We prove that if the Mathias forcing is followed by a forcing with the Laver Property, then any V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {V}$$\end{document}-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {q}$$\end{document}-point is isomorphic via a ground model bijection to the canonical V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {V}$$\end{document}-Ramsey ultrafilter added by the Mathias real. This improves a result of Shelah and Spinas.

We give topological characterizations of filters${\cal F}$onωsuch that the Mathias forcing${M_{\cal F}}$adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

We present and analyze \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{ACA}_0}$$\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}$${\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}$$\end{document}, whereas Mathias forcing does not. We also show that the needed reals (...) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing. (shrink)

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...) adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (shrink)

Classical dispersion relations are derived from a time-asymmetric constraint. I argue that the standard causal interpretation of this constraint plays a scientifically legitimate role in dispersion theory, and hence provides a counterexample to the causal skepticism advanced by John Norton and others. Norton ([2009]) argues that the causal interpretation of the time-asymmetric constraint is an empty honorific and that the constraint can be motivated by purely non-causal considerations. In this paper I respond to Norton's criticisms and argue that Norton's skepticism (...) derives its force partly by holding causal principles to a standard too high to be met by other scientifically legitimate constraints. (shrink)

The so-called .Karlsschulrede. (1793) of the German naturalist Carl Friedrich Kielmeyer can be considered as a keystone to the understanding of"Naturphilosophie" both in German idealism (Schelling) and the romantic period.Kielmeyer's work considers life as the result of specific forces in the organic realm and thereby searches to explain the harmony of organic existence anddevelopment. Taking into account Kant.s outlines for a lifescience in the "Kritik der Urteilskraft" (1790), Kielmeyer's notion of teleological processes in nature is sketched. The historical and epistemological (...) relevance of this "vital-materialistic" (Lenoir) theory of life can be characterized by three major transformations in the understanding of nature in the "Karlsschulrede": First, the development of a holistic, organological view on the world, second, the emphasis on phenomena of life as historical processes and third the analogy between organism and mind. These issues found the strong influence of Kielmeyer's text on philosophy and science in the early 19th-century. (shrink)

ALTHOUGH majority rule finds ready acceptance whenever groups make decisions, there are surprisingly few philosophically interesting arguments in support of it.1 Jeremy Waldron’s The Dignity of Legislation contains the most interesting recent defense of majority rule. Waldron combines his own argument from respect with May’s influential characterization of majority rule, tying both to a reinterpretation of a well-known passage from Locke’s Second Treatise (“the body moves into the direction determined by the majority of forces”). Despite its impressive resourcefulness, Waldron’s defense (...) is deficient, and one goal of this essay is to show how. Yet our main concern is not to criticize Waldron, but to demonstrate general deficiencies of arguments for majority rule and to suggest a strategy for a more adequate and more complete defense. Such arguments tend to have one of two weaknesses: Either they assume that collective decisionmaking is done in terms of ranking options and thus neglect both aggregation methods using more information than the relative standing of options in rankings (such as so-called positional methods) and rules that are not aggregation methods at all (such as fair-division procedures); or they also constitute arguments for other decision rules. In the first case, the argument is too narrow, in the second it is too broad. The narrowness problem is bigger than stated so far because arguments for majority rule tend to assume not only that decisions are made by ranking options, but also that only two options are to be ranked. Both problems arise for Waldron’s defense and leave it incomplete. Yet such incompleteness also characterizes the state of the art in arguing for majority rule. So in addition to.. (shrink)

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and Shoenfield’s unramified (...) forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

Cet essai à teneur philosophique s'intéresse à la force d'évocation et d'invocation de la musique. à son charme magique dirait Nietzsche. Comment la musique me parle-t-elle '? Que suggère-t-elle? Pourquoi parle-t-elle de moi? Quelle fonction a-t-elle dans l'humanité '? L'argumentation de l'auteur utilise la philosophie et la psychologie musicales afin d'expliquer la scandaleuse disproportion entre la puissance à dire de la musique et l'inévidence foncière de ce qu'elle dit (pour reprendre les mots de Jankelé?itch). disproportion en laquelle réside selon l'auteur (...) tout son mystère, en laquelle réside tout son sens. Ce texte court mais dense s'adresse aux musicologues ainsi qu'à tout lecteur ou musicien éclairé voulant densifier son horizon de compréhension de la musique et de ses effets. Il puise ses exemples dans de nombreux corpus : musiques classique et romantique, avant-gardes, jazz, musiques de film."--Page 4 of cover. (shrink)

I examine Harvey Brown’s account of relativity as dynamic and constructive theory and Michel Janssen recent criticism of it. By contrasting Einstein’s principle-constructive distinction with a related distinction by Lorentz, I argue that Einstein's distinction presents a false dichotomy. Appealing to Lorentz’s distinction, I argue that there is less of a disagreement between Brown and Janssen than appears initially and, hence, that Brown’s view presents less of a departure from orthodoxy than it may seem. Neither the kinematics-dynamics distinction nor Einstein’s (...) principle- and constructive theory distinction ultimately capture their disagreement, which may instead be a disagreement about the role of modality in science and the explanatory force of putatively nomic constraints. (shrink)

This essay proposes that What Is Philosophy? (1991), written in collaboration with Félix Guattari, not only presents a summary of Gilles Deleuze’s late creative period and, to some extent, a recapitulation of his entire oeuvre but also constitutes his third masterwork. The essay begins by tracing Deleuze’s three periods, especially the development of his thought during the last period, and the process of writing his final book. Then it explores the inextricable connection between the method of creation that results from (...) What Is Philosophy? and its stylistic devices. Through its refined composition and succinct prose, the book puts itself at the service of the approach that leads to bringing forth philosophical concepts, and it attunes the reader to the creative activity. Thus, like The Logic of Sense (1969) and A Thousand Plateaus (1980), it forms a masterwork in which Deleuze’s philosophical efforts that spanned a whole period are condensed and mobilized such that while reading one finds oneself exposed to these efforts and is forced to continue them. (shrink)

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse $$2^\omega $$ 2 ω to $$\omega $$ ω, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martinʼs number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.

We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.

In this article we give a forcing characterization for the Ramsey property of Σ 1 2 -sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of Σ 1 2 -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.

We present a version for κ-distributive ideals over a regular infinite cardinal κ of some of the combinatorial results of Mathias on happy families. We also study an associated notion of forcing, which is a generalization of Mathias forcing and of Prikry forcing.

We study ultrafilters produced by forcing, obtaining different combinatorics and related Rudin-Keisler ordering; in particular we answer a question of Baumgartner and Taylor regarding tensor products of ultrafilters. Adapting a method of Blass and Mathias, we show that in most cases the combinatorics satisfied by the ultrafilters recapture the forcing notion in the Lévy model.

In this paper, we answer a question asked in Koepke et al. regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.

We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy [Formula: see text][Formula: (...) see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing. (shrink)

We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We generically construct a model in which the [math]-separation property is true, i.e. every pair of disjoint [math]-sets can be separated by a [math]-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) [math]-separation property is true.

We give proofs of Ramsey’s and Hindman’s theorems in which the corresponding homogeneous sets are found with a forcing argument. The object of this paper is the study of the partial order involved in the proof of Hindman’s theorem. We are going to denote it by PFIN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P}_{FIN}}$$\end{document}. As a main result, we prove that Mathias forcing does not add Matet reals, which implies that PFIN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P}_{FIN}}$$\end{document} is not equivalent to Mathias forcing. (shrink)

Es conocido que el método de forcing es una de las técnicas de construcción de modelos más importantes de la Teoría de conjuntos en la actualidad, siendo el mismo muy útil para investigar problemas de matemática y/o de fundamentos de la matemática. El destacado matemático Joan Bagaria afirma lo siguiente sobre el método de forcing en su artículo "Paul Cohen y la técnica del forcing" (Gaceta de la Real Sociedad Matemática Española, Vol. 2, Nº 3, 1999, págs (...) 543-553) : "Aunque Cohen recibió la medalla Fields por su demostración de la independencia de la hipótesis del continuo y del axioma de elección, su contribución va mucho más allá de estos problemas. Su nuevo método, el forcing, no sólo ha permitido resolver un sinfín de problemas importantes en prácticamente todas las áreas de las matemáticas, si no que ha cambiado para siempre nuestra concepción de la matemática como ciencia". El objetivo principal del siguiente trabajo es estudiar el método de forcing describiendo algunas de sus aplicaciones (forcing de Cohen, forcing aleatorio, forcing de Mathias, forcing de Sacks, forcing de Silver, etc.), y ofreciendo una aproximación a sus fundamentos metamatemáticos. Se aspira que estas notas sirvan de apoyo para aprender dicho método. (shrink)

We investigate the effect of adding a single real on cardinal invariants associated with the continuum. We show:1. adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1;2. Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH;3. Miller's rational perfect set forcing preserves the axiom MA.

This book is a stimulating and engaging discussion of philosophical issues in the foundations of classical electromagnetism. In the rst half, Frisch argues against the standard conception of the theory as consistent and local. The second half is devoted to the puzzle of the arrow of radiation: the fact that waves behave asymmetrically in time, though the laws governing their evolution are temporally symmetric. The book is worthwhile for anyone interested in understanding the physical theory of electromagnetism, as well for (...) the views it presents on philosophical issues such as causation, counterfactuals, laws, scienti c theories, models, and explanation. While philosophers of physics tend to focus on quantum mechanics and relativity, Frisch’s book shows that there are deep foundational issues in classical physics, equally worthy of attention. That said, let me lodge disagreement on some key points. Frisch argues from an alleged inconsistency in classical electromagnetism— that Maxwell’s equations, the Lorentz force law, and the conservation of energy cannot be jointly true—to the conclusion that the standard view of scienti c theories as a formalism plus an interpretation is incorrect. Consistency is a necessary condition of any view on which scienti c theories give us an account of “ways the world could be” (Frisch, , ). Since classical electromagnetism is successfully used by practicing physicists, consistency must be just one criterion of theory choice weighed equally among others. This is an intriguing idea, but I am not sure that consistency can be given up so easily. That road leads dangerously close to accepting orthodox ‘Copenhagen’ quantum mechanics. Surely the inconsistency of.. (shrink)

If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in (...) all the “classical” models of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle $S_{\mathrm{fin}}$ on subsets of the Cantor space. (shrink)

An analogue of Mathias forcing is studied in connection of free Boolean algebras and nowhere dense ultrafilters. Some applications to rigid Boolean algebras are given.

We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k -colored Milliken-Taylor ultrafilters have at least k +1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model.

1. Setting off: An introduction. 1.1. A measure of motivation. 1.2. Computable mathematics. 1.3. Reverse mathematics. 1.4. An overview. 1.5. Further reading -- 2. Gathering our tools: Basic concepts and notation. 2.1. Computability theory. 2.2. Computability theoretic reductions. 2.3. Forcing -- 3. Finding our path: Konig's lemma and computability. 3.1. II[symbol] classes, basis theorems, and PA degrees. 3.2. Versions of Konig's lemma -- 4. Gauging our strength: Reverse mathematics. 4.1. RCA[symbol]. 4.2. Working in RCA[symbol]. 4.3. ACA[symbol]. 4.4. WKL[symbol]. 4.5. (...) [symbol]-models. 4.6. First order axioms. 4.7. Further remarks -- 5. In defense of disarray -- 6. Achieving consensus: Ramsey's theorem. 6.1. Three proofs of Ramsey's theorem. 6.2. Ramsey's theorem and the arithmetic hierarchy. 6.3. RT, ACA[symbol], and the Paris-Harrington theorem. 6.4. Stability and cohesiveness. 6.5. Mathias forcing and cohesive sets. 6.6. Mathias forcing and stable colorings. 6.7. Seetapun's theorem and its extensions. 6.8. Ramsey's theorem and first order axioms. 6.9. Uniformity -- 7. Preserving our power: Conservativity. 7.1. Conservativity over first order systems. 7.2. WKL[symbol] and II[symbol]-conservativity. 7.3. COH and r-II[symbol]-conservativity -- 8. Drawing a map: Five diagrams -- 9. Exploring our surroundings: The world below RT[symbol]. 9.1. Ascending and descending sequences. 9.2. Other combinatorial principles provable from RT[symbol]. 9.3. Atomic models and omitting types -- 10. Charging ahead: Further topics. 10.1. The Dushnik-Miller theorem. 10.2. Linearizing well-founded partial orders. 10.3. The world above ACA[symbol]. 10.4. Still further topics, and a final exercise. (shrink)

In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all classical results known for the Ramsey property also hold for the dual-Ramsey property. We will also show that the dual-Ramsey property is closed under a generalized Suslin operation (the similar result for the Ramsey property was proved by Matet). Further (...) we compare two notions of forcing, the Mathias forcing and a dual form of it, and will give some symmetries between them. Finally we give some relationships between the dual-Mathias forcing and the dual-Ramsey property. (shrink)

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are $${\mathcal {B}}$$ B -Canjar for any countably directed unbounded family $${\mathcal {B}}$$ B of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that $${\mathfrak {b}}=\omega _1$$ b = ω 1 in every extension by (...) the above forcing notions. (shrink)

A model of ZFC is constructed in which the distributivity cardinal h is 2 ℵ 0 = ℵ 2 , and in which there are no ω 2 -towers in [ω] ω . As an immediate corollary, it follows that any base-matrix tree in this model has no cofinal branches. The model is constructed via a form of iterated Mathias forcing, in which a mixture of finite and countable supports is used.

We prove that Martin’s Axiom implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while \ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel’s conjecture.

Henle, Mathias, and Woodin proved in [21] that, provided that${\omega }{\rightarrow }({\omega })^{{\omega }}$holds in a modelMof ZF, then forcing with$([{\omega }]^{{\omega }},{\subseteq }^*)$overMadds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model$M[\mathcal {U}]$, where$\mathcal {U}$is a Ramsey ultrafilter, with many properties of the original modelM. This begged the question of how important (...) the Ramseyness of$\mathcal {U}$is for these results. In this paper, we show that several classes of$\sigma $-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme,k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras$\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$,$2\le {\alpha }<{\omega }_1$, forcing non-p-points also produce barren extensions. (shrink)

We prove that RCA₀ + RT $RT\begin{array}{*{20}{c}} 2 \\ 2 \\ \end{array} $ ̸͢ WKL₀ by showing that for any set C not of PA-degree and any set A, there exists an infinite subset G of A or ̅Α, such that G ⊕ C is also not of PA-degree.

A generalization of Příkrý's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkrý generic sequences reminiscent of Mathias' criterion for Příkrý genericity is provided, together with a maximality theorem which states that a generalized Příkrý sequence almost contains every other one lying in the same extension.This forcing can be used to falsify the covering (...) lemma for a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ. (shrink)

This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L$, JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of (...) $\omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show that there are no infinite mad families in $L$ under large cardinals and that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$, respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L$. Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T, then $L$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.prepared by Zach Norwood.E-mail: [email protected]. (shrink)

This article challenges the general thesis that an unconditional basic income, set at the highest sustainable level, is required for maximizing the income-leisure opportunities of the least advantaged, when income varies according to the responsible factor of labor input. In a linear optimal taxation model (of a type suggested by Vandenbroucke 2001) in which opportunities depend only on individual productivity, adding the instrument of a uniform wage subsidy generates an array of undominated policies besides the basic income maximizing policy, including (...) a “zero basic income” policy which equalizes the post-tax wage rate. The choice among such undominated policies may be guided by distinct normative criteria which supplement the maximin objective in various ways. It is shown that most of these criteria will be compatible with, or actually select, the zero basic income policy and reject the basic income maximizing one. In view of the model's limited realism, the force of this main conclusion is discussed both in relation to Van Parijs' argument for basic income in Real Freedom for All (1995) and to some key empirical conditions in the real world. Footnotes1 I am grateful for useful comments on various stages of this paper by Marc Fleurbaey, Susan Hurley, Mathias Hild, Dirk Van de gaer, Frank Vandenbroucke, Alex Voorhoeve, Roland van der Veen, Andrew Williams, Chris Woodard and an anonymous referee. Above all I thank Loek Groot for his contribution. The research has also been supported by the Netherlands Organization for Scientific Research (NWO), and the Social and Political Theory Program of the Research School of Social Sciences, Australian National University. (shrink)

We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$. Also, we discuss the associated variants $$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y| \omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$, and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; (...) and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal. (shrink)

I criticize two accounts of the temporal asymmetry of electromagnetic radiation - that of Huw Price, whose account centrally involves a reinterpretation of Wheeler and Feynman's infinite absorber theory, and that of Dieter Zeh. I then offer some reasons for thinking that the purported puzzle of the arrow of radiation does not present a genuine puzzle in need of a solution.

The grounds of justice -- "Un pouvoir ordinaire": shared membership in a state as a ground of -- Justice -- Internationalism versus statism and globalism: contemporary debates -- What follows from our common humanity? : the institutional stance, human rights, and nonrelationism -- Hugo Grotius revisited : collective ownership of the Earth and global public reason -- "Our sole habitation" : a contemporary approach to collective ownership of the earth -- Toward a contingent derivation of human rights -- Proportionate use (...) : immigration and original ownership of the Earth -- "But the earth abideth for ever" : obligations to future generations -- Climate change and ownership of the atmosphere -- Human rights as membership rights in the global order -- Arguing for human rights : essential pharmaceuticals -- Arguing for human rights : labor rights as human rights -- Justice and trade -- The way we live now -- "Imagine there's no countries" : a reply to John Lennon -- Justice and accountability : the state -- Justice and accountability : the World Trade Organization. (shrink)

Aratus has been notorious for his wordplay since the first decades of his reception. Hellenistic readers such as Callimachus, Leonidas, or ‘King Ptolemy’ seem to have picked up on the pun on the author's own name atPhaenomena2, as well as on the famous λεπτή acrostic atPhaen.783–6 that will be revisited here. Three carefully placed occurrences of the adjective have so far been uncovered in the passage, but for a full appreciation of its elegance we must note that Aratus has set (...) his readers up to notice a fourth. (shrink)

Mathias Frisch provides the first sustained philosophical discussion of conceptual problems in classical particle-field theories. Part of the book focuses on the problem of a satisfactory equation of motion for charged particles interacting with electromagnetic fields. As Frisch shows, the standard equation of motion results in a mathematically inconsistent theory, yet there is no fully consistent and conceptually unproblematic alternative theory. Frisch describes in detail how the search for a fundamental equation of motion is partly driven by pragmatic considerations (...) (like simplicity and mathematical tractability) that can override the aim for full consistency. The book also offers a comprehensive review and criticism of both the physical and philosophical literature on the temporal asymmetry exhibited by electromagnetic radiation fields, including Einstein's discussion of the asymmetry and Wheeler and Feynman's influential absorber theory of radiation. Frisch argues that attempts to derive the asymmetry from thermodynamic or cosmological considerations fail and proposes that we should understand the asymmetry as due to a fundamental causal constraint. The book's overarching philosophical thesis is that standard philosophical accounts that strictly identify scientific theories with a mathematical formalism and a mapping function specifying the theory's ontology are inadequate, since they permit neither inconsistent yet genuinely successful theories nor thick causal notions to be part of fundamental physics. (shrink)

Much has been written on the role of causal notions and causal reasoning in the so-called 'special sciences' and in common sense. But does causal reasoning also play a role in physics? Mathias Frisch argues that, contrary to what influential philosophical arguments purport to show, the answer is yes. Time-asymmetric causal structures are as integral a part of the representational toolkit of physics as a theory's dynamical equations. Frisch develops his argument partly through a critique of anti-causal arguments and (...) partly through a detailed examination of actual examples of causal notions in physics, including causal principles invoked in linear response theory and in representations of radiation phenomena. Offering a new perspective on the nature of scientific theories and causal reasoning, this book will be of interest to professional philosophers, graduate students, and anyone interested in the role of causal thinking in science. (shrink)