We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters.

We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly-̰Σ₃¹ absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for Σ₃¹ absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence (...) we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions. (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 continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such.

We further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω 1 , 1)-morass, then there exists a ccc forcing of size ω 1 that adds an ω 2 -Suslin tree. (2) If there is a simplified (ω 1 , 2)-morass, then there exists a ccc forcing of size ω 1 that adds a 0-dimensional Hausdorff topology τ on ω (...) 3 which has spread s(τ) = ω 1 . While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserve GCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈X α | α < ω 2 〉 such that X α ⊆ ω 1 , X β — X α is finite and X α — X β has size ω 1 for all β < α < ω 2. (shrink)

We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin+. We introduce transitive nep and present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is a Suslin ccc ideal (e.g. Lebesgue-null or meager) and P is a transitive nep forcing (e.g. P is Suslin+) and P does not make any I-positive Borel set small, then P does not make any I-positive set small.

We prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal J (coded in the ground model) a cofinal subset of (J, ⊆*) is order isomorphic to (Q, ≤).

We give a proof of Hechler's theorem that any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\aleph_1$\end{document}-directed partial order can be embedded via a ccc forcing notion cofinally into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega^\omega$\end{document} ordered by eventual dominance. The proof relies on the standard forcing relation rather than the variant introduced by Hechler.

The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost (...) disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\mathcal {B\subseteq U}$ there is $X\in $ $\mathcal {U}$ such that $X\setminus B$ is finite for every $B\in \mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them.In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $\mathfrak {s\leq a}.$ None of the results in this chapter are due to the author.The second chapter is dedicated to a principle of Sierpiński. The principle $\left $ of Sierpiński is the following statement: There is a family of functions $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ such that for every $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ there is $n\in \omega $ for which $\varphi _{n}\left [ I\right ] =\omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=\left \{ f_{\alpha }\mid \alpha \alpha $ then $f_{\beta }\cap g$ is infinite. Miller showed that the principle of Sierpiński implies that non $\left =\omega _{1}.$ He asked if the converse was true, i.e., does non $\left =\omega _{1}$ imply the principle $\left $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $\left =\omega _{1}$ is enough to construct an $\mathcal {IE}$ -Luzin set. It is not hard to see that the $\mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $\left =\omega _{1}$ and every $\mathcal {IE}$ -Luzin set is meager.The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $\left $ or at least cov $\left $. Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of $\omega _{1}^{\omega }$ ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal.In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than $\mathfrak {c}$ can be extended to a Cohen indestructible MAD family is equivalent to $\mathfrak {b=c}$. A MAD family $\mathcal {A}$ is Shelah–Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ either there is $A\in \mathcal {I}\left $ such that $s\cap A\neq \emptyset $ for every $s\in X$ or there is $B\in \mathcal {I}\left $ that contains infinitely many elements of X $ denotes the ideal generated by $\mathcal {A}$ ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals. According to the author’s best knowledge, this is the strongest notion that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them.The fifth chapter is one of the most important chapters in the thesis. A MAD family $\mathcal {A}$ is called $+$ -Ramsey if every tree that branches into $\mathcal {I}\left $ -positive sets has an $\mathcal {I}\left $ -positive branch. Michael Hrušák’s first published question is the following: Is there a $+$ -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions.In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct several MAD families with different properties.In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský.prepared by Osvaldo Guzmán GonzálezE-mail : oguzman@matmor.unam.mxURL : https://arxiv.org/abs/1810.09680. (shrink)

We prove that the Sacks forcing collapses the continuum onto ${\frak d}$ , answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses $\omega_2$ then it forces $\diamondsuit_{\omega_{1}}$.

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)

There is a partial order ${\mathbb{P}}$ preserving stationary subsets of ω 1 and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω 1 over V also collapses ω 1 over ${V^{\mathbb{P}}}$ . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the (...) proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B. (shrink)

It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from (...) [3]. If A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)

In this paper we analyse some notions of amoeba for tree forcings. In particular we introduce an amoeba-Silver and prove that it satisfies quasi pure decision but not pure decision. Further we define an amoeba-Sacks and prove that it satisfies the Laver property. We also show some application to regularity properties. We finally present a generalized version of amoeba and discuss some interesting associated questions.

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

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. (...) We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports. (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).

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.

In lieu of an abstract, here is a brief excerpt of the content:Montaigne and the Coherence of EclecticismPierre ForceSince the publication of Pierre Hadot's essays on ancient philosophy by Arnold Davidson in 1995,2 Michel Foucault's late work on "the care of the self"3 has appeared in a new light. We now know that Hadot's work was familiar to Foucault as early as the 1950s.4 It is also clear that Foucault's notion of "techniques of the self" is very close to what (...) Hadot calls "spiritual exercises." At the same time, there are important differences between the views of these two philosophers, and Hadot has often expressed his regret that Foucault's untimely death prevented them from exploring these differences.5 One important point of disagreement was the status of eclecticism. In Foucault's interpretation of ancient philosophy, the "constitution of the self" implied a personal choice among disparate philosophical references: "Writing as a personal exercise done by the self and for the self, is an art of disparate truth."6 In a 1989 article Hadot argued from a historical point of view that, so far as Stoicism and Epicureanism were concerned, [End Page 523] eclecticism had no place in a mature practice of philosophy. Foucault's favorite example, Seneca's Letters to Lucilius, in which Stoic arguments were invoked along with Epicurean arguments, was a work for beginners. In a mature practice of Stoicism, one would stick to the arguments of the Stoic school, choose them for their coherence, and instead of trying to forge a spiritual identity for oneself through writing, one would "liberate oneself from one's individuality in order to raise oneself up to universality."7 Hadot added that "it is only in the New Academy—in the person of Cicero, for instance—that a personal choice is made according to what reason considers as most likely at a given moment."8 This discussion of Foucault may give the impression that Hadot was somewhat dismissive of eclecticism. Yet in a 2001 interview Hadot claimed that he had always admired Cicero's intellectual independence, and he proposed a reappraisal "of an attitude that has always had a bad reputation: eclecticism."9Something very similar is a stake in the work of Alexander Nehamas, as Lanier Anderson and Joshua Landy showed in their study of his Art of Living. In the conception of philosophy as a "way of life" (Hadot) or "care of the self" (Foucault) or "self-fashioning" (Nehamas), a central issue is the relationship between theoretical coherence and the coherence of the person who theorizes. Elaborating on Nehamas's chapter on Montaigne, Anderson and Landy argue that there is a move in Montaigne away from doctrinal coherence and towards a coherence of the self. As they put it, "the harmonious whole Montaigne commends [... ] is not a coherent body of fact or theory; it is the unified self of the theorizer."10 Anderson and Landy show that this brings up all kinds of difficulties: can this unified self be other than a fiction? If that is the case, how can philosophy be a way of life? Nehamas's answer is very much in the spirit of the late Foucault: there is a coherence of the self in Montaigne, and this coherence is the product of writing as a philosophical activity. The self is fashioned through writing.Because Montaigne writes in the ancient tradition of philosophy as a way of life, one may recall Hadot's suggestion that Foucault's notion of [End Page 524] "writing the self" is an intriguing but historically inaccurate description of ancient philosophical practice. But perhaps Hadot agrees with Foucault after all, since in his most recent interviews, he speaks favorably of eclecticism, a notion that is central to Foucault's analysis of self-fashioning through writing. The case of Montaigne is particularly interesting for these purposes, not only because the Essays seem to be the prototypical example of "writing the self," but also because eclecticism is both discussed and practiced throughout the Essays. I propose to take a fresh look at this issue by investigating the status of eclecticism in Montaigne's Essays. This must start with an examination of the philosophical tradition most closely associated with... (shrink)

We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree. In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, (...) and Radin forcing. (shrink)

Heinrich Hertz dedicated the last four years of his life to a systematic reformulation of mechanics. One of the main issues that troubled Hertz in the customary formulation of mechanics was a "logical obscurity" in the notion of force. However, it is unclear what this logical obscurity was, hence it is unclear how Hertz took himself to have avoided it. -/- In this paper, I argue that a subtle ambiguity in Newton's original laws of motion lay at the basis of (...) Hertz's concerns; an ambiguity which led to the development of two slightly different notions of force. I then show how Hertz avoided this ambiguity by deriving a unitary notion of force, thus dispelling the obscurity that lurked in the customary representation of mechanics. (shrink)

This is an encyclopedia entry on the notion of force as entering into physical science.

Dealing with conflicts seems to be a great challenge in society today. But not only in society. Higher education displays an air of resoluteness with certainty and security that disguises the conflicts and the fear of conflicts in a substantial number of subjects. If not in a state of denial, higher education avoids taking up conflicts over issues, for learning. The detailed investigation of Tagore’s pedagogical writings, with a focus on the importance of conflicts in education, reveals a genuine embrace (...) of conflicts for education. Conflicts are natural and necessary for the development and change of both the individual and society and the start of a ‘creative imagination’ to solve the problems we face in life. Contradictions in conflicts are not incompatible incongruities that are irreconcilable and mutually exclusive, but to the contrary, in need of each other. Contradictions do not represent different worlds but are substantial parts of one world: together they form a unity. Conflicting forces are necessary to create harmony. Creativity, imagination, love, art, and critical encounters are key elements in Tagore’s practical education aimed at finding similarities among people instead of emphasizing differences. Relations between people over the Identity of people. In other words, we need conflicts to become creative and imaginative human beings. The paper continues discussing conceptual and practical issues that seem necessary to get the teaching of conflicts in education off the ground. On the conceptual level, in particular our dealing with uncertainty and fear, the valuation of conflict and the need for uncertainty-researching education. On a practical level, I propose a certain teaching model, a supportive curriculum, a way of choosing genuine conflicts for education and finally, I argue for specific support and education of teachers, acknowledging the vital role teachers play in delivering the education that we need. (shrink)

The Lévy-Solovay Theorem [8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

The extender based Prikry forcing notion is being generalized to super compact extenders.

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We (...) also study the 1—1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

Abstract“Force dynamics” refers to a previously neglected semantic category—how entities interact with respect to force. This category includes such concepts as: the exertion of force, resistance to such exertion and the overcoming of such resistance, blockage of a force and the removal of such blockage, and so forth. Force dynamics is a generalization over the traditional linguistic notion of “causative”: it analyzes “causing” into finer primitives and sets it naturally within a framework that also includes “letting,”“hindering,”“helping,” and still further (...) class='Hi'>notions. Force dynamics, moreover, appears to be the semantic category that uniquely characterizes the grammatical category of modals, in both their basic and epistemic usages. In addition, on the basis of force dynamic parameters, numerous lexical items fall into systematic semantic patterns, and there exhibit parallelisms between physical and psychosocial reference. Further, from research on the relation of semantic structure to general cognitive structure, it appears that the concepts of force interaction that are encoded within language closely parallel concepts that appear both in early science and in naive physics and psychology. Overall, force dynamics thus emerges as a fundamental notional system that structures conceptual material pertaining to force interaction in a common way across a linguistic range: the physical, psychological, social, inferential, discourse, and mental-model domains of reference and conception. (shrink)

For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over and the core model . We use our results to answer open questions from set theory of the reals.

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.

I argue that scientific and poetic modes of objectivity are perspectival duals: 'views' from and onto basic natural forces, respectively. I ground this analysis in a general account of objectivity, not in terms of either 'universal' or 'inter-subjective' validity, but as receptivity to basic features of reality. Contra traditionalists, bare truth, factual knowledge, and universally valid representation are not inherently valuable. But modern critics who focus primarily on the self-expressive aspect of science are also wrong to claim that our knowledge (...) is or should be ultimately mediated by ourselves. In objective science, we represent nature as a field of phenomena determined by and within explanatorily-basic physical structures that encode the causal impact of elemental physical forces. These causally-basic forces hence ground the possibility of objective scientific theory, even though (I argue) they are systematically excluded from its representational contents. I show how my notions of force and field emerge partly through my effort to naturalize concepts of 'God' and 'Laws' in 17th-century natural philosophy. In science, we use the structure of basic forces' impact to see corresponding objects. We do not see these forces themselves—and this relates to the 'transcendence' of 'God'. I supplement this historical analysis by critiquing standard accounts of objectivity, and by synthesizing elements from neo-Kantian epistemology and current 'structural' realism into a view that I call 'dynamic structuralism'. Finally, I argue that physics systematically relies on passive characterizations of phenomena like inertia and impressed force. Dynamic alternatives that are just as empirically adequate are possible, but they are non-scientific. Poetic objectivity involves a direct presentation of nature as an order of interacting forces. I elaborate first by developing a novel theory of the relation between beauty and sublimity, mediated by a basic category of 'sensory force' (e.g. a 'singing' tone in a beautiful melody). I then describe the interaction between sensory and affective dimensions of aesthetic experience, in relation to an original historical analysis of passion and 'disinterestedness' in Kant, Schiller and Nietzsche. I synthesize these lines of analysis into a critique of the role of 'expression' in art. I also resist 'spectator'-centric aesthetics, in this connection, by developing an account of artistic objectivity in dialogue with Ruskin's critique of the 'pathetic fallacy'. I contrast artistic objectivity with Romantic ideals of self-expression, while elaborating its positive relation to certain currents within Victorian and Modernist aesthetics, including Imagists' stress on 'exteriority'. My project creatively extends Nietzsche's view that philosophical and scientific theories are 'perspectival' expressions of theorists' drives. It is a virtue of Nietzsche's account that he takes the outlet of strong human drives in creative acts of theorizing to be just one among many higher modes of 'will to power' in nature—many of which are totally impersonal. But Nietzsche is still too focused on self-expression or willing one's own power. Objective science involves external forces' 'expression' in and through us, not just 'expressing' our own powers. And Nietzsche's vitalistic 'will to power' must be replaced by a basic ontology that better accommodates inorganic phenomena, like the 'lawful' order of celestial motions. Thus I aim to de-anthropomorphize and 'de-organicize' post-Kantian notions of willful striving and organic development, while preserving for force the primary ontological and evaluative status that theorists from Schiller and Goethe to Schelling and Schlegel to Emerson and Nietzsche attributed to will or to life. (shrink)

Teleological terms such as "function" and "design" appear frequently in the biological sciences. Examples of teleological claims include: A (biological) function of stotting by antelopes is to communicate to predators that they have been detected. Eagles' wings are (naturally) designed for soaring. Teleological notions were commonly associated with the pre-Darwinian view that the biological realm provides evidence of conscious design by a supernatural creator. Even after creationist viewpoints were rejected by most biologists there remained various grounds for concern about (...) the role of teleology in biology, including whether such terms are: 1. vitalistic (positing some special "life-force"); 2. requiring backwards causation (because future outcomes explain present traits); 3. incompatible with mechanistic explanation (because of 1 and 2); 4. mentalistic (attributing the action of mind where there is none); 5. empirically untestable (for all the above reasons). Opinions divide over whether Darwin's theory of evolution provides a means of eliminating teleology from biology, or whether it provides a naturalistic account of the role of teleological notions in the science. Many contemporary biologists and philosophers of biology believe that teleological notions are a distinctive and ineliminable feature of biological explanations but that it is possible to provide a naturalistic account of their role that avoids the concerns above. Terminological issues sometimes serve to obscure some widely-accepted distinctions. (shrink)

Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions . We close with a detailed investigation (...) of iterated Sacks indestructibility. (shrink)

It is difficult to evaluate the role of activity - of force or of that which has causal efficacy - in Descartes’ natural philosophy. On the one hand, Descartes claims to include in his natural philosophy only that which can be described geometrically, which amounts to matter (extended substance) in motion (where this motion is described kinematically).’ Yet on the other hand, rigorous adherence to a purely geometrical description of matter in motion would make it difficult to account for the (...) interactions among the particles that constitute Descartes’ universe, since the notions of extension and kinematical motion do not in themselves imply any causal agency. There is, after all, no reason to expect that a particle whose single essence is extension, even if we suppose it to be moving, should impart motion to another particle, while conversely, there is no reason to expect a resting particle to hinder the motion of an impinging one. Descartes' hankering for an austere ontology of matter in motion is in danger of excluding causal agency (force, conceived dynmically) from matter. To the modern reader it may seem obvious that Descartes did not get himself into such a fix, since matter in motion so readily reminds us of kinetic energy or of some more primitive notion of force. Yet by no means is it obvious that Descartes attributed causal efficacy to matter in motion per se. Serious scholars have held opposite positions on this issue. Westfall, Gabbey, and others’ have argued that although on the metaphysical plane Descartes attempted to eliminate force from his mechanical universe, nonetheless ‘force is a real feature’ of his mechanical world. The thrust of this view is that Descartes conceived of force as ‘the capacity of a body in motion to act’, by means of impact, upon other bodies. An opposing interpretation, defended here, is that Descartes did in fact deny causal agency to moving matter per se, restricting agency to immaterial substances such as the human mind, angels, and God. The intention of this interpretation is not to de-emphasize the role of matter in motion in Descartes’ explanation of nature, but rather to stress the fact that Descartes did not conceive of this moving matter dynamically. -/- Reprinted in Descartes, ed. by J. Cottingham, Oxford Readings in Philosophy (Oxford: Oxford University Press, 1998), 281–310. (shrink)

We consider forcing axioms for suitable families of μ‐complete ‐c.c. forcing notions. We show that some form of the condition “ have a in ” is necessary. We also show some versions are really stronger than others.

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}. The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important (...) conclusions from the Proper Forcing Axiom, for example, the P-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects. (shrink)

We study the projective posets and their properties as forcing notions. We also define Martin's axiom restricted to projective sets, MA, and show that this axiom is weaker than full Martin's axiom by proving the consistency of ZFC + ¬lCH + MA with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists κ < 20 such that 20 < 2k”.

The three notions of illocutionary force, sentence-meaning, and speaker-meaning (what a speaker means by an utterance) have been bandied about, misused and confused in some influential papers about speech acts and, I presume, in quieter corners as well. My object here is to disentangle these notions.

In this paper we will study four forcing notions, two of them giving a minimal degree of constructibility. These constructions give answers to questions in [Ih].

By using finite support iteration Suslin c.c.c forcing notions we construct several models which satisfy some ♢-like principles while other cardinal invariants are larger than ω1.