From XIX Century, the controversy over the authenticity of the Metaphysics’ Kappa has been a constant in the context of studies on the Metaphysics. In regard to the redaction of the aporias of K 1-2, the differences in style and content are remarkable. However, after analyzing the differences, we can assert quite safely that the author of K 1-2 is Aristotle. Indeed, K 1-2 do not contains non-Aristotelian philosophy in approach of the aporias nor in the doctrines involved in (...) them. In this sense, the differences may be explained by the renewed interest of Aristotle for the supersensible, in the face of its research on Λ 6-10. Secondly, with respect to the redaction of aporias, researches on style and morphometry show that the use of some expressions change significantly from one work to another, so differences in style, use of particles and unusual expressions in K 1-2 can be understood as normal. (shrink)

Answering a question of Sakai :29–45, 2013), we show that the existence of an ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document}-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with □ω1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\omega _1, 2}$$\end{document}. By a result of Donder, volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to (...) the incompatibility of □λ,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\lambda, 2}$$\end{document} and ↠\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \twoheadrightarrow $$\end{document} for uncountable κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. (shrink)

It is proved that $\Pi^1_1$ -indescribability in $P_{\kappa}\lambda$ can be characterized by combinatorial properties without taking care of cofinality of $\lambda$ . We extend Carr's theorem proving that the hypothesis $\kappa$ is $2^{\lambda^{<\kappa}}$ -Shelah is rather stronger than $\kappa$ is $\lambda$ -supercompact.

Arthur Madigan presents a clear, accurate new translation of the third book (Beta) of Aristotle's Metaphysics, together with two related chapters from the eleventh book (Kappa). Madigan's accompanying introduction and commentary give detailed guidance to these texts, in which Aristotle setsout what he takes to be the main problems of metaphysics or 'first philosophy' and assesses possible solutions to them; he takes his starting-point from the work of earlier philosophers, especially Plato and some of the Presocratics. These texts serve (...) as a useful introduction both to Aristotle's ownwork on metaphysics and to classical metaphysics in general; they are also a good example of Aristotle's dialectical method, which reasons not from known truths but from reputable opinions. (shrink)

Normal 0 21 false false false ES-CO X-NONE X-NONE MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} The very core of the Aristotelian philosophy, where the great metaphysical issues tackled by his predecessors (especially Plato and the philosophers from the Academy) were reconsidered, is structured around the notions of being (einai, tó ón), substance (...) (ousía) and form (eidos, tó ti en einai), largely discussed by Aristotle in the central books of Metaphysics. The understanding of the solutions provided by Aristotle to the aporethic questions formulated and developed by him in the Beta and Kappa 1-2 books concerning the proper object of the first philosophy, the determination of the existent substances as well as the determination of the nature of the first principles of reality, depend all of them on the correct interpretation of such notions. (shrink)

The main result of this paper is that if $\kappa$ is not a weakly Mahlo cardinal, then the following two conditions are equivalent: 1. $\mathscr{P}(\kappa)/ \mathscr{J}$ is $\kappa^+$-complete. 2. $\mathscr{J}$ is a prenormal ideal. Our result is a generalization of an announcement made in [Z]. We say that $\mathscr{J}$ is selective iff for every $\mathscr{J}$-function $f: \kappa \rightarrow \kappa$ there is a set $X \in \mathscr{J}$ such that $f\mid(\kappa - X)$ is one-to-one. Our theorem (...) provides a positive partial answer to a question of B. Weglorz from [BTW, p. 90], viz.: is every selective ideal $\mathscr{J}$, with $\mathscr{P}(\kappa)/\mathscr{J} \kappa^+$-complete, isomorphic to a normal ideal? The theorem is also true for fine ideals on $\lbrack\lambda\rbrack^{<\kappa}$ for any $\kappa \leq \lambda$, i.e. if $\kappa$ is not a weakly Mahlo cardinal then the Boolean algebra $\mathscr{P} (\lbrack\lambda\rbrack^{<\kappa})/\mathscr{J}$ is $\lambda^+$-complete iff $\mathscr{J}$ is a prenormal ideal (in the sense of $\lbrack\lambda\rbrack^{<\kappa})$. (shrink)

Assuming an inaccessible cardinal $\kappa$ , there is a generic extension in which $MA + 2^{\aleph_0} = \kappa$ holds and the reals have a $\Delta^2_1$ well-ordering.

We study descriptive set theory in the space $^{\omega_1}\omega_1$ by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of $\Pi^1_1$-sets of $^{\omega_1}\omega_1$. We call a family $\mathscr{U}$ of trees universal for a class $\mathscr{V}$ of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in $\mathscr{V}$ can be order-preservingly mapped into a tree in $\mathscr{U}$. It is well (...) known that the class of countable trees with no infinite branches has a universal family of size $\aleph_1$. We shall study the smallest cardinality of a universal family for the class of trees of cardinality $\leq\aleph_1$ with no uncountable branches. We prove that this cardinality can be 1 (under $\neg$CH) and any regular cardinal $\kappa$ which satisfies $\aleph_2 \leq \kappa \leq 2^{\aleph_1}$ (under CH). This bears immediately on the covering property of the $\Pi^1_1$-subsets of the space $^{\omega_1}\omega_1$. We also study the possible cardinalities of definable subsets of $^{\omega_1}\omega_1$. We show that the statement that every definable subset of $^{\omega_1}\omega_1$ has cardinality $<\omega_n$ or cardinality $2^{\omega_1}$ is equiconsistent with ZFC (if $n \geq 3$) and with ZFC plus an inaccessible (if $n = 2$). Finally, we define an analogue of the notion of a Borel set for the space $^{\omega_1}\omega_1$ and prove a Souslin-Kleene type theorem for this notion. (shrink)

We investigate the partition property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document}. Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} carries a (λκ, 2)-distributive normal ideal without the partition property, then λ is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^1_n}$$\end{document}-indescribable for all n < ω but not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^2_1}$$\end{document} -indescribable. (2) If cf(λ) ≥ κ, then every ineffable subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} has the partition property. (shrink)

Assuming the existence of a supercompact cardinal, we construct a model of ZFC + (There exists a nonsplitting stationary subset of $\mathscr{P}_|kappa(\kappa^+)$). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes $\mathscr{P}_{\aleph_1}(\aleph_2) - V$ stationary.

This research was partially supported by Grant-in-Aid for Scientific Research (No. 06640178 and No. 06640336), Ministry of Education, Science and Culture of Japan Mathematics Subject Classification: 03E05 --> Abstract. Following Carr's study on diagonal operations and normal filters on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal P}_{\kappa}\lambda$\end{document} in [2], several weakenings of normality have been investigated. One of them is to consider normal filters without \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\ (...) class='Hi'>kappa$\end{document}-completeness, for example, see DiPrisco-Uzcategui [3]. The other is weakening normality itself while keeping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document}-completeness such as in Mignone [10] and Shioya [11]. We take the second one so that all filters are assumed to be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document}-complete. In Sect. 1 a hierarchy of filters on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal P}_{\kappa}\lambda$\end{document} is presented which corresponds to the length of diagonal intersections under which the filters are closed. It turns out that many ranks exist between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $FSF_{\kappa\lambda}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $CF_{\kappa\lambda}$\end{document}. We consider seminormal ideals in Sect. 2 and determine the minimal seminormal ideal extending Johnson's result in [6]. Its precise descripti on changes according to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $cf(\lambda )$\end{document} although we can write it in a single form as well. We also prove that a nonnormal seminormal ideal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $I\supset NS_{\kappa\lambda}$\end{document} exists if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document} is regular. (shrink)

In this paper, the following are proved:Theorem A. The quotient algebra ${\cal P} (\kappa )/I$ is complete if and only if the only non-trivial I -closed ideals extending I are of the form $I\lceil A$ for some $A\in I^+$ .Theorem B. If $\kappa$ is a stationary cardinal, then the quotient algebra ${\cal P} (\kappa )/ NS_\kappa$ is not complete.Corollary. (1) If $\kappa$ is a weak compact cardinal, then the quotient algebra ${\cal P} (\kappa )/NS_\ (...) class='Hi'>kappa$ is not complete.(2) If $\kappa$ bears $\kappa$ -saturated ideal, then the quotient algebra ${\cal P} (\kappa )/NS_\kappa$ is not complete.Theorem C. Assume that $\kappa$ is a strongly compact cardinal, I is a non-trivial normal $\kappa$ -complete ideal on $\kappa$ and B is an I -regular complete Boolean algebra. Then if ${\cal P} (\kappa )/I$ is complete, it is B -valid that for some $A\subseteq\check\kappa$ , ${\cal P} (\kappa )/({\bf J}\lceil A)$ is complete, where J is the ideal generated by $\check I$ in $V^B$ .Corollary. Let M be a transitive model of ZFC and in M , let $\kappa$ be a strongly compact cardinal and $\lambda$ a regular uncountable cardinal less than $\kappa$ . Then there exists a generic extension M [ G ] in which $\kappa =\lambda^+$ and $\kappa$ carries a non-trivial $\kappa$ -complete ideal I which is completive but not $\kappa^+$ -saturated. (shrink)

Given an uncountable cardinal κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} that satisfies κκ=κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{\kappa}=\kappa}$$\end{document}, we provide a forcing that is <κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${<\kappa}$$\end{document} -closed, has size 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^\kappa}$$\end{document} and is κ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^+}$$\end{document} -cc, to introduce a Σ1-definable wellorder of (...) H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm H}}$$\end{document}. This improves the main result of Holy and Lücke :221–236, 2014), where such a wellorder is introduced by a forcing which potentially collapses cardinals, and where the additional requirement that 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^\kappa}$$\end{document} be regular is needed. As an application, we use this to infer that Σ1-definable wellorderings of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm H}}$$\end{document} can be introduced for many different cardinals κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} simultaneously, while preserving a lot of ground model structure, improving results of Friedman and Holy :133–166, 2011) and Friedman and Lücke. Moreover the results of this paper answer Holy and Lücke. (shrink)

A $\Sigma _{1}^{2}$ truth for λ is a pair 〈Q, ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ Hλ+ such that (Hλ+; ε, B) $(H_{\lambda +};\in ,B)\vDash \psi [Q]$ . A cardinal λ is $\Sigma _{1}^{2}$ indescribable just in case that for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $\overline{\lambda}<\lambda $ so that $\overline{\lambda}$ is a cardinal and $\langle Q\cap H_{\overline{\lambda}},\psi \rangle $ is a (...) $\Sigma _{1}^{2}$ truth for $\overline{\lambda}$ . More generally, an interval of cardinals [κ, λ] with κ ≤ λ is $\Sigma _{1}^{2}$ indescribable if for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $??\leq \overline{\lambda}<\kappa,??\subseteq H_{\overline{\lambda}}$ , and π: $H_{\overline{\lambda}}\rightarrow H_{\lambda}$ so that $??$ is a cardinal, $\langle ??,\psi \rangle $ is a $\Sigma _{1}^{2}$ truth for $??$ , and π is elementary from $(H_{\overline{\lambda}};\in,??,??)$ with $\pi \,|\,??={\rm id}$ . We prove that the restriction of the proper forcing axiom to c-linked posets requires a $\Sigma _{1}^{2}$ indescribable cardinal in L, and that the restriction of the proper forcing axiom to c⁺-linked posets, in a proper forcing extension of a fine structural model, requires a $\Sigma _{1}^{2}$ indescribable 1-gap [κ, κ⁺]. These results show that the respective forward directions obtained in " Hierarchies of Forcing Axioms I" by Neeman and Schimmerling are optimal. (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 is strong over a partition $p:[\kappa ]^2\to \theta $. A coloring f is strong over p if for every $A\in [\kappa ]^{\kappa }$ there is $i<\theta $ so that for every color $\unicode{x3b3} <\kappa $ is attained by $f\restriction )$.We prove that whenever $\kappa \nrightarrow [\kappa ]^2_{\kappa }$ holds, also $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ holds for an arbitrary finite partition p. Similarly, arbitrary finite p-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 $\operatorname {Pr}_1_p$ or $\operatorname {Pr}_0_p$, also hold for an arbitrary partition p to $\theta $ parts.The symbols $$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0_p \end{gather*} $$ hold for an arbitrary countable partition p under the Continuum Hypothesis and are independent over ZFC $+ \neg $ CH. (shrink)

We make comments on some problems Erdős and Hajnal posed in their famous problem list. Let X be a graph on $\omega _1$ with the property that every uncountable set A of vertices contains a finite set s such that each element of $A-s$ is joined to one of the elements of s. Does then X contain an uncountable clique? We prove that both the statement and its negation are consistent. Do there exist circuitfree graphs $\{X_n:n<\omega \}$ on $\omega _1$ (...) such that if $A\in [\omega _1]^{\aleph _1}$, then $\{n<\omega :X_n\cap [A]^2=\emptyset \}$ is finite? We show that the answer is yes under CH, and no under Martin’s axiom. Does there exist $F:[\omega _1]^2\to 3$ with all three colors appearing in every uncountable set, and with no triangle of three colors. We give a different proof of Todorcevic’ theorem that the existence of a $\kappa $ -Suslin tree gives $F:[\kappa ]^2\to \kappa $ establishing $\kappa \not \to [\kappa ]^2_{\kappa }$ with no three-colored triangles. This statement in turn implies the existence of a $\kappa $ -Aronszajn tree. (shrink)

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$ -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $ : $u_2$, and secondly to give the consistency strength of a property of Lücke’s.TheoremThe following are equiconsistent:There exists $\kappa $ which is stably measurable;for some cardinal $\kappa $, $u_2=\sigma $ ;The $\boldsymbol {\Sigma }_{1}$ (...) -club property holds at a cardinal $\kappa $.Here $\sigma $ is the height of the smallest $M \prec _{\Sigma _{1}} H $ containing $\kappa +1$ and all of $H $. Let $\Phi $ be the assertion: TheoremAssume $\kappa $ is stably measurable. Then $\Phi $.And a form of converse:TheoremSuppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: $\mbox {``}\exists \kappa \Phi \mbox {''}$ is -generically absolute ${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.When $u_2 < \sigma $ we give some results on inner model reflection. (shrink)

Let A and B be subsets of the reals. Say that A κ ≥ B, if there is a real a such that the relation "x ∈ B" is uniformly Δ 1 (a, A) in L[ ω x,a,A 1 , x,a,A]. This reducibility induces an equivalence relation $\equiv_\kappa$ on the sets of reals; the $\equiv_\kappa$ -equivalence class of a set is called its Kleene degree. Let K be the structure that consists of the Kleene degrees and the induced (...) partial order K ≥. A substructure of K that is of interest is P, the Kleene degrees of the Π 1 1 sets of reals. If sharps exist, then there is not much to P, as Steel [9] has shown that the existence of sharps implies that P has only two elements: the degree of the empty set and the degree of the complete Π 1 1 set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in P; in the context of V = L, Hrbacek has shown that P is dense and has no minimal pairs. The Hrbacek results led Simpson [6] to make the following conjecture: if V = L, then p forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if V = L, then Godel's maximal thin Π 1 1 set is the infimum of two strictly larger elements of P. The second main result deals with the notion of jump in K. Let A' be the complete Kleene enumerable set relative to A. Say that A is low-n if A (n) has the same degree as $\varnothing^{(n)}$ , and A is high-n if A (n) has the same degree as $\varnothing^{(n + 1)}$ . Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete Π 1 1 set in L. They have also shown that various other Π 1 1 sets are neither high nor low in L. Legrand [5] extended their results by showing that, if there is a real x such that x # does not exist, then there is an element of P that, for all n, is neither low-n nor high-n. In § 2, ZFC is used to show that, for all n, if A is Π 1 1 and low-n then A is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp]. (shrink)

We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$, we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible, and $\kappa _2$ ’s strong compactness is (...) indestructible under $\mathrm {Add}$ for any ordinal $\delta $. This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness. (shrink)

The recent in vitro derivation of gamete‐like cells from mouse embryonic stem (mES) cells is a major breakthrough and lays down several challenges, both for the further scientific investigation and for the bioethical and biolegal discourse. We refer here to these cells as gamete‐like (sperm‐like or oocyte‐like, respectively), because at present there is still no evidence that these cells behave fully like bona fide sperm or oocytes, lacking the fundamental proof, i.e. combination with a normally derived gamete of the opposite (...) sex to yield a normal individual. However, the results published so far do show that these cells share some defining features of gametes. We discuss these results in the light of the bioethical and legal questions that are likely to arise would the same process become possible with human embryonic stem (hES) cells. (shrink)

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: 1. If it is consistent that there is a 1-inaccessible cardinal (...) then it is consistent that ${\sf BC}_{\aleph_1}$.2. If it is consistent that ${\sf BC}_{\aleph_1}$, then it is consistent that there is an inaccessible cardinal.3. If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} + (\forall n < \omega){\sf BC}_{\aleph_n}$ is consistent.4. If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$.5. If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ for a proper class of cardinals $\kappa$ of countable cofinality. (shrink)

We prove new upper bound theorems on the consistency strengths of SPFA (θ), SPFA(θ-linked) and SPFA(θ⁺-cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(c-linked) and PFA(c-linked) are each equiconsistent with the existence of a $\Sigma _{1}^{2}$ -indescribable cardinal. Our (...) upper bound for SPFA(c-c.c.) is a $\Sigma _{2}^{2}$ -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(c⁺-linked) is a cardinal κ that is $(\kappa ^{+},\Sigma _{1}^{2})$ -subcompact, which is strictly weaker than κ⁺-supercompact. The axiom MM(c) is a consequence of SPFA(c⁺-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(c⁺⁺-c.c.) is a cardinal κ that is $(\kappa ^{+},\Sigma _{2}^{2})$ -subcompact, which is also strictly weaker than κ⁺-supercompact. (shrink)

According to the Einstein-Mayer theory of the Riemanniann space-time with Einstein-Cartan teleparallelism, the local Lorentz invariance is broken by the gravitational field defining Machian reference systems. This breaking of symmetry implies the occurrence of “hidden matter” in the Einstein equations of gravity. The hidden matter is described by the non-Lorentz-invariant energy-momentum tensor $\hat \Theta _{ik}$ satisfying the relation $\hat \Theta _{i;k}^k = 0$ . The tensor $\hat \Theta _{ik}$ is formed from the Einstein-Cartan torsion field given by the anholonomy objects, (...) FAik=2hA[i,k], and appears together with Hilbert’s energy-momentum tensor T* ik and Poincaré’s pressure λgik on the right-hand side of Einstein’s equations so that one has $$R_{ik} - (1/2)g_{ik} R = - \kappa T*_{ik} - \lambda g_{ik} - \hat \Theta _{ik}$$ According to this theory, in the universe and in cosmic systems one must excep “invisible masses” described by the Poincaré and Einstein-Cartan terms to exist. The torsion field FAik makes the space-time a Machian universe; it is of the same nature as the “weak interacting matter” discussed in astrophysics. (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)

Abstract(1)We show that it is possible to add $\kappa ^+$ -Cohen subsets to $\kappa $ with a Prikry forcing over $\kappa $. This answers a question from [9].(2)A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author [5].(3)A situation with Extender-based Prikry forcings is examined. This relates to a question of H. Woodin.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:1 & 2 Thessalonians by Douglas FarrowAnna Silvas1 & 2 Thessalonians by Douglas Farrow (Grand Rapids, MI: Brazos, 2020), xx + 336 pp.1 and 2 Thessalonians are probably the very first written testimonies of early Christianity. When Paul wrote 1 Thessalonians in AD 50, Our Lord Jesus Christ had "accomplished his exodus in Jerusalem" (see Luke 9:31) not twenty years before. Here we find the paradosis of the (...) Mystery of Christ among the Gentiles, the first articulations of Trinitarian faith (see 1 Thess 1:2-5), expressions of longing for the coming Parousia of Christ, warnings of a strange lawlessness already at work ripening towards a great apostasy and the coming of the "man of lawlessness."Douglas Farrow has pondered 1 and 2 Thessalonians for many years, attested by his numerous expositions and essays over three decades. He brings to his task a ready command of Greek, Hebrew, and Latin, and a vast reading of historical commentary and controversy, an intense personal and ecclesial faith, and a sense of historical, prophetic, and pastoral urgency.Throughout the book are excurses on themes opened up by the words and phrases of St Paul's text: Satan (82-86); the sexual body and virtue in sex, divorce, and remarriage (95-114); the remedial pains of purgatory (beginning in this life); the reality of final retribution, the finality of hell; the untenability of so-called "universalism" (131-41,178-84,192-96); the imminency and the delay of the Parousia (199-204); the apostasy / man of lawlessness / Antichrist; the temple in which the Antichrist seats himself (205-42); the mystery of lawlessness at work, especially in undermining the Church (245-61).In studying Paul's eschatological themes, Farrow necessarily ponders other scriptural texts leading in the same direction: Dani 7-12; Jesus's Olivet Discourse in its most extensive form at Matt 24:1-25:46 (though it really begins at 23:23); 2 Pet 2:1-5, 3:3-15; much of the Rev 6:1 through 19:21; [End Page 398] Deut 32:1-43 (Farrow says the Song of Moses encapsulates the whole of human history); Psalms 97 (LXX 96); 110 (LXX 109); Wis 4-5, 10-19; Sirach 50; Joel; Zech 8-14; Mai 3-4; 2 Mace 5-9; Isa 66:5,14-16; and so on. Farrow holds that Paul worked from knowledge of Jesus's apocalyptic teachings in aversion closest to Matthew's (138,142,184, 189, 205).Farrow takes companions in his exegesis: Irenaeus, Hippolytus, Cyril of Jerusalem, John Chrysostom, Augustine, Haimo, Anselm, Aquinas, and others. He knows Luther's hermeneutics and Calvinist disputes about supra- and infra-sessionism and double predestinationism. He consults more recent prophets, quoting at length from Newman's amazingly prescient 1838 discourses on the last times, Solovyovin 1899, and Fulton Sheen in 1948 and 1958. Another exegete is an artist: Luca Signorelli, whose Disputation of the Antichrist in Orvieto Cathedral Farrow refers to several times.Farrow shows his affinity with Irenaeus. That means cleaving to the visible and invisible truths of the Incarnation of God the Word, respecting concrete and embodied realities, keeping in view the transcendent goal ("The glory of God is a living man—and the life of man is in the vision of God"), instinctively flinching at gnosticizing tendencies—sophisticated re-interpretations of Scripture that elide the rule of faith and dissolve truth. It means a ruling sense of the Kingdom of God as an intervention from heaven rather than generating from man or from the earth. It means a spirit of obedience to the commandments, to the paradosis that descends to us from above in Christ, effected by the Spirit in the ecclesia, the company of brethren destined to "gather to Christ" at his coming (1 Thess 2:1), who practice that final gathering to Christ by participating in the Eucharist on the "Eighth Day," the Lord's Day.Farrow deplores the patchy reading of 1 and 2 Thessalonians in the lectionaries: "While 1 Thessalonians fares a little better, barely half of 2 Thessalonians is read in the Churches" (11). Missing is the eschatologically pivotal 2 Thess 2:3-15, where Paul, distinguishing... (shrink)

This paper is a response to the opening of new lines of research on CSR and SMEs (Thompson & Smith, 1991; Spence, 1999; Moore & Smith, 2006; Spence, 2007). It seeks to explore the business case for CSR in this corporate segment. The paper, which is based on four case studies of medium-sized firms in the automotive sector, took the distinctive approach of trying to understand the nature of CSR-like activities developed not by best-in-class CSR-driven companies but by purely competitiveness-driven (...) firms. The case studies provide explicit evidence that the CSR activities of SMEs and the notion of social capital are interrelated, turning social capital into a powerful instrument to better explain what academic literature has called silent CSR practices . The analysis that follows questions some of the basic tenets that underpin the branch of business ethics that deals with the nature of SMEs' approach to CSR. Four basic concerns, which take the form of propositions for further research, serve as the basis for this analysis: a) A definition of CSR that includes most of the actions taken by all companies in the territory contributes no academic value to the discipline b) Any study of the motivation behind these CSR actions must reflect their essentially pragmatic nature. Actions are linked to social values but also, more importantly, to the nature of the competitive environment c) Business ethics must seek common ground with other more sociological disciplines if it is to explain the reasons behind this type of action d) Any study of this kind of practice requires a dual approach: a) normative when using tools developed by CSR; and b) descriptive and instrumental using the notion of social capital. (shrink)

Antonio Argandoña, Norbert Bilbeny, Victòria Camps, Miquel Calsina, Àngel Castiñeira, Cristian Palazzi, Ferran Requejo, Raimon Ribera, Begoña Román, Ferran Sàez, Miquel Seguró, Francesc Torralba, Josep Maria Vallès, Rosamund Thomas Ramon Llull Journal of Applied Ethics 2012 3(3):9-16.

Rhet. Her. 1.2 quoad eius fieri poterit contains the surprising reading quoad eius. Earlier scholarship has debated the authenticity of this reading and its relationship to quod eius. A survey of the sources shows that quod eius appears in a number of inscriptions as well as in the transmitted text of nine passages within surviving Latin literature. So that phrase must be authentic; it appears to have arisen as a limiting formula in the language of the law. In two other (...) passages, quoad eius appears in inferior manuscripts that lack authority, while the reading transmitted by authoritative textual sources is quod eius. Rhet. Her. 1.2 is the only passage in which quoad eius is the transmitted reading. This phrase is also linguistically problematic. Hence it is very likely to be corrupt. It probably arose as a conflation of quod eius with quoad, both of which are attested in similar contexts. On balance, it seems more likely that the original reading in this passage was quoad. (shrink)

1. Consistent inequality [We prove the consistency of irr $(\displaystyle\prod_{i < \kappa} B_i/D) < \displaystyle\prod_{i < \kappa}$ irr(B i )/D where D is an ultrafilter on κ and each B i is a Boolean algebra and irr(B) is the maximal size of irredundant subsets of a Boolean algebra B, see full definition in the text. This solves the last problem, 35, of this form from Monk's list of problems in [M2]. The solution applies to many other properties, e.g. (...) Souslinity.] 2. Consistency for small cardinals [We get similar results with κ=ℵ1 (easily we cannot have it for κ=ℵ0) and Boolean algebras B i (i<κ) of cardinality $< \beth_{\omega_1}$ .] This article continues Magidor Shelah [MgSh:433] and Shelah Spinas [ShSi:677], but does not rely on them: see [M2] for the background. (shrink)