent with ZFC that there exists a hyperfinite type $\mathrm{II}_1$-factor of density character $\aleph_1$ which is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.
We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A intersect C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.
We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.
We show that it is consistent with ZFC that L^infty (Y,B, nu) has no linear lifting for many non-complete probability spaces (Y,B, nu), in particular for Y=[0,1]^A, B= Borel subsets of Y, nu = usual Radon measure on B .
We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszyński and Judah.
We show that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O_2, or of the CAR algebra.
A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals
Instituto de Matemática e Estatística, Universidade Federal da Bahia（巴伊亚联邦大学数学与统计学院）；Department of Mathematics and Statistics, University of Ottawa（渥太华大学数学与统计系）
It is consistent (relative to ZFC) that the union of max{b,g} many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U, the cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.
Generalizing [ShSi:494], for every n< omega we construct a ZFC-model where the distributivity number of r.o. (P(omega)/fin)^{n+1}, h(n+1), is smaller than the one of r.o.(P(omega)/fin)^{n}. This answers an old problem of Balcar, Pelant and Simon. We also show that Laver and Miller forcing collapse the continuum to h(n) for every n<omega, hence by the first result, consistently they collapse it below h(n)
A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $å$ in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G-delta set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology in inherits a subset of the Power set of {0,1,2,..}, ie the Cantor set.
This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph $(μ,R)$ we can find a family $\{\mathbb G_α: α< μ\}$ of abelian groups such that for each $α,β\inμ$: $$Ext(\mathbb G_α, \mathbb G_β) = 0 \Longleftrightarrow(α,β) \in R.$$ In this regard, we present four results. First, we give a connection to Quillen's small object argument which helps $Ext$ vanishes and uses to present useful criteria to the question. Suppose $λ= λ^{\aleph_0}$ and $μ= 2^λ$. We apply Jensen's diamond principle along with the criteria to present $λ$-free abelian groups represe…
A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $μ$ on $G$ such that $μ(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $μ$ such that $μ(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This resul…
This paper is concerned with the isomorphic structure of the Banach space $\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is consistent but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $\ell_\infty/c_0$ does not have an orthogonal \break $\ell_\infty$-decomposition that is, it is not of the form $\ell_\infty(X)$ for any Banach space $X$. The main local result is that it is consistent that $\ell_\infty(c_0(\mathfrak{c}))$ does not embed isomorphically into $\ell_\infty/c_0$, where $\mathfrak{c}$ is the cardinality of…
