It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $\mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $\mathbb {R}^3$ has uncountable chromatic number.

If [Formula: see text] is a closed Noetherian graph on a [Formula: see text]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[Formula: see text][Formula: see text][Formula: see text]DC that [Formula: see text] is countably chromatic and there is no Vitali set.

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. If [math] is a closed Noetherian graph on a [math]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[math][math][math]DC that [math] is countably chromatic and there is no Vitali set.

We prove a partition theorem for analytic sets, namely, if X is an analytic set in a Polish space and [X]n = K0 ∪ K1 with K0 open in the relative topology, and the partition satisfies a finitary condition, then either there is a perfect K0-homogeneous subset or X is a countable union of K1-homogeneous subsets. We also prove a partition theorem for analytic sets in the three-dimensional case. Finally, we give some applications of the theorems.

Let ⪯R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order . We show that ⪯R has fairly high complexity with respect to Borel reducibility , although its exact classification remains open.

Let X be an infinite internal set in an ω1-saturated nonstandard universe. Then for any coloring of [X] k , such that the equivalence E of having the same color is countably determined and there is no infinite internal subset of [X] k with all its elements of different colors (i.e., E is condensating on X), there exists an infinite internal set Z⊆X such that all the sets in [Z] k have the same color. This Ramsey-type result is obtained (...) as a consequence of a more general one, asserting the existence of infinite internal Q-homogeneous sets for certain Q ⊆ [[X] k ] m , with arbitrary standard k≥ 1, m≥ 2. In the course of the proof certain minimal condensating countably determined sets will be described. (shrink)

A strong coloring on a cardinal$\kappa $is a function$f:[\kappa ]^2\to \kappa $such that for every$A\subseteq \kappa $of full size$\kappa $, every color$\unicode{x3b3} <\kappa $is attained by$f\restriction [A]^2$. The symbol$$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$asserts the existence of a strong coloring on$\kappa $.We introduce the symbol$$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$which asserts the existence of a coloring$f:[\kappa ]^2\to \kappa $which isstrong over a partition$p:[\kappa ]^2\to \theta $. A coloringfis strong overpif for every$A\in [\kappa ]^{\kappa }$there is$i<\theta $so that for every (...) color$\unicode{x3b3} <\kappa $is attained by$f\restriction ([A]^2\cap p^{-1}(i))$.We prove that whenever$\kappa \nrightarrow [\kappa ]^2_{\kappa }$holds, also$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$holds for an arbitrary finite partitionp. Similarly, arbitrary finitep-s can be added to stronger symbols which hold in any model of ZFC. If$\kappa ^{\theta }=\kappa $, then$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$and stronger symbols, like$\operatorname {Pr}_1(\kappa,\kappa,\kappa,\chi )_p$or$\operatorname {Pr}_0(\kappa,\kappa,\kappa,\aleph _0)_p$, also hold for an arbitrary partitionpto$\theta $parts.The symbols$$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p \end{gather*} $$hold for an arbitrary countable partitionpunder the Continuum Hypothesis and are independent over ZFC$+ \neg $CH. (shrink)

We say that a coloring ${c: [\kappa]^n\to 2}$ is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some ${\aleph_1}$ -preserving extension of the set-theoretic universe. Given an arbitrary coloring ${c:[\kappa]^n\to 2}$ , we define a forcing notion ${\mathbb P_c}$ that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that ${\mathbb P_c}$ is c.c.c. if (...) and only if c is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding ${\aleph_1}$ Cohen reals to any model of set theory introduces a coloring ${c:[\aleph_1]^2 \to 2}$ which is potentially continuous but not continuous. ${\aleph_1}$ has no uncountable c-homogeneous subset in the Cohen extension, but such a set can be introduced by forcing. The potential continuity of c can be destroyed by some c.c.c. forcing. (shrink)

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in (...) work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica26(2) (2006) 183–205], the Ramsey theory of H3 had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin.2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math.185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph G, there is a finite number T(G) such that for any coloring of all copies of G in H3 into finitely many colors, there is a subgraph of H3 which is again universal homogeneous triangle-free in which the coloring takes no more than T(G) colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code H3 and the development of their Ramsey theory. (shrink)

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically saturated (...) models of Peano Arithmetic such that Aut(M1)≅ Aut(M2).ThenSSy(M1) = SSy(M2).We show that ifMis a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1,M2be countable arithmetically saturated models of Peano Arithmetic such thatAut(M1) ≅ Aut(M2).Then for every n<ωHere RT2nis Infinite Ramsey's Theorem stating that every 2-coloring of [ω]nhas an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic. (shrink)

Let G be a group, and let X be an infinite transitive G-space. A free ultrafilter U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U, or there is a member of U which meets each cell of P in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a G-selective ultrafilter on X. We describe all G-spaces X such (...) that each free ultrafilter on X is G-selective, and we prove that a free ultrafilter U on ω is selective if and only if U is G-selective with respect to the action of any countable group G of permutations of ω. A free ultrafilter U on X is called G-Ramsey if, for any G-invariant coloring χ:[X]2→{0,1}, there is U∈U such that [U]2 is χ-monochromatic. We show that each G-Ramsey ultrafilter on X is G-selective. Additional theorems give a lot of examples of ultrafilters on Z that are Z-selective but not Z-Ramsey. (shrink)

We obtain an improvement of some coloring theorems from Eisworth (Fund Math 202:97–123, 2009; Ann Pure Appl Logic 161(10):1216–1243, 2010), Eisworth and Shelah (J Symb Logic 74(4):1287–1309, 2009) for the case where the singular cardinal in question has countable cofinality. As a corollary, we obtain an “idealized” version of the combinatorial principle Pr1(μ +, μ +, μ +, cf(μ)) that maximizes the indecomposability of the associated ideal.

We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the least number of colors (...) which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs. (shrink)

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. (...) Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (shrink)

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a (...) Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA.Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1.We refute various reasonable generalizations of these results to hypergraphs. (shrink)

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman’s Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some (...) nontrivial countable field of sets. This amounts to a positive result that addresses Carlson’s question in some way. (shrink)

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa (...) > \mathfrak{d}}$ is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals. (shrink)

We prove that colouring of pairs from 2 with strong properties exists. The easiest to state problem it solves is: there are two topological spaces with cellularity 1 whose product has cellularity 2; equivalently, we can speak of cellularity of Boolean algebras or of Boolean algebras satisfying the 2-c.c. whose product fails the 2-c.c. We also deal more with guessing of clubs.

The paper argues that colouring is a conventional ingredient of literal meaning characterized by a considerable degree of semantic under-determination and a high degree of context-sensitivity. The positive, though tentative, suggestion made in the paper is that whereas in the case of words such as "but" and "damn" we are dealing with words lacking in specificity, in the case of pejoratives in general, and racist jargon in particular, we are dealing with words that express concepts that purport to describe the (...) world as being in a certain way. The circumstance that in certain contexts of utterance colouring can be cancelled out, does not show that it forms a detachable part of a word's literal meaning. It only shows that to account for the interplay between context, literal meaning and assertoric content is much trickier than meets the eye. (shrink)

© The Authors 2017. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: [email protected] Colour is a welcome work in history and philosophy of science.1 The opening chapters offer a fresh take on the history of perceptual theory and a broad overview of contemporary philosophy of colour. This is followed by the central fourth chapter, which introduces readers to a cluster of empirical data that to this point have not received explicit (...) treatment in philosophy. The core message is that instead of thinking of colour vision as filling in or painting the experienced world of primary qualities, we should think of colour vision as helping us to experience primary qualities. The colour visual system does not colour in an experiential world of primary qualities, it uses colour for the experience of those qualities. This colouring... (shrink)

In this essay I consider Kaplan’s challenge to Frege’s so-called dictum: “Logic (and perhaps even truth) is immune to epithetical color”. I show that if it is to challenge anything, it rather challenges the view (attributable to Frege) that logic is immune to pejorative colour. This granted, I show that Kaplan’s inference-based challenge can be set even assuming that the pejorative doesn’t make any non-trivial truth-conditional (descriptive) contribution. This goes against the general tendency to consider the truth-conditionally inert logically irrelevant. (...) But I take it that Kaplan is right and take his examples to show that truth-conditional inertness need not entail inferential inertness. I end up assessing the Kaplan-Frege “debate” as giving edge to the former to the extent that clarity is achieved through Kaplanian inferences on what should be considered part of the explanandum. (shrink)

This article examines the early modern household's importance for producing experimental knowledge through an examination of the Halifax household of Margery and Henry Power. While Henry Power has been studied as a natural philosopher within the male-dominated intellectual circles of Cambridge and London, the epistemic labour of his wife, Margery Power, has hitherto been overlooked. From the 1650s, this couple worked in tandem to enhance their understanding of the vegetable world through various paper technologies, from books, paper slips and recipe (...) notebooks to Margery's drawing album and Henry's published Experimental Philosophy. Focusing on Margery's practice of hand-colouring flower books, her copied and original drawings of flowers and her experimental production of ink, we argue that Margery's sensibility towards colour was crucial to Henry's microscopic observations of plants. Even if Margery's sophisticated knowledge of plants never left the household, we argue that her contribution was nevertheless crucial to the observation and representation of plants within the community of experimental philosophy. In this way, our article highlights the importance of female artists within the history of scientific observation, the use of books and paperwork in the botanical disciplines, and the relationship between household science and experimental philosophy. (shrink)

Many scholars claim that Frege's theory of colouring is committed to a radical form of subjectivism or emotivism. Some other scholars claim that Frege's concept of colouring is a precursor to Grice's notion of conventional implicature. I argue that both of these claims are mistaken. Finally, I propose a taxonomy of Fregean colourings: for Frege, there are purely aesthetic colourings, communicative colourings or hints, non-communicative colourings.

Colour plays a fundamental role in the philosophical treatments of painting. Colour while it is an essential part of the work of art cannot be divorced from the account of painting within which it is articulated. This paper begins with a discussion of the role of colour in Schelling's conception of art. Nonetheless its primary concern is to develop a critical encounter with Jean-François Lyotard's analysis of the Dutch painter Karel Appel. The limits of Lyotard's writings on painting, which this (...) paper will attribute in part to Lyotard's ‘empiricism’, becomes most apparent in his treatment of colour. (shrink)

A common view in both philosophy and the vision sciences is that, in human vision, wavelength information is primarily ‘for’ colouring: for seeing surfaces and various media as having colours. In this article we examine this assumption of ‘colour-for-colouring’. To motivate the need for an alternative theory, we begin with three major puzzles from neurophysiology, puzzles that are not explained by the standard theory. We then ask about the role of wavelength information in vision writ large. How might wavelength information (...) be used by any monochromat or dichromat and, finally, by a trichromatic primate with object vision? We suggest that there is no single ‘advantage’ to trichromaticity but a multiplicity, only one of which is the ability to see surfaces and so on as having categorical colours. Instead, the human trichromatic retina exemplifies a scheme for a general encoding of wavelength information given the constraints imposed by high spatial resolution object vision. Chromatic vision, like its partner, luminance vision, is primarily for seeing. Viewed this way, the ‘puzzles’ presented at the outset make perfect sense. 1 Reframing the Problem1.1 Introduction1.2 Three puzzles1.2.1 Why is trichromatic vision an anomaly in diurnal mammals?1.2.2 Why does the colour system occupy such a large and central part in human vision?1.2.3 Why are the blue cones so rare?1.3 Recasting the question2 The Costs and Benefits of Spectral Vision2.1 Spectral information and object vision2.2 Encoding the spectral dimension of light2.2.1 ‘General’ versus ‘specific’ encoding2.2.2 Spectral encoding and the monochromat2.2.3 Spectral encoding and the dichromat2.2.4 Spectral encoding and the human trichromat2.3 Three puzzles revisited2.3.1 Why is trichromatic vision an anomaly in diurnal mammals?2.3.2 Why does the colour system occupy such a large and central part in human vision?2.3.3 Why are the blue cones so rare3 Conclusion. (shrink)

Colour shapes our world in profound, if sometimes subtle, ways. It helps us to classify, form opinions, and make aesthetic and emotional judgements. Colour operates in every culture as a symbol, a metaphor, and as part of an aesthetic system. Yet archaeologists have traditionally subordinated the study of colour to the form and material value of the objects they find and thereby overlook its impact on conceptual systems throughout human history.This book explores the means by which colour-based cultural understandings are (...) formed, and how they are used to sustain or alter social relations. From colour systems in the Mesolithic, to Mesoamerican symbolism and the use of colour in Roman Pompeii, this book paints a new picture of the past. Through their close observation of monuments and material culture, authors uncover the subtle role colour has played in the construction of past social identities and the expression of ancient beliefs. Providing an original contribution to our understanding of past worlds of meaning, this book will be essential reading for archaeologists, anthropologists and historians, as well as anyone with an interest in material culture, art and aesthetics. (shrink)

Recent evidence in cognitive neuroscience indicates that the visual system is influenced by the outcome of an early appraisal mechanism that automatically evaluates what is seen as being harmful or beneficial for the organism. This indicates that there could be valence in perception. But what could it mean for one to see something positively or negatively? Although most theories of emotions accept that valence involves being related to values, the nature of this relation remains highly debated. Some explain valence in (...) terms of evaluative content, others in terms of evaluative attitude. Here I shall argue that an account of affective perception in terms of attitude has more chance of succeeding. To do so, I will first highlight the difficulties that a content-based approach faces, considering the many forms that it might take. I will conclude that seeing the world positively or negatively involves more than a positive or negative content; it involves a distinctive attitude, but which one? Should it be conceived of in imperative or evaluative terms? And what makes this attitude distinct from a proper emotion? (shrink)

Studies of linguistic synaesthesias in English have shown a range of fine-grained language mechanisms governing the associations between colours on the one hand, and graphemes, phonemes and words on the other. However, virtually nothing is known about how synaesthetic colouring might operate in non-alphabetic systems. The current study shows how synaesthetic speakers of Mandarin Chinese come to colour the logographic units of their language. Both native and non-native Chinese speakers experienced synaesthetic colours for characters, and for words spelled in the (...) Chinese spelling systems of Pinyin and Bopomo. We assessed the influences of lexical tone and Pinyin/Bopomo spelling and showed that synaesthetic colours are assigned to Chinese words in a non-random fashion. Our data show that Chinese-speaking synaesthetes with very different native languages can exhibit both differences and similarities in the ways in which they come to colour their Chinese words. (shrink)

Summary Colourful plant images are often taken as the icon of natural history illustration. However, so far, little attention has been paid to the question of how this beautiful colouring was achieved. At a case study of the eighteenth-century Nuremberg doctor and botanist, Christoph Jacob Trew, the process of how illustrations were hand-coloured, who was involved in this work, and how the colouring was supervised and evaluated is reconstructed, mostly based on Trew's correspondence with the engraver and publisher of his (...) books, Johann Jacob Haid in Augsburg. Furthermore, the question of standardizing colours, their uses and their recipes is discussed at two examples of the same time period: the colour charts of the Bauer brothers, arguably the most renowned botanical draughtsmen of the period, and the colour tables by the Regensburg naturalist, Jacob Christian Schaeffer. Hand-colouring botanical illustrations, it is argued, was far from a straightforward task but confronted botanists and their employees with a plethora of practical and methodological problems, to which different solutions were developed in the course of time. Analysing these problems and solutions reveals some new and interesting aspects of the practices of eighteenth-century botany and of the production of scientific illustrations in general. (shrink)

We observe a number of connections between recent developments in the study of constraint satisfaction problems, irredundant axiomatisation and the study of topological quasivarieties. Several restricted forms of a conjecture of Clark, Davey, Jackson and Pitkethly are solved: for example we show that if, for a finite relational structure M, the class of M-colourable structures has no finite axiomatisation in first order logic, then there is no set (even infinite) of first order sentences characterising the continuously M-colourable structures amongst compact (...) totally disconnected relational structures. We also refute a rather old conjecture of Gorbunov by presenting a finite structure with an infinite irredundant quasi-identity basis. (shrink)

At a very fundamental level an individual (or a computer) can process only a finite amount of information in a finite time. We can therefore model the possibilities facing such an observer by a tree with only finitely many arcs leaving each node. There is a natural field of events associated with this tree, and we show that any finitely additive probability measure on this field will also be countably additive. Hence when considering the foundations of Bayesian statistics we may (...) as well assume countable additivity over a σ-field of events. (shrink)

I criticise a paper by Peter Forrest in which he argues that a principle of unrestricted countable fusion has paradoxical consequences. I argue that the paradoxical consequences that he exhibits may be due to his Whiteheadean assumptions about the nature of spacetime rather than to the principle of unrestricted countable fusion.

This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory -- in particular, Savage's theory of subjective expected utility and personal probability. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealised assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational (...) value of sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage's theory and explore some technical properties in relation to other axioms of the system. (shrink)

To what extent are countability distinctions subject to systematic semantic variation? Could there be a language with no countability distinctions—in particular, one where all nouns are count? I argue that the answer is no: even in a language where all NPs have the core morphosyntactic properties of English count NPs, such as combining with numerals directly and showing singular/plural contrasts, countability distinctions still emerge on close inspection. I divide these distinctions into those related to sums and those related to parts. (...) In the Sahaptian language Nez Perce, evidence can be found for both types of distinction, in spite of the absence of anything like a traditional mass–count division in noun morphosyntax. I propose an extension of the Nez Perce analysis to Yudja, analyzed by Lima as lacking any countability distinctions. More generally, I suggest that at least one countability distinction may be universal and that languages without any countability distinctions may be unlearnable. (shrink)

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.

All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $. Our main result shows that, given a cardinal $\kappa $, the following conditions are equivalent: • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $. • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$. In fact, in the case $\mathfrak {d}\leq \kappa \leq (...) \mathfrak {c}$, every countable dense subspace of $2^\kappa $ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\kappa $ has pseudocharacter at most $\kappa $ in any compactification. This will allow us to calculate $\mathsf {Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space X. (shrink)

We use methods of reverse mathematics to analyze the proof theoretic strength of a theorem involving the notion of coloring number. Classically, the coloring number of a graph $G=$ is the least cardinal $\kappa$ such that there is a well-ordering of $V$ for which below any vertex in $V$ there are fewer than $\kappa$ many vertices connected to it by $E$. We will study a theorem due to Komjáth and Milner, stating that if a graph is the union (...) of $n$ forests, then the coloring number of the graph is at most $2n$. We focus on the case when $n=1$. (shrink)

It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.

Two countable filters on ω are incompatible if they have no common infinite pseudointersection. Letting α(P f ) denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on ω, we prove the consistency of t $.

The idea that an utterance of a basic (nondeviant) declarative sentence expresses a single true-or-false proposition has dominated philosophical discussions of meaning in this century. Refinements aside, this idea is less of a substantive theses than it is a background assumption against which particular theories of meaning are evaluated. But there are phenomena (noted by Frege, Strawson, and Grice) that threaten at least the completeness of classical theories of meaning, which associate with an utterance of a simple sentence a truth-condition, (...) a Russellian proposition, or a Fregean thought. And it may well be the case that a framework within which utterances express sequences of propositions provides much of what is needed to account for the relevant phenomena, a better overall picture of the way language works, and an enticingly uniform perspective on a variety of semantic problems. I do not myself take to theories that multiply propositions by appealing to propositions “presupposed” or to pairs of Fregean and Russellian propositions, or theories that show no respect for a distinction between semantics and pragmatics— where the former is the study of propositions whose general form and character is determined by word meaning and syntax—or for theories that blithely abandon general principles of composition and semantic innocence. I would like to sketch a package based on four interconnected ideas: (i) the meaning of an individual word is a sequence of instructions for generating a sequence of propositions (in conjunction with compositional instructions (syntax) and elements of context); (ii) utterances themselves are not bearers of truth or falsity; (iii) judgements of truth, falsity, commitment, and conflict are shaped, in part, by the weights attached to individual 1 propositions that occur in sequences expressed by utterances, weights that may be set (and reset) by contextual considerations; (iv) Fregean senses are superfluous; propositions might as well be Russellian (Mont Blanc and all its snow fields will do as well as any mode of presentation).. (shrink)