We will construct several models where there are no strongly meager sets of size 2ℵ0.

A set X⊆ℝ is strongly meager if for every measure zero set H, X+H ≠ℝ. Let [Formula: see text] denote the collection of strongly meager sets. We show that assuming [Formula: see text], [Formula: see text] is not an ideal.

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.

We study a strengthening of the notion of a perfectly meager set. We say that a subset A of a perfect Polish space X is countably perfectly meager in X, if for every sequence of perfect subsets $\{P_n: n \in \mathbb N\}$ of X, there exists an $F_\sigma $ -set F in X such that $A \subseteq F$ and $F\cap P_n$ is meager in $P_n$ for each n. We give various characterizations and examples of countably perfectly (...) class='Hi'>meager sets. We prove that not every universally meager set is countably perfectly meager correcting an earlier result of Bartoszyński. (shrink)

We prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems (...) from [KL]. We prove that the set K = {1,...,H}, where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is U-meager for some cut U with min{cf (U), ci (U)} ≥ ω1. Also, every Borel, Σ1 m or countably determined set C which is U-meager for every cut U is a countable union of Borel, Σ1 m or countably determined sets respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager sets for min{cf(U), ci(U)} ≥ ω1 coincides with the least family of sets containing internal U-meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase "U-meager sets" is replaced by "U-meager for every cut U.". (shrink)

We prove that the cofinality of the smallest covering of R by meager sets is bigger than the additivity of measure.

We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω 1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary (...) ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal. (shrink)

In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID=""ID="" Mathematics Subject Classification (2000): 03e35 RID=""ID="" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658.

By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion (...) akin to strong measure zero.We investigate the validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants. (shrink)

Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221–242, 2001). We will prove that for a partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document} of the real line into meager sets and for any sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}_n}$$\end{document} of subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document} one can find a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{B}_n}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{B}_{n}}$$\end{document}’s are pairwise disjoint and have the same “outer measure with respect to the ideal of meager sets”. We get also generalization of this result to a class of σ-ideals posessing Suslin property. However, in that case we use additional set-theoretical assumption about non-existing of quasi-measurable cardinal below continuum. (shrink)

We improve results of Marczewski, Frankiewicz, Brown, and others comparing the σ-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of σ-ideals to include (...) the completely Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non(CRU 0); and a third concerning the size of add(CRU 0). We also study those classes of perfect sets which are bases for the class of always first category sets. (shrink)

We consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after ω innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent n moves of ONE only, then TWO has a winning strategy depending (...) on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] and clarifies some of the results in [B-J-S] and in [S1]. (shrink)

Bartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 93–110. We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is (...) equal to the additivity of the ideal of meager sets. (shrink)

We show that for an uncountable κ in a suitable Cohen real model for any family {A ν } ν<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets A ν into the algebra of Baire subsets of 2 κ modulo meager sets such that for all Borel B, B is meager iff h(B) = 0. The proof is uniform, works also for random reals (...) and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property. (shrink)

Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.

We study some properties of the quotient forcing notions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{tr(I)} = \wp(2^{< \omega})/tr(i)}$$\end{document} and PI = B(2ω)/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2ω. We show that the remainder forcing RI = Qtr(I)/PI is σ-closed in these cases. We also study the cardinal invariant of the continuum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, the distributivity number of the quotient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Dense(\mathbb{Q})/nwd}$$\end{document}, in order to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\wp(\mathbb{Q})/nwd}$$\end{document} collapses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, thus answering a question addressed in Balcar et al. (Fundamenta Mathematicae 183:59–80, 2004). (shrink)

In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...) line H, which is a hyperfinite initial segment of the hyperintegers. A subset B of H is called a U-Lusin set in H if B is uncountable and for any Loeb-Borel U-meager subset X of H, B ∩ X is countable. Here a Loeb-Borel set is an element of the σ-algebra generated by all internal subsets of H. In this paper we answer some questions of Keisler and Leth about the existence of U-Lusin sets by proving the following facts. (1) If $U = x/\mathbb{N} = \{y \in \mathscr{H}: \forall n \in \mathbb{N}(y < x/n)\}$ for some x ∈ H, then there exists a U-Lusin set of power κ if and only if there exists a Lusin set of the reals of power κ. (2) If U ≠ x/N but the coinitiality of U is ω, then there are no U-Lusin sets if CH fails. (3) Under ZFC there exists a nonstandard universe in which U-Lusin sets exist for every cut U with uncountable cofinality and coinitiality. (4) In any ω2-saturated nonstandard universe there are no U-Lusin sets for all cuts U except U = x/N. (shrink)

We will prove that there exists a model of ZFC+"c = ω 2 " in which every $M \subseteq \mathbb{R}$ of cardinality less than continuum c is meager, and such that for every $X \subseteq \mathbb{R}$ of cardinality c there exists a continuous function f: R → R with f[X] = [0, 1]. In particular in this model there is no magic set, i.e., a set $M \subseteq \mathbb{R}$ such that the equation f[M] = g[M] implies f = g (...) for every continuous nowhere constant functions f, g: R → R. (shrink)

We prove that consistently the Lebesgue null ideal is not Tukey reducible to the Silver null ideal. This contrasts with the situation for the meager ideal which, by a recent result of the author, Spinas [Silver trees and Cohen reals, Israel J. Math. 211 473–480] is Tukey reducible to the Silver ideal.

Assume that there is no quasi-measurable cardinal not greater than 2ω. We show that for a c. c. c. σ -ideal [MATHEMATICAL DOUBLE-STRUCK CAPITAL I] with a Borel base of subsets of an uncountable Polish space, if [MATHEMATICAL SCRIPT CAPITAL A] is a point-finite family of subsets from [MATHEMATICAL DOUBLE-STRUCK CAPITAL I], then there is a subfamily of [MATHEMATICAL SCRIPT CAPITAL A] whose union is completely nonmeasurable, i.e. its intersection with every non-small Borel set does not belong to the σ (...) -field generated by Borel sets and the ideal [MATHEMATICAL DOUBLE-STRUCK CAPITAL I]. This result is a generalization of the Four Poles Theorem and a result from [3]. (shrink)

We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set R of real numbers have the covering property of Menger if, and only if, each filter on ω associated with (...) its countable ω-cover is a P + filter. (shrink)

The principle of Sierpinski is the assertion that there is a family of functions $\left\{ {{\varphi _n}:{\omega _1} \to {\omega _1}|n \in \omega } \right\}$ such that for every $I \in {[{\omega _1}]^{{\omega _1}}}$ there is n ε ω such that ${\varphi _n}[I] = {\omega _1}$. We prove that this principle holds if there is a nonmeager set of size ω1 answering question of Arnold W. Miller. Combining our result with a theorem of Miller it then follows that is equivalent (...) to $non\left = {\omega _1}$. Miller also proved that the principle of Sierpinki is equivalent to the existence of a weak version of a Luzin set, we will construct a model where all of these sets are meager yet $non\left = {\omega _1}$. (shrink)

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of (...) indices of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)

This paper develops and motivates a unification theory of metaphysical explanation, or as I will call it, Metaphysical Unificationism. The theory’s main inspiration is the unification account of scientific explanation, according to which explanatoriness is a holistic feature of theories that derive a large number of explananda from a meager set of explanantia, using a small number of argument patterns. In developing Metaphysical Unificationism, I will point out that it has a number of interesting consequences. The view offers a (...) novel conception of metaphysical explanation that doesn’t rely on the notion of a “determinative” or “explanatory” relation; it allows us to draw a principled distinction between metaphysical and scientific explanations; it implies that naturalness and fundamentality are distinct but intimately related notions; and perhaps most importantly, it re-establishes the unduly neglected link between explanation and understanding in the metaphysical realm. A number of objections can be raised against the view, but I will argue that none of these is conclusive. The upshot is that Metaphysical Unificationism provides a powerful and hitherto overlooked alternative to extant theories of metaphysical explanation. (shrink)

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...) prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$ . In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property " $F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$ . In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$. (shrink)

Reviewed Works:Tomek Bartoszynski, Marion Scheepers, Set Theory, Annual Boise Extravaganza in Set Theory Conference, March 13-15, 1992, April 10-11, 1993, March 25-27, 1994, Boise State University, Boise, Idaho.R. Aharoni, A. Hajnal, E. C. Milner, Interval Covers of a Linearly Ordered Set.Eyal Amir, Haim Judah, Souslin Absoluteness, Uniformization and Regularity Properties of Projective Sets.Tomek Bartoszynski, Ireneusz Reclaw, Not Every $\gamma$-Set is Strongly Meager.Andreas Blass, Reductions Between Cardinal Characteristics of the Continuum.Claude Laflamme, Filter Games and Combinatorial Properties of Strategies.R. Daniel (...) Mauldin, Analytic non-Borel Sets Modulo Null Sets.Janusz Pawlikowski, Laver's Forcing and Outer Measure.Marion Scheepers, Meager Sets and Infinite Games.Saharon Shelah, On Some Problems in General Topology.Saharon Shelah, Remarks on $\aleph_1$-CWH not CWH First Countable Spaces.Juris Steprans, Cardinal Invariants Associated with Hausdorff Capacities. (shrink)

Rosłanowski and Shelah [Small-large subgroups of the reals, Math. Slov. 68(3) (2018) 473–484] asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with [Formula: see text] 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.

We show that it is consistent with ZFC that the additivity number add of the ideal of meager sets of the real line is strictly greater than the tower number t of the reals. MSC: 03E35, 54D20.

We consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function $u \colon T \to \mathbb {R}$ such that $f(x) = \limsup _{t \to \infty } u(x_{0},\dots,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non-meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak ωω (...) -bounding property, and the property “the set of reals of the ground model has positive outer measure”. (shrink)

We show that it is relatively consistent with ZFC that there is a non-meager set of reals [Formula: see text] such that for every non-meager [Formula: see text], there exist distinct [Formula: see text] such that [Formula: see text] is computable from the Turing join of [Formula: see text] and [Formula: see text].

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of (...) meager sets via Banach-Mazur games. We study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager. (shrink)

Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...) 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory. (shrink)

We define a σ-ideal J D on the set of functions ω ω with the property that a real x ∈ ω ω is a Hechler real over V if and only if x omits all Borel sets in J D . In fact we define a topology D on ω ω related to Hechler forcing such that J D is the family of first category sets in D. We study cardinal invariants of the ideal J D.

We work in the Baire space $$\mathbb {Z}^\omega $$ equipped with the coordinate-wise addition $$+$$. Consider a $$\sigma -$$ ideal $$\mathcal {I}$$ and a family $$\mathbb {T}$$ of some kind of perfect trees. We are interested in results of the form: for every $$A\in \mathcal {I}$$ and a tree $$T\in \mathbb {T}$$ there exists $$T'\in \mathbb {T}, T'\subseteq T$$ such that $$A+\underbrace{[T']+[T']+\dots +[T']}_{\text {n--times}}\in \mathcal {I}$$ for each $$n\in \omega $$. Explored tree types include perfect trees, uniformly perfect trees, Miller (...) trees, Laver trees and $$\omega -$$ Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $$\sigma $$ -ideal $$\mathcal {M}$$ of meager sets, we also analyze $$\mathcal {M}_-$$ and fake null sets $$\mathcal {N}$$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space. (shrink)

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable (...) and continuum-sized or locally compact Hausdorff. We also show that every extension of $\mathsf {S5}$ is the modal logic of a subalgebra of $\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality. (shrink)

We propose a reformulation of the ideal $\mathcal {N}$ of Lebesgue measure zero sets of reals modulo an ideal J on $\omega $, which we denote by $\mathcal {N}_J$. In the same way, we reformulate the ideal $\mathcal {E}$ generated by $F_\sigma $ measure zero sets of reals modulo J, which we denote by $\mathcal {N}^*_J$. We show that these are $\sigma $ -ideals and that $\mathcal {N}_J=\mathcal {N}$ iff J has the Baire property, which in turn is (...) equivalent to $\mathcal {N}^*_J=\mathcal {E}$. Moreover, we prove that $\mathcal {N}_J$ does not contain co-meager sets and $\mathcal {N}^*_J$ contains non-meager sets when J does not have the Baire property. We also prove a deep connection between these ideals modulo J and the notion of nearly coherence of filters (or ideals). We also study the cardinal characteristics associated with $\mathcal {N}_J$ and $\mathcal {N}^*_J$. We show their position with respect to Cichoń’s diagram and prove consistency results in connection with other very classical cardinal characteristics of the continuum, leaving just very few open questions. To achieve this, we discovered a new characterization of $\mathrm {add}(\mathcal {N})$ and $\mathrm {cof}(\mathcal {N})$. We also show that, in Cohen model, we can obtain many different values to the cardinal characteristics associated with our new ideals. (shrink)

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both (...) mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. (shrink)

We study the role the axiom of choice plays in the existence of some special subsets of ℝ and its power set ℘.

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal (...) was established by Bartoszyński and Kada. (shrink)

We consider sets of reals that are “adequate” in various senses, for example dominating or unbounded or splitting or non-meager. Call a real x “needed” if every adequate set contains a real in which x is recursive. We characterize the needed reals for numerous senses of “adequate.” We also consider, for various notions of forcing that add reals, the problem of characterizing the ground-model reals that are recursive in generic reals.

Reproduced with the kind permission of the author. Currently available in the collection I Looked Alive . © 2010 The Brooklyn Rail/Black Square Editions | ISBN 978-1934029-07-7 Originally published 2003 Four Walls Eight Windows. continent. 1.1 (2011): 43-51. Introduction Ben Segal What interests me is instigated language, language dishabituated from its ordinary doings, language startled by itself. I don't know where that sort of interest locates me, or leaves me, but a lot of the books I see in the stores (...) seem to lack language entirely. Gary Lutz (2006). This issue of continent.'s fiction section features a reprint of “This Is Nice of You” from the first edition of Gary Lutz's collection I Looked Alive . That book was originally released by Four Walls Eight Windows in 2003 and has since been reissued in 2010 by The Brooklyn Rail/Black Square Editions. Calamari Press has also reissued Lutz's Stories in the Worst Way (originally published by Knopf, largely forgotten and dismissed, and then re-issued by the too-short-lived 3rd Bed). “This Is Nice of You” was edited for the re-issue of I Looked Alive and, as the original book is out of print, the earlier version of the story has been hard to come by for some time. This initial version is a bit longer than the new version—Lutz is a master of tightening down his texts to their most linguistically intensified cores—but it is in no way sloppy. It is a great privilege for continent. to be able to make available rare texts of (first) aesthetic and (second) scholarly value. For those readers unfamiliar with Gary Lutz, I believe this story will serve as a worthy introduction. Like Raymond Carver, Diane Williams, Christine Schutt, and so many other important twentieth century and contemporary short story writers, Lutz studied with and was edited by Gordon Lish. This connection is apparent in his attention to sound and the sentence as compositional unit, but Lutz's is a unique voice. “This Is Nice of You” bears Lutz's trademark torqued and turning syntax, his sexed-but-nameless characters, his rendering of the human body and its emissions as utterly discomfiting. Lutz's preoccupations with the mundane raise ordinariness, via startling language and observation, to grandiosity and grotesquery. In the fall of 2008, Lutz delivered a lecture entitled “The Sentence is a Lonely Place” to writing students at Columbia University. The lecture was later printed in The Believer (and is linked below). In it, he spoke about the importance of the sentence as a compositional unit, as even the distinguishing formal characteristic of prose. Specifically, Lutz explained that, from Lish, he had developed a "poetics of the sentence." Accordingly, Lutz constructs his stories at the level of the sentence, paying extreme attention to the material of words- their letters and their sounds- and the way they abut one another and interact on this material level even before they are employed in the service of traditional aims of fiction like meaning or story. Lutz often uses a(n almost constraint-related) technique called consecution. Letters or sounds from one word are carried forward into the following, producing drama and movement in the language itself at a level entirely apart from that of plot. For example, you can see how the 't's in the first sentence of “This Is Nice of You” repeat fast and hard through met and then soften into the 'th's of brothers and they. This close attention to the specific material and characteristic conditions of prose means that Gary Lutz's work always involves both a happening in language and of language. “This Is Nice of You” has a richness and depth that is hidden not in symbolism or obscure reference but on the surface of the page, in the language of every sentence. FURTHER READING Lutz, Gary. “ The Sentence Is a Lonely Place. ” The Believer January, 2009. Web. Short stories and excerpts at Web del Sol . Web. Taylor, Justin. “ An Interview with Gary Lutz .” Bookslut. July, 2006. Web. THIS IS NICE OF YOU [156] [1] I was a man dropping already well through my forties, filthy with myself, when, taking a turn at the toilets one afternoon, I met two brothers-they said they were brothers-who swore they had a sister, a schoolteacher, an officer of instruction at the county college, a whirlwind midlife turmoil of everything already put to ruin, who had gone off from a new marriage in an old car, an upkept and ennobling sedan, but had returned now to the apartment and [157] was living there alone with the little runoff there was from the marriage-some outcurved appliances, apparently, and low-posted furniture promoting its own mystery but becoming figurable in certain concentrations of TV light-and, above all, a telephone (on a pedestal, they insisted), the handpiece of which she gripped in lieu of exercise, or in fury, and I thus let out my little, reliable cry that I was in fact a student of the telephone, that it was a debasing apparatus in the main, with its meager economy of bells and tones, and the intimacy of the mouthpiece that sent your breath, tiny aftervapors of it, back toward your lips, so that regardless of the party accepting the outgoing products of your voice, you were, at most, in a further, rivaling exchange with yourself alone, and this is what must have brought the two of them around, the men who proclaimed brotherhood with the woman, because they offered me her phone number, put it at my disposal on a piece of paper one of them had already committed it to, a tearing from a menu, and the looks the men were now giving me had deletions in them, already, of my exact, beanpole shape and size. So off I went to a pay phone, the nearest canopied one I could find. The woman answered after the second ring and said she needed a lift right that very moment into the little, unlevel city close by. She was idling in the doorway of the building when I pulled up in front, and I helped her into the car, then got back in myself. I had always had a way of not having to look at people that nonetheless brought them to me in full, and so I still am certain of the susceptive and impressible comp-[158]lexion, the shimmer on the mouth, a lipstick of low brilliance, a difficulty around the eyes, the hair short and rayed out exclamation ally, skin bagged up over the elbow bone, conflict even in how her arms stayed at her sides—in sum, a spinal loveliness for me, an off-blonde quantity with shadowed, thumbworn hollows that put me out of as much as I might have ever known of women before. I set the two of us into the narrow traffic, and I remember telling her, by way of explaining the little burden which I had shifted, by now, from the shelf of the dashboard and onto my lap, that when you lived in filth, as I then did, a daily newspaper came to count for a lot, although instead of the thick-supplemented local paper I bought a trimmer one from a backlying town—not, of course, for any affluences of native data it carried, but as an article of house ware: a rough immaculacy in four lank sections, a set of fresh, hygienic surfaces to come between the table, say, and whatever I had going for me at the table, if the table was where I was going forward—because what else so cheap comes so clean and far-spreading? The woman told me that her own trouble with paper was that through a modest hole, no larger than a quarter, that had been drilled a foot or so above the floor (the standard height, she had reasoned, of legitimate electrical outlets), and by means of which her faculty office had at last gained communication with the roomier but unoccupied office next door, more and more often shot a single sheet of paper, plain copier paper she had rolled just barely into a tube, [159] so that after landing on the floor of the neighboring office the paper would preserve little if any of the curl, and there would be nothing written or typed on it, of course, and it was always a blank sheet that had been ageing on her desk for some time and had already been moved around, or advanced, from station to station on the desktop, coming into further creaselets and crimps and other infirmities—paper, in short, still too bare and unfraught to be thrown responsibly and forgettably away, and yet too seasoned and beset with irritations of the surface (a molelike blemish, say, or what looked like a tiny hair, an eyelash, sunk into it, or frecklings, or notational pressings of a fingernail), too wrought, in sum, for the paper to be appointed to any secure curricular purpose. Her office, she claimed, was in fact full of such paper, much-handled and singularized sheets of it by the loose, functionless hundred. I had to get it across to the woman that I myself no longer had an office, or any other place to divide me reliably from everybody else, and that for much of the daylight I thus appeared to be among people because I kept putting myself where people came together into even closer-fitting assortments, the viewing areas and showrooms and rotundas and such: I took in the lean-to look of the women, the tongues coming and going in what the men kept thinking of to say—whole families of low knees for me to bark my shins against during the crowded and involving way out. At home afterward, in the one room where my life was packed down, I would keep my nose stuck in the safehold of the phone [160] book, where the names of people suffered reduction to mere episodes of the alphabet and underwent humbling declensions down every column (Lail, Lain, Laine, Lainerd), and he names of streets, of the towns and townships, got docked in crude, heedless abbreviations (the vowels almost always the first to get poked out), and I would run my eyes over the telephone numbers themselves, each sequence of digits another fallible run of the infinite. I thus corrected my feelings for people and assembled myself emotionally into whatever else I had at hand—the obligating arms of the clothes hangers, usually, or the keen-angled understructure, the guardian legginess, of the ironing board. The woman said that in her case, though, it was more a matter of making slow circuits of the classroom where she had to put across the Emporial Sciences, retail theory and methods, to heat-giving and suggestive young women, some of them world cruelties already. There was the cooing of empty stomachs in the hour just before lunch, and the braying and fizzle of loaded stomachs in the low hours of the afternoon. She would recite her notes in a voice barely loyal to anyone octave, a tiny alluvium of slaver hardening at the corners of her mouth on the days she gave the glassy lozenges a slow, warming suck, and she would take lowering notice of how whatever she said succumbed at once to freak spellings and razzing paraphrase in the big, dividered notebooks; and because in midafternoon light the world looked as thorough, as filled in, as it was ever going to get, a better way to set about ruining her eyes was to review how hair had [161] established itself on the arms of the young women, because almost every arm had brought across itself a welcome and diversifying shadow. On one girl it would be a fine, driftless haze afloat above the white of the arm, never seeming to touch down on the skin itself. (An atmosphere, at most, of chestnut brown.) On another, it was as if copper wire, the narrowmost lengthlets of it, had been stuck into the fleshy batter of the thick, freckled forearms. On a third: a field of it–wheat-colored, thin-spun. On a fourth; a differencing, darkish updrift that shaded off as it approached the inner bend of the elbow, then re-emerged at the base of the upper arm as whiskery fringe. On others it was a brassy or rust-colored frizz, or it was as coarse as cornsilk, or it looked fussed on, as if the arm had been slowly stroked with charcoal. But here the woman broke off, or I may well have made an interruption of my own—I think I must have asked whether she was hungry, and she said if I was, and so on one of the lesser streets I parked the car and led her down into a belowstairs eating house I still remembered. Sandwiches were presently lowered in front of us. I watched her remove the festooned toothpick from hers and then play her fingers over the toasted planes before she took a fond first bite. "This is nice of you," she said. I must have looked at her in the way I then had of getting people to speak so they would not seem to be dwelling any longer on my features, because if on the well-set face the mouth and eyes are said to seem frozen in elegant orbit about the til) of the nose, then mine was a face that behold-[162]ers, regarders, could not help trying to round off with greater success, to goad the particulars of it back into the arcs they had wandered away from—the mouth, for instance, having been pursed and pinched suchwise that it seemed resident more on one side of the face than the other—and there were other signs of original strife to be busied with (slapdash eyebrows unbunched, it appeared, from reserves of hair elsewhere on my person; a showing of adult acne, a shrivelly little relevance of it, confined to the declivity of my nose); and so to be polite, the woman thus sank her gaze into her sandwich, and told me, in a voice lowered accordingly, that, one late-childhood summer, she had devoted herself to collecting postage stamps: it was a tongue-involving sideline to early-arriving puberty, and she liked having to lick the pale, gummed hinges instead of the sticky backs of the stamps themselves before entering everything into the hosting album; and once, during some foul weather between her and a brother (the older, thrown-over one, who had already made a habit of fooling the underside of his arm across the top of hers and calling her "pussified"), she reached for the shoe box in which she had let duplicate stamps accumulate—Spanish ones, mostly, of a fading orange—and sent the box slooshing through the lower air so that the stamps showered onto the brother's bare legs with a full, delicate harm. The woman was now touching up the surface of her iced tea with tiny activities, initiatives, of the longspun spoon. I myself was good at getting my touch onto things, although in a way that seemed to mix up the motive atoms inside [163] them, but I was satisfied that for the moment my sandwich, the unbitten-at half of it, was displayable and fit and local to my plate. The woman went on to say that, as a child, she had been bundled off, many an afternoon, to the slope-ceilinged quarters of a bachelor uncle who, when speaking of anybody not immediately present, could not bring himself to use the person's name but instead would say "an acquaintance of mine in ..." and then mention the name of some lapsed homeland, or little-loved rural orchestra, or backset building about to come down; and it was never a riddle, this device of his (not once could the girl have been expected to identify any of the subjects), and no matter how often and aloud he insisted that particularizing persons any further—bringing even a first name down upon anyone of them—would have been indecent, he claimed, much like doing things to people while they slept, the girl accepted all of the uncle's prim and extravagant evasion for what he surely must have intended it to be: a neat, protective trick to space the world out a little further in her favor, to scatter the population so that wherever her hand might at last come down, it would have to be on herself alone. And here I could sense that the woman wanted from my mouth an account of as much as I myself might have ever managed of attachment, so I told her I had once owned a house (a rising, really, of much-fingered, handwrought architecture that amounted to a little family of rooms above garages: a boxlike building with a rattly thorax of downspouts [164] and drainpipes and an unfolded but full-toned fire escape), and I had had for a time a boarder, a student, a high-colored, loose-packed representative of declining girlhood, hung with necklaces and barrettes, a girl of precise but shifting leanings and inclinations; and the afternoon she had come round to ask after the room, I stood in the entranceway, handshaken and asweat, and from what would later be my memory of the girl, I made off with, first, how every pore of her nose seemed to be sheltering within itself a tiny dark seedlet, a grain of something immediately, enormously valuable. And an almost lipless mouth (just a slit, practically), the teeth inside looking wet, watered-it was my life's chore, at that instant, to keep from sending the back of my thumb blotterlike across the line of them (I was later to learn she drank everything cold and through the narrowest of straws). And her hair: it was tea-colored hair she had, long and reachful, an unstopped downcome of it. Tall for a girl, but she managed to stay out of much of her height and put herself across as somebody backward, or behind. I must have told the girl, as best I could, that I of course had a wife, a full-faced, imperishable partner, though for the moment she was gone otherwhere in the marriage, and here the woman, my present companion, my tablemate, whose feet were now parked, in parallel, on the grade of my upper leg, interrupted to say that her husband, too, had been such a liar, and what could I bring up by way of reply other than that a lie is a truth struck through with other, further truth, or that a lie is the present multiplied by the past or that a lie [165] is an outcry of borrowed hope? The woman gave me an allowed look of disgust, her eyes lowered but still popular with me, and on I went with what had now become the girl of my story. For there had been a great, gainful carpet in the room I put the girl into, a matty expanse of coarse, grabby piles, an engrossing affair that took things into itself and held them tight, misered them, and I of course insisted that the girl not bother herself with its upkeep, that I enjoyed weekly access to a prestigious, upstanding vacuum cleaner; but no sooner was the girl out of the house each morning than I would withdraw from my room, where time was unportionable, and loose myself into the ticktock impertinence of the girl's room and get down on my knees, and, going after the carpet first with my fingers, then with a forceps, and finally by unspooling lengths of clear package-sealing tape and pressing them against the tufts in neat rectangles to catch what I might have missed, I brought vast tracts of the carpet to depletion, recovering not simply the girlinesses, the girleries, one would expect (buttons, straight pins, downed jewelry), but flirtier personalia in the form, say, of a stray confetto brought into the world when a page had been wrung with out caution from a spiral-bound notebook, or some pleated paper shells of the chocolates she required, or one of the bargain antihistamines she took to get her naps going, or a trash-bag tie ragged enough to show the kinked line of the wire within (this I would get wound around my finger), or a cough drop enwrapped like a bonbon (I would undo the [166] wings of the wrapper and have to decide whether to suck the drop all down or begin chewing it midway)—I became the following, the public, that these things, these off-faIlings, had come to have; but mostly there was hair, afloat above the uppermost pushings of the fabric of the carpet an almost continuous haziness of loose hairs of all lengths and sources, and I would have to set them out on a fresh sheet of paper and assort them according to the regions of the body they had taken their departure from, and in no time I had nest-like filiations of broken filaments and smaller involvements of the hairs that made me think of hooks, of barbs, of treble clefs, and each pile required a separate envelope, to be filed in a separate shoe box for every sector of the body until, I hoped, the boxes themselves would no longer be enough and I would have raised something semblable, brought up something equal in volume to the comprehensive girl herself. And her wastebasket! For every bit of rubbish, every dreariment she tossed into the thing, I would, in secret, deposit a reciprocal discard of my own, matching a spotted, confessory tissue of hers with a lurid throwaway after my own heart–the cardboard substructions of a fresh parcel of underwear, maybe, or tearings from pantyhose I now and again pressed against a span of my forearm to work onto it the complications of female shading I otherwise made do, choosily, without. I thus built the two of us up together in her trash! One afternoon-it was another of those lifeful, unsam-[167]pled days on which the world humors us each a little differently to keep us nicely on our last legs—I discovered in the wastebasket an inch-deep textbook of hers, a paperback with a celery-colored cover that had come partly unglued, and this dilapidation I paired off, naturally, with a name-your baby guide to whose pages, during my recurrent turnings of them, in bed or at table, I had contributed dried produces of my person, a chemical splendor entirely mine. This coupling sent a sudden spigoty thrill from me that forced an unbuckling and an unzipping and a cleanup with a handkerchief I then ventured responsively into the wastebasket as well. It was in the bathroom afterward that I found a suds-clouded puddle on the unlevel floor of the tub, a little undrained remainder, rimmed with offscum, of the girl's prolonged early-morning soak, and this was as much as I needed to get on my hands; I pressed them flat against the wet porcelain, then flapped them around in the air, and that was when I noticed it—in the amphitheater of the toilet bowl, an orange-yellow tint, or value, to the waters. When the girl arrived home that evening, I told her, of course, that I had discovered fresh, unforeseen trouble within the tank of thelet (a misalignment of the trip lever, a waywardness of the float ball, a ulation of the lift wires, kinks and defects, really, throughout the entire system) and that in fine it was an apparatus now operable only by means of advanced and strenuous equilibrial manipulations that it would be unseemly, inhospitable, of me to pre-[168]sume to burd to burden her with–so that from here on out, following any leak or evacuation she need merely lower the lid and then, before quitting the room, ring the handbell that had been placed on the would seeeverythinghing else. But the bell never rang, not even once, and from my window the next morning I watched the girl carry from the house a little plastic bag distended balloonishly, much like those bags you will remember having seen in the hands of children bearing homeward their solitary carnival-prize goldfish. In fact, I never ran across the likes of the girl again. The man who came to collect her things-not the father, apparently, but an advocate, an upholder-l found to be dull-eared and lax in his speech, and the better part of his face seemed to have already begun making tiny, rotational departures from whatever it was that the eyes, themselves impressively mobile, were just that moment having to take in. (Was there a lamp in the house that was not that night slopping its wattage over everything?) I guess I was waiting for the man to take a laggard, last-minute interest in me, and by now I was pushing everything out into the paired first person–it was, I said, “Our night shot,” and I began including the two of us in whatever it might be doing out, the expected sprinkles and such—but he was no friendship buff, and he paid no heed to my telling him that the only dress of hers I'd ever twidged myself into even part of the way had been the simplest of them all, a large-buttoned wonder of depthless blue, and then only on the principle that one naturally fits whatever one has into whatever somebody else [169] had first, or how else would the world keep getting any taller with people? The man just went about the removal of the girl's things without having to be reminded too noticeably, I guess, of how every dick hangs by a thread. My listener, though, had by this time brought about some becoming slowings of her arm—it was an arm inclinable to langorous diagonals and magicianly swoops through the air above the tabletop—but it no longer was involving itself with her plate, so I suggested we shove off, I made payment for the food, and on our way to the car, and then in the car as I took to following her pointings, the directional tilts of her head, she said that you naturally kept putting more and more of yourself into another person, at first wondering how much she can take, how much of you is accumulable and how much she can hold, and you're letting things out, disbursing yourself, and you've soon got things set up in her, and room is being made for even more of you, and if you bring this off with enough people, even two or three, what you've got is at first a comfort, because you can pass yourself along and move a little more widely through the world and leave it to these others to man your grievances, your disappointments; and what brought this to mind, the woman said, was a term of financial hardship she had contrived for herself a few years back, an unpaid leave of absence from scrupling letter grades onto quiz papers (propped-up As, and upended Bs made to look, rather, like fannies; all As and Bs, no Cs or anything lower, the difference between an A and a B having less to do with the accuracy of whatever facts might [170] have been impounded in provided for an answer than with anything recallable about the way the enrollee ad conformed her body to the confining perpendiculars of her chair and the navel-level writing surface that projected from it, or the way there might one day have been an unignorable blush on the instep of a once-moseying foot, disburdened of its shoe, that had got itself trapped in the grillwork of that cagelike involvement, intended for books, that was welded to, or otherwise schemed into, the underworks of the chair)—this had been a duration, in short, of controlled difficulty when the misexpenditure of even a twenty-dollar bill had set her thrilling, gloating, over everything she would miss out on, and one afternoon she had made an engagement for a haircut, just a trim, and very early in the session the haircutter, a woman ill-defined in the face but otherwise full of conspicuities of emotion, set down the prevailing scissors and pressed the flat of a lukewarm hand against the woman's cheek, held it there for a good minute or longer, while the other hand eventually found its way into a drawer, a shallow treasury of slender specialty scissors, one pair of which the cutter withdrew and began routing deductively through the woman's hair, the other hand staying put on the cheek, longer and longer, and the woman went home and for afterward the bathtub was now a more likely destination than any of the upright furniture, and it got easier to fill the tub with further clarifying volumes than to clear space on the difficult heights of the sofa, and she was hardly claiming to have become a cleaner [171] person in result—she in fact would often discover, voyaging about her body, a browned, fractionary detail of a larger crepe of toilet tissue that must have got itself stuck in some assy crevice and was impossible to get plucked out of the revolving suds—she was saying only that she spent more and more of her time thus immersed, ill off in water, and the haircutter had surely had a hand in it, the woman was doing some of the cutter's life now, coming into some of its wrong, because you sometimes have to look to somebody else's life to get dimensions set back even part of the way around your own, and it should not have to be any less your own life when it comes from somebody else, and you could surely fudge a society out of anyone available person and get this person doubling for the many, so that in the little run of things perpetuable from one person to the next, every loose moment stood to become a complete, active finality. But by now this was a new day, with only an hour or so off it already, and the place the woman had made me bring us to, the man's place, with a promise that the man was elsewhere—this was on a little offshoot of a street, a stewy efficiency apartment the color had long ago gone out of; and when, once in bed, still clothed, I found among the sheets and blankets a spoiling pair of the man's underpants, one of the leg openings of which was puckered into an avid, sloppy mouth, I held myself accountable for redisposing the fabric until I got a befitting featurelessness back onto it; but all the while, I am sure I had to make myself go over again in my mind that if the body is the porter of as many organs of affec-[172]tion as there might one day turn out to be, then the idea was to let the thing carry you to where you would otherwise never have any reason to arrive, because I listened for the unmelodiof the the thede ofthe woman's zipper, and then the woman made me put myself out of my own clothes, the attritional corduroys and overshirt, and got herself up at last onto the topic not of the man whose apartment it was (because his story was scarcely the story of how the boy who decides he is half a girl no sooner starts to worry about where the other half might than he gets careless with where he rests his eyes, or what he gives even the feeblest of fingerholds to, and anything, even a crumbly triangle of pie offered on a saucer instead of on a pie plate proper, comes in easy, ready, wronging answer), but of her husband, and how, no more than a couple of months into the marriage, he had begun snugging away in his undershorts a little source of chanc, reliable frictions to nudge him onward through the workday—anything company-keeping that could be counted on not to slide out of the elasticky leg holes: a half-dollar packet of chocolate tittles, maybe, that was barely noticeable in the baggy surround of the widecut trousers so popular at the time. For by now the woman had at last brought what is usually called the other mouth to within only inches of my lips, but it is not a mouth, obviously, although I let myself go along with the goodwill behind the comparison, the way I will remain loyal to anything deliberately and faithfully misunderstood, and I fussed my tongue against the vital trifles hung inside of her, as much of the curtailed finery as I could find, [173] and gave the whole insimplicity of it a slow-circling, examinational lick, until I was taking a sudden tepid downwash on the tongue. It was a familiar, latrine in dribble that must have tasted, no doubt, like trouble just starting out.p [1] Editor's Note: Original pagination from first edition print denoted with bracketed page numbers. (shrink)

The structure Dense /nwd and gaps in analytic quotients of equation image have been studied in the literature 2, 3, 1. We prove that the structures Dense /nwd and equation image have gaps of type equation image, and there are no -gaps for equation image, where equation image is the additivity number of the meager ideal. We also prove the existence of -gaps in these structures. Finally we characterize the cofinality of the meager ideal equation image using families (...) of sets in Dense /nwd. (shrink)

This paper seeks to uncover and rationally reconstruct four theoretical prescriptions that H. L. A. Hart urged philosophers to observe and follow when investigating and theorizing about the nature of law. The four prescriptions may appear meager and insignificant when each is seen in isolation, but together as an inter-connected set they have substantial implications. In effect, they constitute a central part of Hart's campaign to put philosophical investigations about the nature of law onto a path to a genuine (...) research program. The paper takes note of certain prevalent and robust trends in contemporary legal philosophy that detract its practitioners from the four prescriptions, and that have them revert to the some older modes of thinking from which Hart sought a decisive break. A number of contemporary legal philosophers' views and commitments are taken up and assessed, and in particular those of John Gardner and Leslie Green. ‘Yet the answer is a prosaic one: don’t ask what time is but how the word ‘time’ is being used’.Friedrich Waismann.I miss the future.Jaron Lanier. (shrink)

Our only channel of information about the world is the impact of external forces on our sensory surfaces. So says science itself. There is no clairvoyance. How, then, can we have parlayed this meager sensory input into a full-blown scientific theory of the world? This is itself a scientific question. The pursuit of it, with free use of scientific theory, is what I call naturalized epistemology. The Roots of Reference falls within that domain. Its more specific concern, within that (...) domain, is reference to concrete and abstract objects: what such reference consists in, and how we achieve it. (shrink)

This paper seeks to uncover and rationally reconstruct four theoretical prescriptions that H. L. A. Hart urged philosophers to observe and follow when investigating and theorizing about the nature of law. The four prescriptions may appear meager and insignificant when each is seen in isolation, but together as an inter-connected set they have substantial implications. In effect, they constitute a central part of Hart’s campaign to put philosophical investigations about the nature of law onto a path to a genuine (...) research program. The paper takes note of certain prevalent and robust trends in contemporary legal philosophy that detract its practitioners from the four prescriptions, and that have them revert to the some older modes of thinking from which Hart sought a decisive break. A number of contemporary legal philosophers’ views and commitments are taken up and assessed, and in particular those of John Gardner and Leslie Green. (shrink)