Levada, William Cardinal; Ladaria, Luis F The supreme law of the Church is the salvation of souls. As such, throughout its history, the Church has always found the pastoral and juridical means to care for the good of the faithful. With the Apostolic Construction Anglicanorum coetibus, promulgated on 4 November 2009, the Holy Father, Pope Benedict XVI, provided for the establishment of Personal Ordinariates through which Anglican faithful may enter, even in a corporate manner, into full communion (...) with the Catholic Church. On that same date, the Congregation for the Doctrine of the Faith published Complementary Norms relating to such Ordinariates. In conformity with what is established in Art, I and 1 and 2 of the Apostolic Coinstitution Anglicanorum coetibus, having received requests from a considerable number of Anglican faithful, and having consulted with the Australian Catholic Bishops Conference, the Congregation for the Doctrine of the Faith. (shrink)

The March 2002 symposium Human Dignity and Reproductive Technology brought together philosophers, theologians, scientists, lawyers, and scholars from across the United States. The essays of this book are the contributions of the symposium's participants.

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa $ is strongly compact if and only if the strong tree property holds at $\kappa $, and supercompact if and only if ITP holds at $\kappa $. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals (...) simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph _n$ for all $2 \leq n < \omega $ simultaneously with the strong tree property at $\aleph _{\omega +1}$ ; we also show that it is consistent for ITP to hold at $\aleph _n$ for all $3 < n < \omega $ and at $\aleph _{\omega +1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously. (shrink)

This book assesses how Vatican II opened up the Catholic Church to encounter, dialogue, and engagement with other world religions. Opening with a contribution from the President of the Pontifical Council for Interreligious Dialogue, Cardinal Jean-Louis Tauran, it next explores the impact, relevance, and promise of the Declaration Nostra Aetate before turning to consider how Vatican II in general has influenced interfaith dialogue and the intellectual and comparative study of world religions in the postconciliar decades, as well as the (...) contribution of particular past and present thinkers to the formation of current interreligious and comparative theological methods. Additionally, chapters consider interreligious dialogue vis-à-vis theological anthropology in conciliar documents; openness to the spiritual practices of other faith traditions as a way of encouraging positive interreligious encounter; the role of lay and new ecclesial movements in interreligious dialogue; and the development of Monastic Interreligious Dialogue. Finally, it includes a range of perspectives on the fruits and future of Vatican’s II’s opening to particular faiths such as Judaism, Islam, Hinduism, and Buddhism. (shrink)

Assume that there is no transitive class model of [Formula: see text] with a Woodin cardinal. Let [Formula: see text] be a singular ordinal such that [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] is a regular cardinal in K. Then [Formula: see text] is a measurable cardinal in K. Moreover, if [Formula: see text], then [Formula: see text].

P. T. Geach, notoriously, holds the Relative Identity Thesis, according to which a meaningful judgment of identity is always, implicitly or explicitly, relative to some general term. ‘The same’ is a fragmentary expression, and has no significance unless we say or mean ‘the same X’, where ‘X’ represents a general term (what Frege calls a Begriffswort or Begriffsausdruck). (P. T. Geach, Mental Acts (London: Routledge and Kegan Paul, 1957), p. 69. I maintain that it makes no sense to judge whether (...) things are ‘the same’, or remain ‘the same’, unless we add or understand some general term - ‘the same F’. (P. T. Geach, Reference and Generality, third Edition (Ithaca, N.Y.: Cornell University Press, 1980), pp. 63f. I am arguing for the thesis that identity is relative. When one says ‘x is identical with y’, this, I hold, is an incomplete expression; it is short for ‘x is the same A as y’, where ‘A’ represents some count noun understood from the context of utterance - or else, it is just a vague expression of a half-formed thought. (P. T. Geach, ‘Identity,’ Review of Metaphysics 21 (1967-8), p. 3.) One of the ways Geach seeks to support this is by tying it to the well nigh universally admired Fregean thesis about cardinality. (shrink)

We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model Kc which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and is universal with respect to set sized premice.

Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on (...) partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text]. (shrink)

Epicurus, though popularly and indeed nominally associated with a doctrine advocating the procurement of rather expensive pleasure, lived very simply in his garden with a circle of friends. The 14th of his Sovran Maxims or Cardinal Tenets , as collected by Diogenes Laertius, reads: ‘When tolerable security against our fellowmen is attained, then on a basis of power sufficient to afford support and of material prosperity arises in most genuine form the security of a quiet private life withdrawn from (...) the multitude’ R. D. Hicks, the translator, gives, as an alternative to ‘power sufficient to afford support’, ‘power to expel’, but on either reading, we are to think of Epicurus's garden, both the real place and the conceptualized, or tropologized, topos of wisdom and earthly felicity, as enclosed and exclusive. This enclosure, exclusivity, and—what is also implied—abstention of its inmates from participation in the affairs of state are given apophthegmatic expression in ‘Live hidden!’ We contemplate this adage, or slogan, from an enormous cultural distance and, for most of us, ‘Live hidden!’ is much more likely to trigger secondary reflections about the relations between ethics and politics, private and public, perfection of self and improvement of the world, than it is to constitute an actual, or potential, piece of advice we might think about whether , and how , to follow. (shrink)

In this article we characterize all those sequences of cardinals of length ω1 which are cardinal sequences of some compact scattered space . This extends the similar results from [R. La Grange, Concerning the cardinal sequence of a Boolean algebra, Algebra Universalis, 7 307–313] for such sequences of countable length. For ordinals between ω1 and ω2 we can only give a sufficient condition for a sequence of that length to be a cardinal sequence of a compact scattered (...) space. This condition is, arguably, the most one can expect in ZFC. In any case, ours is a significant extension of the sufficient conditions given in [J.C. Martinez, A consistency result on thin-tall superatomic Boolean algebras, Proc. Amer. Math. Soc. 115 473–477] and [J. Bagaria, Locally generic Boolean algebras and cardinal sequences, Algebra Universalis 47 283–302]. (shrink)

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2κ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf > ω then it follows that o λ, and if cf = ωthen either o λ or {α: K o α+n} is (...) confinal in κ for each n ε ω.We also prove several results which extend or are related to this result, notably Theorem. If 2ω ω1 then there is a sharp for a model with a strong cardinal.In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders. (shrink)

In [4], Kunen used iterated ultrapowers to show that ifUis a normalκ-complete nontrivial ultrafilter on a cardinalκthenL[U], the class of sets constructive fromU, has only the ultrafilterU∩L[U] and this ultrafilter depends only onκ. In this paper we extend Kunen's methods to arbitrary sequencesUof ultrafilters and obtain generalizations of these results. In particular we answer Problem 1 of Kunen and Paris [5] which asks whether the number of ultrafilters onκcan be intermediate between 1 and 22κ. If there is a normalκ-complete ultrafilterUonκsuch (...) that {α <κ: α is measurable} ∈Uthen there is an inner model with exactly two normal ultrafilters onκ, and ifκis super-compact then there are inner models havingκ+ +,κ+or any cardinal less than or equal toκnormal ultrafilters.These methods also show that several properties ofLwhich had been shown to hold forL[U] also hold forL[U]: using an idea of Silver we show that inL[U] the generalized continuum hypothesis is true, there is a Souslin tree, and there is awell-ordering of the reals. In addition we generalize a result of Kunen to characterize the countaby complete ultrafilters ofL[U]. (shrink)

In a seminal article, Simo Knuuttila and Anja Inkeri Lehtinen drew attention to a “curious doctrine” holding that contradictories can be true at the same temporal instant, and identified the major defenders of the doctrine as John Baconthorpe, Landolfo Caracciolo, and Hugh of Novocastro. Normann Kretzmann later asserted as fact the suggestion by Knuuttila and Inkeri Lehtinen that the doctrine comes from a misreading of a passage from Aristotle’s Physics. In fact, a study of the relevant texts reveals that Hugh (...) of Novocastro first elaborated the doctrine by building on the Scotist doctrines of synchronic contingency and simultaneous causation. As these doctrines require at the same instant of time an order of priority and posteriority between possibility and actuality, cause and effect, so, Hugh says, there must be prior and posterior different states of affairs. Landolfo Caracciolo made this doctrine notorious outside the Franciscan convent by using it in his principia debates, directly engaging the circle of Cardinal Iacopo Stefaneschi. John Baconthorpe, the first to cite the Physics passage, did not have any noticeable effect on the development of the doctrine. (shrink)

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal (...) companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. (shrink)

If there is no inner model with a cardinal κ such that o(κ) = κ ++ then the set K ∩ H ω 1 is definable over H ω 1 by a Δ 4 formula, and the set $\{J_\alpha[\mathscr{U}]: \alpha of countable initial segments of the core model K = L[U] is definable over H ω 1 by a Π 3 formula. We show that if there is an inner model with infinitely many measurable cardinals then there is a (...) model in which $\{J_\alpha [\mathscr{U}]: \alpha is not definable by any Σ 3 formula, and K ∩ H ω 1 is not definable by any boolean combination of Σ 3 formulas. (shrink)

We reprove Gitik's theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every $\nu \in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik. the iterated forcing $R_{\lambda +1}$ used in this paper has the property that if λ is a cardinal less then κ then $R_{\lambda + 1}$ (...) can be factored in V as $R_{\kappa + 1} = R_{\lambda + 1} \times R_{\lambda + 1, \kappa}$ where $\mid R_{\lambda +1}\mid \leq \lambda^+$ and $R_{\lambda + 1, \kappa}$ does not add any new subsets of λ. (shrink)

My purpose in this talk is to give an overview of the rediscovery of Frege's theorem together with certain of the issues that this rediscovery has raised concerning the evaluation of Frege's logicism—the ‘old doctrine’ of my title.The contextual definition of the cardinality operator, suggested in §63 ofGrundlagen— what, after George Boolos, has come to be known as Hume's principle—assertsThe number of Fs = the number of Gs if, and only if, F ≈ G,where F ≈ G (the Fs and (...) the Gs are in one-to-one correspondence) has its usual, second order, explicit definition. The importance of this principle for the derivation of Peano's second postulate (‘Every natural number has a successor’) was emphasized by Crispin Wright (1983, §xix) who presented an extended argument showing that, in the context of the system of second-order logic of Frege'sBegriffsschrift,Peano's second postulate is derivable from Hume's principle. (shrink)

We reprove Gitik’s theorem that if the GCH holds and $o=\gk+1$ then there is a generic extension in which $\gk$ is still measurable and there is a closed unbounded subset C of $\gk$ such that every $ν\in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik, the iterated forcing $\radin\gl+1$ used in this paper has the property that if $\gl$ is a cardinal less then $\gk$ then $\radin\gl+1$ can be factored in V as $\radin\gk+1=\radin\gl+1\times\radin\gl+1,\gk$ where (...) $\card{\radin\gl+1}\le\gl+$ and $\radin\gl+1,\gk$ does not add any new subsets of $\gl$. (shrink)

If there is no inner model with a cardinal $\kappa$ such that $o = \kappa^{++}$ then the set $K \cap H_{\omega_1}$ is definable over H$_{\omega_1}$ by a $\Delta_4$ formula, and the set $\{J_\alpha[\mathscr{U}] : \alpha < \omega_1\}$ of countable initial segments of the core model $K = L[\mathscr{U}]$ is definable over $H_{\omega_1}$ by a $\Pi_3$ formula. We show that if there is an inner model with infinitely many measurable cardinals then there is a model in which $\{J_\alpha [\mathscr{U}] : (...) \alpha < \omega_1\}$ is not definable by any $\Sigma_3$ formula, and $K \cap H_{\omega_1}$ is not definable by any boolean combination of $\Sigma_3$ formulas. (shrink)

This paper investigates the principles □λ,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{{\rm ta}}}_{\lambda,\delta}}$$\end{document}, weakenings of □λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_\lambda}$$\end{document} which allow δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta}$$\end{document} many clubs at each level but require them to agree on a tail-end. First, we prove that □λ,<ωta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm {ta}}}_{\lambda,< \omega}}$$\end{document} implies □λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_\lambda}$$\end{document}. Then, (...) by forcing from a model with a measurable cardinal, we show that □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} does not imply □λ,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm{ta}}}_{\lambda,\delta}}$$\end{document} for regular λ\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}, and □δ+,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm{ta}}}_{\delta^+,\delta}}$$\end{document} does not imply □δ+,<δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\delta^+,< \delta}}$$\end{document}. With a supercompact cardinal the former result can be extended to singular λ, and the latter can be improved to show that □λ,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm {ta}}}_{\lambda,\delta}}$$\end{document} does not imply □λ,<δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\lambda,< \delta}}$$\end{document} for δ<λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta < \lambda}$$\end{document}. (shrink)

We use a $\kappa^{+}-Mahlo$ cardinal to give a forcing construction of a model in which there is no sequence $\langle A_{\beta} : \beta \textless \omega_{2} \rangle$ of sets of cardinality $\omega_{1}$ such that $\{\lambda \textless \omega_{2} : \existsc \subset \lambda & (\bigcupc = \lambda otp(c) = \omega_{1} & \forall \beta \textless \lambda (c \cap \beta \in A_{\beta}))\}$ is stationary.

We use a κ+-Mahlo cardinal to give a forcing construction of a model in which there is no sequence 〈 Aγ:γ<ω2 〉 of sets of cardinality ω1 such that {λ<ω2:∃ c⊂λ =ω1&∀γ<λ )} is stationary.

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

This book presents and survey of the foundations of mathematics. The emphasis is on a mathematical comparison of systems rather than on any exhaustive development of analysis within a single system. Nevertheless, for most systems considered, enough details are given for the development of arithmetic, and the method of constructing the other notions of analysis is indicated. The elements of the general theory of cardinal and ordinal numbers are also furnished in the course of this work.

St. Omer MS 239 contains the unstudied Lectura of Pastor de Serrescuderio, OFM, who read the Sentences at Paris in 1332-33. The article traces his academic and ecclesiastical career from provincial minister in Provence to cardinal at Avignon, and includes the list of question titles from his Lectura.

We characterize some large cardinal properties, such as $\mu$-measurability and $P^2(\kappa)$-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on $P_\kappa(2^\kappa)$. This leads to the characterization of $2^\kappa$-supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, $\mathrm{Full}_\kappa$, of $P_\kappa(2^\kappa)$, whose elements code measures on cardinals less than $\kappa$. The hypothesis that $\mathrm{Full}_\kappa$ is stationary (a weaker assumption than $2^\kappa$-supercompactness) is equivalent (...) to a higher order Lowenheim-Skolem property, and settles a question about directed versus chain-type unions on $P_\kappa\lambda$. (shrink)

Mitchell, W.J., An infinitary Ramsey property, Annals of Pure and Applied Logic 57 151–160. We prove that the consistency of a measurable cardinal implies the consistency of a cardinal κ>+ satisfying the partition relations κ ω and κ ωregressive. This result follows work of Spector which uses the same hypothesis to prove the consistency of ω1 ω. We also give some examples of partition relations which can be proved for ω1 using the methods of Spector but cannot be (...) proved for cardinals κ>+ without a much stronger hypothesis. (shrink)

Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal \ having an extender of length \.

Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...) T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice. (shrink)

Popular election from the Dynasty.—Jehoahaz and Jehoiakim.—Defeat of Necho at Carchemish.—Jeremiah’s Political Prophecies.—Babylonian invasions.—Firstdeportation of Jews to Babylon.—Rebellion of Zedekiah.—Destruction of Jerusalem.—Gedaliah the Babylonian Satrap.—Prophecies against Egypt.—Later School of Prophecy.—Function of the Jewish Nation.

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 characterize some large cardinal properties, such as μ-measurability and P 2 (κ)-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on P κ (2 κ ). This leads to the characterization of 2 κ -supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, Full κ , of P κ (2 κ ), whose elements code measures on cardinals less than (...) κ. The hypothesis that Full κ is stationary (a weaker assumption than 2 κ -supercompactness) is equivalent to a higher order Lowenheim-Skolem property, and settles a question about directed versus chain-type unions on P κ λ. (shrink)

Levin and Schnorr (independently) introduced the monotone complexity, Km(α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α ↾ n) ≤ Km(β ↾ n) + c for all n. The monotone degree of α is the set of all β such that α ≤Km β and β ≤Km α. We (...) show the monotone degrees contain an antichain of size 2N0, a countable dense linear ordering (of degrees of cardinality 2N0), and a minimal pair. Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K -degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K -degrees was constructed by Csima and Montalbán. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We define a real to be (Km, K)-trivial if for some constant c, Km(α ↾ n) ≤ K(n)+c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km, K)-trivial. Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering. (shrink)

Epicurus, though popularly and indeed nominally associated with a doctrine advocating the procurement of rather expensive pleasure, lived very simply in his garden with a circle of friends. The 14th of his Sovran Maxims or Cardinal Tenets (kuriai doxai), as collected by Diogenes Laertius, reads: ‘When tolerable security against our fellowmen is attained, then on a basis of power sufficient to afford support and of material prosperity arises in most genuine form the security of a quiet private life withdrawn (...) from the multitude’ R. D. Hicks, the translator, gives, as an alternative to ‘power sufficient to afford support’, ‘power to expel’, but on either reading, we are to think of Epicurus's garden, both the real place and the conceptualized, or tropologized, topos of wisdom and earthly felicity, as enclosed and exclusive. This enclosure, exclusivity, and—what is also implied—abstention of its inmates from participation in the affairs of state are given apophthegmatic expression in ‘Live hidden!’ We contemplate this adage, or slogan, from an enormous cultural distance and, for most of us, ‘Live hidden!’ is much more likely to trigger secondary reflections about the relations between ethics and politics, private and public, perfection of self and improvement of the world, than it is to constitute an actual, or potential, piece of advice we might think about whether, and how, to follow. (shrink)

The great medieval philosopher Thomas Aquinas was Dominican regent master in theology at the University of Paris, where he presided over a series of questions - academic debates - on ethical topics. This volume offers translations of disputed questions on the nature of virtues in general, the fundamental or 'cardinal' virtues of practical wisdom, justice, courage, and temperateness, the divinely bestowed virtues of hope and charity, and the practical question of how, when and why one should rebuke a 'brother' (...) for wrongdoing. The introduction explains how Aquinas's theory of virtue fits into his ethics as a whole, and it illuminates Aquinas's views by explaining the institutional and intellectual context in which these disputed questions were debated. (shrink)

We construct a model where every increasing ω-sequence of regular cardinals carries a mutually stationary sequence which is not tightly stationary, and show that this property is preserved under a class of Prikry-type forcings. Along the way, we give examples in the Cohen and Prikry models of ω-sequences of regular cardinals for which there is a non-tightly stationary sequence of stationary subsets consisting of cofinality ω_1 ordinals, and show that such stationary sequences are mutually stationary in the presence of interleaved (...) supercompact cardinals. (shrink)