We study the partition relation $X@>{\rm w}>>[Y]_{p}^{2}$ that is a weakening of the usual partition relation $X\rightarrow [Y]_{p}^{2}$ . Our main result asserts that if κ is an uncountable strongly compact cardinal and $\germ{d}_{\kappa}\leq \lambda ^{<\kappa}$ , then $I_{\kappa,\lambda}^{+}@>{\rm w}>>[I_{\kappa,\lambda}^{+}]_{\lambda <\kappa}^{2}$ does not hold.

Working in the theory “ZF + There is a nontrivial elementary embedding j: V → V ”, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which (...) every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation μ → [μ]ωℵ2. We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j : V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice. (shrink)

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on Pκ that satisfy the square brackets partition relation equation image, where μ is a cardinal ≥2.

In this paper, we investigate the extent to which techniques used in [10], [2], and [3]—developed to prove coloring theorems at successors of singular cardinals of uncountable cofinality—can be extended to cover the countable cofinality case.

We study ultrafilters onω2produced by forcing with the quotient of${\cal P}$(ω2) by the Fubini square of the Fréchet filter onω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is (...) not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters. (shrink)

The notion of an entangled linear order gives a useful method to construct counter examples for such problems as the productivity of chain conditions, the square bracket partition relation and the existence of a large size monotonic subfunction. In particular, if there exists and ℵ1-entangled linear order then some consequences ofMA ℵ 1 or of wOCA fail. So, in which model ofZFC does an ℵ1-entangled linear order exist? Todorcevic [6] has shown if cf2ℵ 0=ω1 then there is an (...) ℵ1-entangled linear order of size ℵ1. And it is widely known that adding κ many Cohen reals adds such a linear order of size κ. In this note it is shown that adding asingle Cohen real is sufficient to get a maximal size ℵ1-entangled linear order. (shrink)

We prove that if ${\mu ^ + } \to \left[ {{\mu ^ + }} \right]_\mu ^2 + $ holds for a singular cardinal μ, then any collection of fewer than cf(μ) stationary subsets of μ⁺ must reflect simultaneously.

This is a transcript of Milne's manuscript notes for a talk which he gave to fellow members of the Cambridge University Natural Science Club in his rooms at Trinity College, Cambridge, on February 6, 1922. The notes are deposited in the Bodleian Library, University of Oxford, Special Collections and Western Manuscripts. The background and essential points of Milne's talk are analysed in the article preceding this one. As far as is known, the text has not hitherto been published. Milne's handwriting (...) is difficult to read, and square brackets [...] denote the addition of a word for clarity of meaning. Meg Weston Smith and Simon Rebsdorf have ventured to fill in some chapter titles just in order to help with the separation of the varying content, hence we have made up all the headlines. All italics correspond to underlined words in the original. The Bergson citations and the Eddington passage at the end of the original text have not been transcribed. (shrink)

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well (...) ordered partition of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)

Building upon earlier work of Donna Carr, Don Pelletier, Chris Johnson, Shu-Guo Zhang and others, we show that a normal ideal J on Pκ is strongly normal if and only if J+→< 2 for every μ < κ, and we describe the least normal ideal J on Pκ such that J+ →< 2.

Using previous work of Baumgartner, Shelah and others, we describe, for each infinite cardinal θκ, the smallest κ-normal ideal J on Pκ such that.

Several canonical partition theorems are obtained, including a simultaneous generalization of Neumer's lemma and the Erdos-Rado theorem. The canonical partition relation for infinite cardinals is completely determined, answering a question of Erdos and Rado. Counterexamples are given showing that in several ways these results cannot be improved.

We consider natural strengthenings of H. Friedman's Borel diagonalization propositions and characterize their consistency strengths in terms of the n -subtle cardinals. After providing a systematic survey of regressive partition relations and their use in recent independence results, we characterize n -subtlety in terms of such relations requiring only a finite homogeneous set, and then apply this characterization to extend previous arguments to handle the new Borel diagonalization propositions.

We consider colorings of the pairs of a family \ of topological type \, for \; and we find a homogeneous family \ for each coloring. As a consequence, we complete our study of the partition relation \^2_{l,m}}\) identifying \ as the smallest ordinal space \^2_{l,4}}\).

The goal of this short note is to interest set theorists in the order type ω*ω1, and to encourage them to work on the question of whether or not the Continuum Hypothesis decides the partition relation τ→2, for τ=ω*ω1 and for τ=ω1ω+2.

It is shown that for any cardinal $\kappa, \dbinom{(2^{ , and if κ is weakly compact $\dbinom{\kappa^+}{\kappa} \rightarrow \dbinom{\kappa}{\kappa}_{.

We investigate the effect of adding $\omega _2$ Cohen reals on graphs on $\omega _2$, in particular we show that $\omega _2 \to (\omega _2, \omega : \omega )^2$ holds after forcing with $\mathsf {Add}(\omega, \omega _2)$ in a model of $\mathsf {CH}$. We also prove that this result is in a certain sense optimal as $\mathsf {Add}(\omega, \omega _2)$ forces that $\omega _2 \not \to (\omega _2, \omega : \omega _1)^2$.

It is a theorem of Rowbottom [12] that ifκis measurable andIis a normal prime ideal onκ, then for eachλ<κ,In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.The set theoretical terminology is standard and background results on the theory of ideals may be found in [5] and [8]. Throughoutκwill denote an uncountable regular cardinal, andIa proper, nonprincipal,κ-complete ideal onκ.NSκis the ideal of nonstationary subsets (...) ofκ, andIκ= {X⊆κ∣∣X∣<κ}. IfA∈I+ −I), then anI-partitionofAis a maximal collectionW⊆,P ∩I+so thatX∩ Y ∈IwheneverX, Y∈W, X≠Y. TheI-partitionWis said to be disjoint if distinct members ofWare disjoint, and in this case, fordenotes the unique member ofWcontainingξ. A sequence 〈Wα∣α<η} ofI-partitions ofAis said to be decreasing if wheneverα<β<ηandX∈Wβthere is aY∈Wαsuch thatX⊆Y.. (shrink)

Working in ZFC, we show that for any infinite cardinal κ and ordinal $\gamma the polarized partition relation $\[\begin{pmatrix} (2^{ → $\[\begin{pmatrix}(2^{ holds. Our proof of this relation involves the use of elementary substructures of set models of large fragments of ZFC.

We prove the independence of a strong partition relation on ℵ ω , answering a question of Erdos and Hajnal. We then give an almost complete answer to the free subset problem.

We study the subtlety of a cardinal κ and of. We show that, under a certain large cardinal assumption, it is consistent that is subtle for some but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of can be characterized by a certain partition relation on. We also study the property of faintness of κ, and the subtlety of with the strong (...) inclusion. (shrink)

We show that if κ is a weakly compact cardinal, then $\left( \matrix \kappa ^{+} \\ \kappa\endmatrix \right)\rightarrow \left(\left( \matrix \alpha \\ \kappa \endmatrix \right)_{m}\left( \matrix \kappa ^{n} \\ \kappa \endmatrix \right)_{\mu}\right)^{1,1}$ for any ordinals α < κ⁺ and µ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partition $\kappa \times \kappa ^{+}=\underset i<m\to{\bigcup }K_{i}\cup \underset j<\mu \to{\bigcup }L_{j}$ of κ × κ⁺ into m + µ pieces (...) either there are A ∈ [κ]κ, B ∈ [κ⁺]α, and i < m with A × B ⊆ Ki or there are C ∈ [κ]κ, D ∈ [κ⁺]α, and j < μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC. (shrink)

A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n ≤ ω, there is a natural number m (n) such that, the following holds. Assume GCH: If $\lambda are regular, there is a cofinality preserving forcing extension in which 2 λ = μ and, for all $\sigma such that η +m(n)-1) ≤ μ, ((η +m(n)-1) ) σ ) → ((κ) σ ) η (1)n . This generalizes results of [3], Section 1, and (...) the forcing is a "many cardinals" version of the forcing there. (shrink)

Kant’s essay Idea for a universal history with a cosmopolitan purpose differs in deep ways from standard Enlightenment views of human history. Although he agrees with many contemporaries that unsocial sociability can drive human progress, he argues that we know too little about the trends of history to offer either metaphysical defence or empirical vindication of the perfectibility of man or the inevitability of progress. However, as freely acting beings we can contribute to a better future, so have grounds for (...) committing ourselves to human progress even if we cannot guarantee or know that it will continue indefinitely. As Kant sees, it, human progress is better seen as a practical assumption—an Idea of Reason—than as a theoretical claim.Keywords: Cosmopolitanism; Historical trends; Ideas of Reason; Immanuel Kant; Philosophy of history; Optimism; Pessimism; Progress; Unsocial sociability. (shrink)

En el ámbito de la lógica matemática existe un problema sobre la relación lógica entre dos versiones débiles del Axioma de elección (AE) que no se ha podido resolver desde el año 2000 (aproximadamente). Tales versiones están relacionadas con ultrafiltros no principales y con Propiedades Ramsey (Bernstein, Polarizada, Subretículo, Ramsey, Ordinales flotantes, etc). La primera versión débil del AE es la siguiente (A): “Existen ultrafiltros no principales sobre el conjunto de los números naturales (ℕ)”. Y la segunda versión débil del (...) AE es la siguiente (B): “Existen ultraflitros sobre ℕ”. Se sabe que A implica B, pero se desconoce si B implica A. Di Prisco y Henle conjeturan en los artículos ([1], [2]) que esto no ocurre, es decir, conjeturan que B no implica A, en otras palabras, conjeturan que A es más fuerte estrictamente que B, que A es independiente de B, pero esto no se ha podido demostrar todavía aunque se ha intentado hacer desde hace aproximadamente 21 años. Una descripción detallada de este problema abierto puede encontrarse en esta ponencia (dictada en el marco del Día Mundial de la Lógica 14-01-2022) y en el artículo [3]. [1] C. Di Prisco y H. Henle. “Doughnuts, Floating Ordinals, Square Brackets, and Ultraflitters”. Journal of Symbolic Logic 65 (2000) 462-473. [2] C. Di Prisco y H. Henle. “Partitions of the reals and choice”. En “Models, algebras and proofs”. X. Caicedo y C.M. Montenegro. Eds. Lecture Notes in Pure and Appl. Math, 203, Marcel Dekker, 1999. [3] F. Galindo. “Tópicos de ultrafiltros”. Divulgaciones Matemáticas. Vol. 21, No 1-2, 2020. (shrink)

Human communication is multi-modal. It is an empirical fact that many of our acts of communication exploit a variety of means to make our communicative intentions recognisable. Scholars readily distinguish between verbal and non-verbal means of communication, and very often they deal with them separately. So it is that a great number of semanticists and pragmaticists give verbal communication preferential treatment. The non-verbal aspects of an act of communication are treated as if they were not underlain by communicative intentions. They (...) are “relegated” to mere aspects of the context. However, several schools of thoughts have a different take on the issue. Thus psychologists or semioticians of gesture have shown how intricately gestures and speech are related in utterances. And, in a different area of the theoretical landscape, so-called “Relevance Theorists” have made the same point. Thus, Robyn Carston writes that “the domain of pragmatics is a natural class of environmental phenomena, that of ostensive stimuli; verbal utterances are the central case, but not the only one, and they themselves are frequently accompanied by other ostensive gestures of the face, hands, voice etc, all of which have to be interpreted together if one is to correctly infer what is being communicated”. This position rests on the assumption that there is a single “pragmatic system” or module at work in the interpretation of “ostensive stimuli”. When it comes to interpreting verbal stimuli, the same mechanisms and resources are used as when it comes to processing non-verbal ones. If there is no distinct “linguistic pragmatic system”, then the scholar who studies communication should not favour the verbal at the expense of the non-verbal. In this paper, I want to make a contribution to the study of multi-modal messages by considering a type of utterances that mix the verbal with the non-verbal in such a way that a piece of non-linguistic communication seems to stand in for a linguistic constituent which remains unrealised. Here is a real-life example : I didn't see the [IMITATION OF FRIGHTENING GRUMPINESS] woman today; will she be back this week? The square-bracketed string in small capitals is meant to capture the facial expressions and gestures performed in the conversational setting. What is intriguing here is that this instance of ostensive mimicry does not come as a mere complement to some linguistic stimulus; it appears to take the place of that stimulus. I shall try to show that a linguistic analysis can indeed be offered for cases like – though, I believe, without succumbing to the pro-linguistic bias that Carston warns against. I will, however, argue against an ellipsis-based account : the structure of sentence does not contain an unrealised adjective phrase. Instead, I shall defend a ‘syntactic-recruitment' account. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Charles S. Peirce: Selected Writings on Semiotics 1894–1912 ed. by Francesco BellucciCornelis de WaalEdited by Francesco BellucciCharles S. Peirce: Selected Writings on Semiotics 1894–1912 Berlin: De Gruyter, 2020; no index.With Francesco Bellucci's Charles S. Peirce: Selected Writings on Semiotics 1894–1912 and the Peirceana series, which Bellucci co-edits with Ahti-Veikko Pietarinen, the center of gravity in Peirce studies has dramatically shifted to Europe. The powerhouses of old, such as (...) the Peirce Edition Project at Indiana University and the Institute for Studies in Pragmaticism at Texas Tech University, appear pretty much dead in the water. And this new center of gravity is by no means limited to secondary literature. Bellucci's edition of Peirce's writings on semiotics and Pietarinen's multi-volume Logic of the Future present us with well-thought-out and responsibly edited volumes of primary literature that will leave their mark on Peirce scholarship for decades to come. And there are other volumes on the way.Bellucci's volume spans almost two decades of Peirce's life—a period during which Peirce had retreated to Milford, Pennsylvania, spending most of his time with his pen on paper. The main purpose of Bellucci's volume is to give us better access to material that has not yet been published—material that is, as Bellucci contends, "essential to understand Peirce's views about signs, their classification, and the relations between semiotics and logical inquiry" (p.1).In line with this, Bellucci starts off where the Writings break off. The last volume of the Writings, which appeared well over a decade ago, ends midway 1892, and Bellucci's volume opens with the sixth chapter of Peirce's 1894 logic book How to Reason, "The Essence of Reasoning." Bellucci thereby skips the second chapter of that same logic book, even though it deals directly with the question "What is a Sign?" He does this on the ground that that chapter is already published in the second [End Page 265] volume of The Essential Peirce (p.3). Bellucci's Selected Writings on Semiotics is thus best considered a professional tool aimed at filling gaps in the literature for Peirce scholars who are already familiar with, or knowledgeable of, what is already out there. Making these texts available is especially important since the manuscripts they are based on are not widely accessible and are in various states of disarray.Each text in Bellucci's volume comes with a (sometimes-extensive) editorial note that is largely bibliographical and, like the Writings, explains which manuscript sheets constitute the text by supplementing the Robin Catalogue designation with the page reference system devised by the Institute for Studies in Pragmaticism. The headnote is followed by explanatory annotations, emendations, and alterations, not unlike the system used for the Writings. In this way the reader can easily ascertain where Bellucci made corrections to the original text and what changes Peirce made himself when he was in the process of writing. Putting the apparatus immediately following each text is a much better choice than hiding everything in the back of the volume, as was done in the Writings. Given the nature of this edition, however, it would have been better to have the editorial note precede the text, as is done, for instance in The Essential Peirce, rather than follow it (or use two notes), as that would have made it easier for the reader to position the text within Peirce's oeuvre before beginning to read it (for instance, by reading "What is a Sign?" before starting "The Essence of Reasoning").Bellucci's Selected Writings on Semiotics includes eighteen selections in total, each of which is described briefly in the volume's Introduction with a rationale for including it. As with the Writings, titles not given by Peirce are printed in italic square brackets. At times a more forceful editorial hand in establishing the titles themselves would have been helpful. Selection 10, for instance, "The Basis of Pragmaticism," suggests that it would include two pieces already found in The Essential Peirce, namely "The Basis of Pragmaticism in Phaneroscopy," and "The Basis of Pragmaticism in the... (shrink)