In lieu of an abstract, here is a brief excerpt of the content:166 HUME'S INTEREST IN NEWTON AND SCIENCE Many writers have been forced to examine — in their treatments of Hume's knowledge of and acquaintance with scientific theories of his day — the related questions of Hume's knowledge of and acquaintance with Isaac Newton and of the nature and extent of Newtonian influences upon Hume's thinking. Most have concluded that — in some sense — Hume was acquainted with and (...) influenced by Newton's thought in particular and scientific thought in general. The genesis of this paper is the recent point of view put forward by Peter Jones which challenges the many permutations of this almost ritualistic standard line by removing Hume entirely from the Newtonian and the scientific scenes of thought. Jones argues that Hume knew less about Newton and science, and needed to know less about Newton and 2 science, than he believes is required by the above interpretation. Indeed, Jones argues that Hume's fundamental assumptions, which, according to Jones, derive ultimately from a form of Ciceronian humanism, drive a "wedge" between Newton's thought and that of 3 Hume. Even Hume's introductory remarks in the Treatise about his universal "science of man" are, for Jones, a declaration of independence from the materialistic trend (as Jones sees it) of Newtonian 4 science and not, as so many commentators have maintained — however tenuously or strongly evidence for linkage of Hume's project with Newtonian or scientific thought. Jones baldly argues that Hume totally lacked interest in science in general and in Newton and Newtonian science in particular. Following J. H. Burton's observation that Hume's work is surprisingly free from the "opinions" of contemporary scientists, 167 Jones states there is no evidence that Hume ever studied science at the University of Edinburgh or that he "pursued" scientific studies of any formal sort. Regarding Newtonian scientific thought, he emphasizes the paucity of specifically Newtonian scientific textbooks in the early eighteenth century 7 which might have been available for Hume to study and argues that nowhere in Hume's writings is there evidence of precise and detailed knowledge of Newton's science beyond what is available in Q Chamber's Cyclopaedia. Jones acknowledges that, in the Introduction to the Treatise, Hume utilizes a "general version" of Newton's "Regulae Philosophandi" from the beginning of Book III of Newton's Principia. Nevertheless, in Jones' view, Hume's fundamentally humanistic orientation separates him completely from g any Newtonian influence. Finally, according to Jones, Hume does not betray the least bit of knowledge of Newton's mathematics and its role in Newton's experimental methodology. On this evidence Jones grounds his central claim of Hume's "total lack of interest in contemporary science." What references there are to Newton and to science in Hume's works Jones finds "traceable to essentially literary predecessors such as Fontenelle or Montesquieu, or to standard works of theologians ] 2 or free-thinkers." The absence of clearly direct references to what Jones feels are scientific works results both from Hume's "total lack of interest in science" and from his commitment to a form of Ciceronian humanism which is "inimical" to what Jones finds to be the obvious materialistic tendencies of 13 science in the early modern period. Jones' account of the Ciceronian and French contexts of Hume's thought is excellent. But his claim that Hume had no interest whatsoever in science 168 is simply too strong and finally forces us to view science in Hume's day as equivalent to science in our own time, a manifestly anachronistic point of view. Throughout this paper, my argument will be conditioned by my view that Hume's interest in science cannot be separated from his epistemology or his religious scepticism. Hume's interest in science was precisely that of a man of letters of the eighteenth century vitally engaged in determining the proper use of scientific methodology in establishing the limits of the secular science of man once it has 14 been freed from the fetters of theology. Hume's interest in theological and epistemological issues inevitably gave rise to a strong interest on his part in the science of his day and in Newton's contributions... (shrink)

We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We (...) also show that the consistency strength of the absoluteness of under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal. (shrink)

We show that L absoluteness for semi-proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L absoluteness for proper forcings. By [7], L absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi-Proper Forcing Axiom is equiconsistent with the Bounded Proper Forcing Axiom , which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum (...) is much stronger than BSPFA in that if BMM holds, then for every X ∈ V , X# exists. (shrink)

The French philosopher and intellectual historian Pierre Hadot (1922-2010) is known primarily for his conception of philosophy as spiritual exercise, which was an essential reference for the later Foucault. An aspect of his work that has received less attention is a set of methodological reflections on intellectual history and on the relationship between philosophy and history. Hadot was trained initially as a philosopher and was interested in existentialism as well as in the convergence between philosophy and poetry. Yet he chose (...) to become a historian of philosophy and produced extensive philological work on neo-Platonism and ancient philosophy in general. He found a philosophical rationale for this shift in his encounter with Wittgenstein's philosophy in the mid-1950s (Hadot was one of Wittgenstein's earliest French readers and interpreters). For Hadot, ancient philosophy must be understood as a series of language games, and each language game must be situated within the concrete conditions in which it happened. The reference to Wittgenstein therefore supports a strongly contextualist and historicist stance. It also supports its exact opposite: presentist appropriations of ancient texts are entirely legitimate, and they are the only way ancient philosophy can be existentially meaningful to us. Hadot addresses the contradiction by embracing it fully and claiming that his own practice aims at a coincidence of opposites (a concept borrowed from the Heraclitean tradition). For Hadot the fullest and truest way of doing philosophy is to be a philosopher and a historian at the same time. (shrink)

Blood supplies have become indexes of national security and the public good. While blood shortages can provoke anxiety, controversies continue to erupt in many countries over proper donor screening, especially with reference to HIV. This article sketches these dynamics in several global settings, focusing especially on activist efforts by gay men to reform exclusionary blood donor guidelines. The contours of the debate recall familiar conflicts between the putative demands of public health and the rights of individuals in the era (...) of AIDS. However, if gay activists marshal a discourse of individual rights vis-a-vis forms of institutional exclusion, they also seek a broader shift in social and cultural understandings of gay identity. To capture this complex interplay of citizenship and sociality, risk and responsibility, the article introduces the notion of `vital publics' to refer to the peculiar associational form represented by blood supplies. Vital publics are kinds of embodied association elicited through the generalized exchange of body — in this case, blood. Hailed to `give life' by the jargon of the pervasive social marketing of varied blood service systems, activists seek to contribute to the life of the `vital public' that transfusion medicine calls into being. (shrink)

We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with (...) parameters is forceable, then it is true. Further, if for every real x, $x^{\sharp}$ exists, and the second uniform indiscernible is less than $\omega_2$ , then the same holds for $\Sigma^1_4$ sentences. (shrink)

I provide indestructibility results for large cardinals consistent with V = L, such as weakly compact, indescribable and strongly unfoldable cardinals. The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by <κ-closed. κ-proper forcing. This class of posets includes for instance all <κ-closed posets that are either κ -c.c, or ≤κ-strategically closed as well as finite iterations of such posets. Since strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, (...) the Main Theorem therefore makes these two large cardinal notions similarly indestructible. Finally. I apply the Main Theorem forcing extension preserving all strongly unfoldable cardinals in which every strongly unfoldable cardinal κ is indestructible by <κ-closed. κ-proper forcing. (shrink)

We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

Suppose that T^∗ is an ω_1-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T^∗) for proper forcings which preserve these properties of T^∗. We prove that PFA(T^∗) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ω_1, and the P-ideal dichotomy. On the other hand, PFA(T^∗) implies some of the consequences of diamond principles, such as (...) the existence of Knaster forcings which are not stationarily Knaster. (shrink)

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper (...) forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.The second example aims to show that the transfer principle ↠ fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs. (shrink)

This paper argues that it will be important for new genomic technologies to recognize the limits of traditional approaches to informed consent, so that other-regarding implications of genomic information can be properly contextualized and individual rights respected. Respect for individual autonomy will increasingly require dynamic consideration of the interrelated dimensions of individual and broader community interests, so that the interests of one do not undermine fundamental interests of the other. In this, protection of individual rights will be a complex interplay (...) between individual and community concerns. (shrink)

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. If there (...) is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal 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}$${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$$\end{document}. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results. (shrink)

The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of $H(\omega _2)$. A broad program in modern Set Theory is searching for strong forcing axioms beyond $\omega _1$. In other words, one would like to figure out the structural properties of taller initial segments of the universe. However, the classical techniques of forcing iterations seem unable to bypass the obstacles, as the resulting forcings axioms beyond $\omega _1$ have not thus far been (...) strong enough! However, with his celebrated work on generalised side conditions, I. Neeman introduced us to a novel paradigm to iterate forcings. In particular, he could, among other things, reprove the consistency of the Proper Forcing Axiom using an iterated forcing with finite supports. In 2015, using his technology of virtual models, Veličković built up an iteration of semi-proper forcings with finite supports, hence reproving the consistency of Martin’s Maximum, an achievement leading to the notion of a virtual model.In this thesis, we are interested in constructing forcing notions with finitely many virtual models as side conditions to preserve three uncountable cardinals. The thesis constitutes six chapters and three appendices that amount to 118 pages, where Section 1 is devoted to preliminaries, and Section 2 is a warm-up about the scaffolding poset of a proper forcing. In Section 3, we present the general theory of virtual models in the context of forcing with sets of models of two types, where we, e.g., define the “meet” between two virtual models and prove its properties.The main results are joint with Boban Veličković, and partly appeared in Guessing models and the approachability ideal, J. Math. Log. 21 (2021).Pure Side Conditions. In Section 4, we use two types of virtual models (countable and large non-transitive ones induced by a supercompact cardinal, which we call Magidor models) to construct our forcing with pure side conditions. The forcing covertly uses a third type of models that are transitive. We also add decorations to the conditions to add many clubs in the generic $\omega _2$. In contrast to Neeman’s method, we do not have a single chain, but $\alpha $ -chains, for an ordinal $\alpha $ with $V_\alpha \prec V_\lambda $. Thus, starting from suitable large cardinals $\kappa <\lambda $, we construct a forcing notion whose conditions are finite sets of virtual models described earlier. The forcing is strongly proper, preserves $\kappa $, and has the $\lambda $ -Knaster property. The relevant quotients of the forcing are strongly proper, which helps us prove strong guessing model principles. The construction is generalisable to a ${<}\mu $ -closed forcing, for any given cardinal $\mu $ with $\mu ^{<\mu }=\mu <\kappa $.The Iteration Theorem. In Section 5, we use the forcing with pure side conditions to iterate a subclass of proper and $\aleph _2$ -c.c. forcings and obtain a forcing axiom at the level of $\aleph _2$. The iterable class is closely related to Asperó–Mota’s forcing axiom for finitely proper forcings.Guessing Model Principles. Section 6 encompasses the main parts of the thesis. We prove the consistency of the guessing principle $\mathrm {GMP}^+(\omega _3,\omega _1)$ that states for any cardinal ${\theta \geq \omega _3}$, the set of $\aleph _2$ -sized elementary submodels M of $H(\theta )$, which are the union of an $\omega _1$ -continuous $\in $ -chain of $\omega _1$ -guessing, I.C. models is stationary in $\mathcal P_{\omega _3}(H(\theta ))$. The consistency and consequences of this principle are demonstrated in the following diagram. We also prove that one can obtain the above guessing models in a way that the $\omega _1$ -sized $\omega _1$ -guessing models remain $\omega _1$ -guessing model in any outer transitive model with the same $\omega _1$, and we denote this principle by $\rm{SGMP}^+(\omega_3,\omega_1)$.In the following diagram, $\mathrm{TP}$ stands for the tree property; $w\mathrm{KH}$ stands for the weak Kurepa Hypothesis; $\mathrm{MP}$ stands for Mitchell property, i.e., the approachability ideal is trivial modulo the nonstationary ideal; $\mathrm{AP}$ stands for the approachability property; $\mathrm {AMTP}(\kappa ^+)$ states that if $2^\kappa <\aleph _{\kappa ^+}$, then every forcing which adds a new subset of $\kappa ^+$ whose initial segments are in the ground model, collapses some cardinal $\leq 2^{\kappa }$. The dotted arrow denotes the relative consistency, while others are logical implications.Appendices. Appendix A includes merely the above diagram. Appendix B presents a proof of the Mapping Reflection Principle with finite conditions under $\mathrm {PFA}$. Appendix C contains open problems. Finally, the thesis’s bibliography consists of 42 items.Abstract prepared by Rahman MohammadpourE-mail: [email protected]: https://theses.hal.science/tel-03209264. (shrink)

. From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact and strong cardinals precisely coincide. In this model, it is the case that no strongly compact cardinal \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} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $2^\kappa = \kappa^+$\end{document} supercompact.

A strong antidiamond principle is shown to be consistent with . This principle can be stated as a “P-ideal dichotomy”: every P-ideal on ω1 either has a closed unbounded subset of ω1 locally inside of it, or else has a stationary subset of ω1 orthogonal to it. We rely on Shelah’s theory of parameterized properness for iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application (...) of the iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic. (shrink)

We construct creature forcings with strong Axiom A that specialize a given Aronszajn tree. We work with tree creature forcing. The creatures that live on the Aronszajn tree are normed and have the halving property. We show that our models fulfill ℵ1=d

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In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including (...) all ω-proper forcing notions and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions. (shrink)

The main purpose of this dissertation is to apply and develop new forcing techniques to obtain models where several cardinal characteristics are pairwise different as well as force many (even more, continuum many) different values of cardinal characteristics that are parametrized by reals. In particular, we look at cardinal characteristics associated with strong measure zero, Yorioka ideals, and localization and anti-localization cardinals.In this thesis we introduce the property “F-linked” of subsets of posets for a given free filter F on (...) the natural numbers, and define the properties “ $\mu $ -F-linked” and “ $\theta $ -F-Knaster” for posets in a natural way. We show that $\theta $ -F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. These kinds of posets led to the development of a general technique to construct $\theta $ - $\textrm {Fr}$ -Knaster posets (where $\textrm {Fr}$ is the Frechet ideal) via matrix iterations of ${<}\theta $ -ultrafilter-linked posets (restricted to some level of the matrix). The latter technique allows proving consistency results about Cichoń’s diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal characteristics associated with it are pairwise different. Another important application is to show that three strongly compact cardinals are enough to force that Cichoń’s diagram can be separated into 10 different values. Later on, it was shown by Goldstern, Kellner, Mejía, and Shelah that no large cardinals are needed for Cichoń’s maximum (J. Eur. Math. Soc. 24 (2022), no. 11, p. 3951–3967).On the other hand, we deal with certain types of tree forcings including Sacks forcing, and show that these increase the covering of the strong measure zero ideal $\mathcal {SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal characteristics of the continuum. Even more, Sacks forcing can be used to force that $\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cov}}(\mathcal {SN})<\operatorname {\mathrm{cof}}(\mathcal {SN})$, which is the first consistency result where more than two cardinal characteristics associated with $\mathcal {SN}$ are pairwise different. To obtain another result in this direction, we provide bounds for $\operatorname {\mathrm{cof}}(\mathcal {SN})$, which generalizes Yorioka’s characterization of $\mathcal {SN}$ (J. Symbolic Logic 67.4 (2002), p. 1373–1384). As a consequence, we get the consistency of $\operatorname {\mathrm{add}}(\mathcal {SN})=\operatorname {\mathrm{cov}}(\mathcal {SN})<\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cof}}(\mathcal {SN})$ with ZFC (via a matrix iteration forcing construction).We conclude this thesis by combining creature forcing approaches by Kellner and Shelah (Arch. Math. Logic 51.1–2 (2012), p. 49–70) and by Fischer, Goldstern, Kellner, and Shelah (Arch. Math. Logic 56.7–8 (2017), p. 1045–1103) to show that, under CH, there is a proper $\omega ^\omega $ -bounding poset with $\aleph _2$ -cc that forces continuum many pairwise different cardinal characteristics, parametrized by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localization and anti-localization cardinals, respectively. This answers several open questions from Klausner and Mejía (Arch. Math. Logic 61 (2022), pp. 653–683).Abstract prepared by Miguel Antonio Cardona-MontoyaE-mail: [email protected]: https://repositum.tuwien.at/handle/20.500.12708/19629. (shrink)

This article continues Rosłanowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ + (for a strongly inaccessible cardinal λ).

With the principle of tour de force, we refer to the use of the power of moral legality, the strength of statutes, and the fairness of judgments. A quantum force of moral legality and legal morality serves as an imperative force in the implementation of fair criminal justice, as well as in the prevention of future victims across the globe. Contrary to positivist ideas, the simple notion of morality contains within itself the very essence of international criminal norms. If the (...) source of morality did not exist in the structure of our criminal justice, we could no longer call it criminal justice and would instead have to call it criminal revenge. Beyond the moral essence of international criminal justice, we also challenge the hidden illegality and immorality of certain attitudes. This is done in order to prevent acts which violate human rights law and criminal justice provisions. The codes of international human rights law cannot properly function without the comprehensive nature of morality which elevates the moral virtue of international criminal justice and strengthens the tour de force. One may observe that the principle of the tour de force of the moral virtue within the system of international criminal justice loses its legal power when governments attempt to justify their unjust decisions. This is the main reason why the major potential value of the tour de force of the moral virtue of human rights law and international criminal justice is not as strong as it should be. In order to achieve a vision of the tour de force of moral virtue, we need to establish a devoted influential ethical vision of the truth beyond political tradition. An unparalleled philosophical piece of literature for libraries. (shrink)

The current paper proves the results announced in [5]. We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and ω-Erdos cardinals. They are characterized by the existence of "O # -like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness for proper forcings. In particular, said absoluteness does not imply Π 1 1 determinacy.

The general thrust of Neil Levy's paper is that a certain amount of paternalism should be viewed as compatible with liberalism.1 I am not quite convinced that what he is defending is properly paternalism. In addition, I am not entirely sure what his proposal is. Here are a few comments about several points in the paper.1. A possibly small question is worth raising when Levy says, ‘That is, the state may not interfere with individuals’ actions, even to promote their own (...) conception of the good. This claim is typically justified on epistemic grounds: individuals are best placed to judge for themselves how to pursue their conception of the good.’But I don't think that this is quite the right picture, though admittedly, it has often been suggested by liberals. Still: if it's a matter of ‘epistemic’ grounds, then that implies that there is something out there to be known, so that the question is, how do we go about knowing it? But this way of putting it is misleading. For liberalism says that whenever anybody, B, other than the individual we are considering, say A, claims that what is good for A is something ‘out there’—something that can strongly contradict what A thinks for his own case—then that outsider has no authority to override A's preference. This isn't epistemic—it's a moral position.If we were to talk of ‘epistemic’ here, presumably what is known in this case is how the individual …. (shrink)

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.

Although Newton carefully eschews questions about gravity’s causal basis in the published Principia, the original version of his masterwork’s third book contains some intriguing causal language. “These forces,” he writes, “arise from the universal nature of matter.” Such remarks seem to assert knowledge of gravity’s cause, even that matter is capable of robust and distant action. Some commentators defend that interpretation of the text—a text whose proper interpretation is important since Newton’s reasons for suppressing it strongly suggest that (...) he continued to endorse its ideas. This article argues that the surface appearance of Newton’s causal language is deceptive. What does New- ton intend with his causal language if not a full causal hypothesis? His remarks actually indicate a way of considering the force mathematically, something he contrasts to the structure of the force as it really is in nature. In explaining that, he identifies a significant disjunction between the physical force itself and mathematical ways of considering it, and the text’s significance lies in its view of the force’s structure and in the questions raised about the relationship between mathematical representations and the physical world. (shrink)

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s (...) found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L. One of the nice things about AD is that the theory ZF + AD + V = L appears as a choiceless “completion” of ZF in that any interesting question seems to find an at least attractive answer in that theory. Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension. (shrink)

We show under that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under if we additionally assume that the set of Borel codes for I‐positive sets is. If we do not assume, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar (...) to the one introduced by Bagaria and Bosch [2], we show under without using that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is strongly proper assuming every set of reals is ∞‐Borel and there is no ω1‐sequence of distinct reals. In particular, the same conclusion holds in a Solovay model. (shrink)

We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at κ then every stationary subset of reflects. It will also (...) be demonstrated that MRP always fails in a generic extension by Prikry forcing. (shrink)

The current paper proves the results announced in [5]. We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and $\omega$-Erdos cardinals. They are characterized by the existence of "O$^#$-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L absoluteness for proper forcings. In particular, said absoluteness does not imply $\Pi^1_1$ determinacy.

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ 1 -definable in (H(ω 2 ), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω 1 , ω 1 * , C, C * where X is any (...) suborder of the reals of size ω 1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at κ unless stationary subsets of S κ + ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω 2 ). (shrink)

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency (...) strength of the bounded proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure. (shrink)

In this paper we show that the Proper Forcing Axiom is preserved under forcing over any poset PP with the following property: In the generalized Banach–Mazur game over PP of length , Player II has a winning strategy which depends only on the current position and the ordinal indicating the number of moves made so far. By the current position we mean: The move just made by Player I for a successor stage, or the infimum of all (...) the moves made so far for a limit stage. As a consequence of this theorem, we introduce a weak form of the square principle and show that it is consistent with PFA. (shrink)

Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

κ-M-proper forcing, introduced in [K00] when κ = ω1, is a very powerful new technique for generic stepping up, subsuming all previous generic steppings up using auxiliary functions. A general framework for using κ-M-proper forcing is set out, and a couple of examples of such forcings, adding κ−-thin-very tall scattered spaces and long chains in P(κ) modulo <κ−, are given. These objects are not currently obtainable by the previously known techniques.

, a notion of forcing over E, the E-closure of L, is said to be effective if every sideways -generic extension preserves E-closure. There are set notions of forcing in E that do not preserve E-closure. The main theorem below asserts that is effective if and only if it is locally proper, a weak variant of Shelah's notion of proper.

We show how the use of a Laver function in the proof of the consistency, relative to the existence of a supercompact cardinal, of both the Proper Forcing Axiom and the Semiproper Forcing Axiom can be eliminated via the use of lottery sums of the appropriate partial orderings.

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω 1 with ω 1 generators, then there exists an uncountable $X \subseteq \omega_1$ , such that either [ X] ω ∩ I = ⊘ or $\lbrack X\rbrack^\omega \subseteq I$.

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.

We examine the differences between three standard classes of forcing notions relative to the way they collapse the continuum. It turns out that proper and semi-proper posets behave differently in that respect from the class of posets that preserve stationary subsets of \.

We show that MRP + MA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + (...) MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω 2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω 1 ) for all λ ≥ ω 2. (shrink)