Current attempts by historians of science to revise the narrative of the Scientific Revolution by using the concept of the Baroque have important implications for art history. Correspondences between baroque art and baroque science gain new complexity when the rational, epistemologically optimistic image of the New Science is put in doubt. Rather than a method of objective observation, early seventeenth‐century science and art share an acceptance of the constructed nature of reality, of human epistemological limitations and of the role of (...) passions in the observation of the world. While Caravaggio has revolutionised art precisely through his interest in questions of knowledge and sensorial perception and by his subversive transformation of Renaissance epistemological values and ideals, this article concentrates on the work of Jusepe de Ribera, who made the senses and their shortcomings a major theme of his pictorial research. Ribera's epistemology is examined in the context of contemporary Neapolitan philosophy and science. Through the confrontation of some of the Spagnoletto's paintings with the work of figures such as Giovanni Battista della Porta, Federico Cesi and particularly Colantonio Stigliola, it becomes clear that early modern Neapolitan faith in rational knowledge was more ambiguous than is sometimes assumed. (shrink)

What sort of epistemic positions are compatible with inquiries driven by interrogative attitudes like wonder and puzzlement? The ignorance norm provides a partial answer: interrogative attitudes directed at a particular question are never compatible with knowledge of the question’s answer. But some are tempted to think that interrogative attitudes are incompatible with weaker positions like belief as well. This paper defends that the ignorance norm is exhaustive. All epistemic positions weaker than knowledge directed at the answer to a question are (...) compatible with having an interrogative attitude towards that question. We offer two arguments for this conclusion. The first is based on considerations about the role of hedging in inquiry. The second is conditional on considerations related to the aim of inquiry as a goal-directed activity. (shrink)

This paper introduces cosmopsychism as a holistic alternative to atomistic panpsychism, and as a general perspective on the metaphysics of consciousness. I begin with some necessary background details concerning contemporary panpsychism and the problems it faces, and then proceed to the theory itself. The starting point of the theory is the assumption that an all pervading cosmic consciousness is the single ontological ultimate. From this assumption, a panpsychist ontology of mind with distinct holistic overtones is developed. In particular, I argue (...) that such universal consciousness serves as the ground for the emergence of individual conscious creatures. The result is a theory with significant conceptual resources which presents novel means for confronting some of the most recalcitrant problems facing contemporary panpsychism: in particular, the subject combination problem, and the problem of entailment associated with it. In so doing, cosmopsychism places itself as an viable alternative to atomi... (shrink)

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove (...) from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ. (shrink)

According to John Searle's connection principle (CP) all intentional states are, necessarily, potentially conscious (Searle 1992). Thus formulated, CP implies that intentionality is ontologically dependent on consciousness. Searle's argument in favour of CP is based on the assumption that, while every intentional state is endowed with an aspectual shape, only conscious intentional states are intrinsically so endowed. In turn, the contention that only conscious intentional states are intrinsically aspectual and perspectival is based on what I call the Cartesian view of (...) subjectivity (CS), according to which subjectivity implies consciousness. I argue that, though intuitive, CS is ultimately untenable. A plea is made for an alternative, `Aristotelian' view of subjectivity (AS), on which the roots of subjectivity lie not in consciousness per se, but in the more basic capacity for autonomous conduct. Searle's failure to appreciate this point, I argue, accounts for the unresolved dualistic tension over- shadowing his conception of mind. (shrink)

This thoroughly intriguing volume explains the many ways our thoughts are manipulated through temptation, distraction, misdirection, and more.

A seminal 1921 work by the linguist Edward Sapir, outlining his influential ideas and hypotheses on language and its speakers.

This paper explores cosmopsychism’s explanatory aspirations from a programmatic perspective. The bulk of the text consists of an argument in favor of the conclusion that cosmopsychism suffers from no insurmountable individuation problem. I argue that the widespread tendency to view IND as a mirror-image of micropsychism’s combination problem is mistaken. In particular, what renders CP insolvable, namely, the commitment to the coupling of phenomenal constitution with phenomenal inclusion, is, from the standpoint of cosmopsychism, an entirely nonmandatory assumption. I proceed to (...) show that severing this unhappy coupling is the key for defending cosmopsychism against the charge of theoretical incoherence. Moreover, I argue that successful defense against such accusation could be mounted regardless of whether or not we assume cosmic consciousness to be perspectival in nature. In addition, the paper touches upon another foundational issue: cautioning against the popular tendency to identify cosmopsychism’s monism with a mereological unity-in-diversity, and motivating an alternative conception which I call generative monism. Finally, as befitting this volume, I pause to reflect upon the question of which schools of Hindu philosophy might tally best, and connect most fruitfully, with cosmopsychism as I understand it. (shrink)

This article attempts to think of thinking as the essence of critical education. While contemporary education tends to stress the conveying of knowledge and skills needed to succeed in present-day information society, the present article turns to the work of Theodor W. Adorno to develop alternative thinking about education, thinking, and the political significance of education for thinking. Adorno touched upon educational questions throughout his writings, with growing interest in the last ten years of his life. Education, he argues following (...) Kant, must enable students to think for themselves and to break free of the authority of teachers, parents and other adults. Nevertheless, in his discussions of education Adorno says little about the nature of thinking, and the secondary literature on his educational theory addresses this question only cursorily. Important claims on the nature of thinking do appear elsewhere in Adorno's work. From his early writings up to Negative Dialectics, Adorno is preoccupied with thinking, sketching the outlines of critical-dialectical thought. Still, these reflections rarely touch upon educational questions, and the Adorno scholarship has yet to establish this link. Unlike studies which read Adorno's educational thought against the backdrop of the history of education and the German Bildung tradition, or in relation to art and aesthetics, the present article brings together Adorno's ideas on education and thinking in an attempt to contribute both to the Adorno scholarship and to the growing field of education for thinking. (shrink)

The aim of this paper is twofold. First, our purpose is to propose and motivate a novel and scientifically informed variant of cosmopsychism, namely, the view that the experiences of ordinary subjects are ultimately grounded in an all-pervading cosmic consciousness. Second, we will demonstrate that this approach generates promising avenues for addressing familiar problems of phenomenal constitution. We use stochastic electrodynamics (SED) as the physical bedrock of our approach, supplementing it with key insights about the nature of consciousness long emphasized (...) in eastern philosophy and other wisdom traditions. We proceed to show that our approach substantiates an intriguing way of thinking about the dynamical emergence of ordinary consciousness from cosmic consciousness, identifying the latter with the vacuum state of quantum field theory. Finally, we argue that the present approach is well suited to address problems of phenomenal constitution, in particular as they pertain to the qualities and structure of experience and to the generation of subjects. (shrink)

In the marketplace of opinions concerning the metaphysics of mind and consciousness panqualityism (PQ) occupies an interesting position. It is a distinct variant of neutral monism, as well as of protophenomenalism, and as such it strives to carve out a conceptual niche midway between physicalism and mentalism. It is also a brand of Russellian monism, advocated by its supporters as a less costly and less extravagant alternative to panpsychism. Being clearly articulated and relatively well-developed it constitutes an intriguing view. Nonetheless, (...) the present paper takes a decisively critical stance towards PQ. In particular, it challenges it on two principal grounds. First, I argue that PQ's analysis of experience, and of the qualities tasked with constituting the phenomenal character of experience, is fundamentally flawed. Second, I argue that PQ's attempt to explain phenomenal consciousness as a function of reflective awareness is equally misguided. Along the way, the paper also points the shortcomings of previously established critiques of PQ. All in all, the discussion identifies some difficulties that are likely to generalize beyond PQ's specific circumstances, raising concerns regarding the viability of a "middle of the road" solution to the mind–body problem. (shrink)

This paper outlines a theory of radical democratic education by addressing a key concept in Laclau and Mouffe’s Hegemony and Socialist Strategy: articulation. Through their concept of articulation, Laclau and Mouffe attempt to liberate Gramsci’s theory of hegemony from Marxist economism, and adapt it to a political sphere inhabited by a plurality of struggles and agents none of which is predominant. However, while for Gramsci the political process of hegemony formation has an explicit educational dimension, Laclau and Mouffe ignore this (...) dimension altogether. My discussion starts with elaborating the concept of articulation and analysing it in terms of three dimensions: performance, connection and transformation. I then address the role of education in Gramsci’s politics, in which the figure of the intellectual is central, and argue that radical democratic education requires renouncing that figure. In the final section, I offer a theory of such education, in which both teacher and students articulate their political differences and identities. (shrink)

According to a widespread view, one of the most important roles of education is the nurturing of common sense. In this article I turn to Gilles Deleuze’s concept of sense to develop a contrary view of education—one that views education as a radical challenge to common sense. The discussion will centre on the relation of sense and common sense to thinking. Although adherents of common sense refer to it as the basis of all thought and appeal to critical thinking as (...) instrumental in eliminating its occasional errors, I shall argue, following Deleuze, that common sense education in fact thwarts thinking, while only education which revolves around making sense may provoke thinking that goes beyond the self-evident. I demonstrate how making sense can become an educational encounter that breaks hierarchies and generates thinking independently of the thinker’s knowledge and place in the sociopolitical order. The present article attempts, therefore, to put some sense into Deleuzian education for thinking, and thereby shed new light on its radical-political, counter-commonsensical power. (shrink)

James A. Diefenbeck, Wayward Reflections on the History ofPhilosophyThomas R. Flynn Sartre, Foucault and Historical Reason. Volume 1:Toward an Existential Theory of HistoryMark Golden and Peter Toohey Inventing Ancient Culture:Historicism, Periodization and the Ancient WorldZenonas Norkus Istorika: Istorinis IvadasEverett Zimmerman The Boundaries of Fiction: History and theEighteenth‐Century British Novel.

Arendt's concept of common sense has generally been misunderstood. It is almost exclusively interpreted in light of Kant's common sense, either as an espousal of the latter or as a distortion of it. This narrow reading of Arendtian common sense has led to a problem, as her uses of the concept do not always fit its Kantian understanding. This has led to accusing her of being inconsistent, or as holding on to several, incompatible concepts of common sense.This article argues that (...) Arendt has one complex concept of common sense, used more or less consistently throughout her writings. Rather than understanding Arendt's common sense in light of Kant's, as most readers do, I demonstrate its links to Aristotle and to the eighteenth‐century Scottish school of common sense. By doing so I turn attention to a difficulty that has thus far not been adequately treated, namely the fact that Arendt presents two, allegedly contradictory pictures of the relation between common sense on the one hand and science and philosophy on the other: a picture in which science and philosophy depend upon common sense, versus one in which they find themselves in a conflict that eventually leads to the loss or demise of common sense. The last part of the article suggests a way to settle the tension between these two pictures, by understanding the political significance Arendt ascribes to them as two distinct yet complementary ways of approaching the modern phenomenon of totalitarianism. (shrink)

Post-critical pedagogy, which offers a significant alternative to the dominant trends in contemporary philosophy of education, objects to seeing education as instrumental to other ends: it attempts to conceive of education as autotelic, namely as having intrinsic value. While there are good reasons for accepting the post-critical reservations with the instrumentalization of education, I argue that its autonomy is equally problematic, as it risks turning the philosophy of education—perhaps education itself—into a privileged activity, out of touch with the most important (...) issues in the contemporary world. In this paper I offer post-critical pedagogy a way out of this dilemma, by drawing on Walter Benjamin’s essay “The Work of Art in the Age of Its Technological Reproducibility”. After presenting the thesis concerning the autonomy of art held by Benjamin’s Frankfurt School friends, and pointing to the similarity between their view and that of post-critical pedagogy, I articulate six Benjaminian theses on post-critical pedagogy. Following Benjamin’s claim that the technological reproducibility of art changes the way it is perceived as well as its political function, I argue that school is an educational technology that reproduces the world to the masses without an “aura”, in a way that allows for political critique that does not reduce education to politics. Next, I highlight the importance of the school to post-critical pedagogy, and contribute to developing the critical-political aspect that remains essential to post-critical pedagogy. Finally, the Benjaminian perspective also makes it possible to reflect on the relation of digital technologies of reproduction to post-critical pedagogy. (shrink)

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than $\aleph_{1}$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important (...) since a proof using finite supports is more amenable to generalizations to cardinals greater than $\aleph_{1}$. (shrink)

The republican political tradition, which originated in Ancient Rome and picked up by several early-modern thinkers, has been revived in the last couple of decades following the seminal works of historian Quentin Skinner and political theorist Philip Pettit. Although educational questions do not normally occupy the center stage in republican theory, various theorists working within this framework have already highlighted the significance of education for any functioning republic. Looking at educational questions through the lens of freedom as non-domination has already (...) yielded important insights to discussions of political education. However, consideration of the existing republican educational discourse in light of the wide range of issues discussed in Pettit’s recent works reveals that it suffers from two major lacunae. First, it does not take into consideration the distinction between democracy and social justice that has become central to Pettit’s republicanism. Thus, the current discussion focuses almost exclusively on education for democratic citizenship and hardly touches upon social justice. Second, the current literature thinks mainly in terms of educating future citizens, rather than conceiving of students also as political agents in the present, and of school itself as a site of non-domination. This paper aims at filling these voids, and it will therefore be oriented along two intersecting axes: the one between democracy and justice, and the other between future citizenship in the state and present citizenship at school. The resulting four categories will organize the discussion: future citizens and democracy; future citizens and social justice; present citizens and democracy; present citizens and social justice. This will not only enable us to draw a clearer line between the civic republican and liberal educational theories, but also make civic republican education a viable alternative to current educational approaches. (shrink)

The Russian financial crisis of August 1998, which occurred between the Asian recession and the devaluation of the Brazilian real in January 1999, put a question-mark over many certainties. It demonstrated that the most destructive speculative activities of financial players can only have free rein when there is a collapse of the state. The crisis proved that the markets need a state and cannot replace it. It was also the moment when people became aware of the dynamics of a globalized (...) financial economy (Stiglitz, 1999a; 1999b), leading to a critical rereading of the policies recommended in the 1990s and a direct questioning of the IMF and its actions. In Russia it caused a paradigm shift that could turn out to be salutary. In this regard the August 1998 crisis looks like the endpoint on the route followed since 1992 by the transition in Russia (Sapir, 1999a). We can see in the crisis both elements directly linked to the financial liberalization of 1985-95 and structural elements peculiar to the consequences of the transition and the policies adopted. This is why it suggests a reinterpretation of the entire development of the transition in Russia starting from 1992. (shrink)

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type (...) and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types. (shrink)

We prove that in $\operatorname {NTP}_{\operatorname {2}}$ theories the dp-rank of a type can be witnessed by indiscernible sequences of tuples satisfying that type. If the type has dp-rank infinity, then this can be witnessed by singletons.

Panpsychism is now a bona fide potential solution to the metaphysical quandary of consciousness. Much of the debate concerning the viability of panpsychism is centered on the combination problem. Intriguingly, the literature analyzing this problem displays two competing interpretations which differ in their modal force. According to the first, which we call the ‘no-necessitation view', CP consists in the absence of a priori necessitation of macro-level phenomenal facts from micro-level phenomenal facts. In contrast, the second interpretation, which we label the (...) ‘incoherence charge’, goes further by insisting that it can be positively demonstrated that micro-experiences cannot combine to yield integral macro-experiences. While conceivability arguments are vulnerable to debates over conflicting intuitions, as well as to the claim that a more complete picture will grant the desired necessitation, IC has the advantage that, if true, it removes these concerns and approaches a more final verdict. Nevertheless, while conceivability arguments are widely discussed IC has not been equally well understood or recognized. In this paper we conduct a systematic investigation and reevaluation of IC, as it applies to constitutive micropsychism. We argue that the problem is real and therefore that constitutive micropsychism is a blind alley. (shrink)

Jullien's indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis. We identify the strength of the theorem relative to standard reverse mathematics markers. We show that it lies strictly between weak [Formula: see text] choice and [Formula: see text] comprehension.

Uniting analytic philosophy with Buddhist, Indian, and Chinese traditions, this collection marks the first systematic cross-cultural examination of one of philosophy of mind's most fascinating questions : can consciousness be conceived as metaphysically fundamental? Engaging in debates concerning consciousness and ultimate reality, emergence and mental causation, realism, idealism, panpsychism, and illusionism, it understands problems through the philosophies of East and South-East Asia, in particular Buddhism and Vedanta. Each section focuses on a specific aspect or theory of consciousness, and examines a (...) particular subject from different disciplinary perspectives including philosophy, psychology, and cognitive science. These different angles allows readers to gain insight into the intellectual challenges and problems of the study of consciousness and its place in the thought traditions of both Eastern and Western philosophy. Raising new questions, it provides a more global and holistic understanding of consciousness, presenting a stimulating and original contribution to contemporary consciousness studies and the metaphysics of mind. (shrink)

The neo-republican political philosophy (sometimes referred to as civic republicanism) advances the idea of freedom as non-domination, in an attempt to provide democracy with a solid normative foundation upon which concrete principles and institutions can be erected so as to make freedom a reality. However, attempts to develop a republican educational theory are still hesitant, and fail to take the republican radical conception of freedom to its full conclusions. This article suggests that dialogue between neo-republicanism and critical pedagogy can be (...) mutually productive. In the first part of the article we present the neo-republican theory, and contrast it with traditional liberalism. In the second we focus on existing neo-republican theories of education, and claim that they do not take the republican presuppositions to their necessary conclusion, namely to an educational theory fully committed to the idea of freedom as dondomination. A republican educational theory, we argue, must take into consideration not only the freedom students will have in the future, but also their freedom in the present: it should think of school as a small-scale republic, which prepares its inhabitants to be future citizens of the state while at the same time treating them as free citizens in their own right. In the third part we use insights taken from critical pedagogy to chart the direction republican education must take by applying three key republican notions—democratic control, civic contestation, and trust. In the fourth and last part we outline four aspects in which neo-republicanism can shed new light on contemporary debates in critical pedagogy: the connection between democracy and justice, the multiplicity of forms of domination, critical education within schools, and work with students from relatively privileged backgrounds. (shrink)

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

The first thing to realize about grading as a psychological process is that it precedes measurement and counting. Judgments of the type “A is larger than B” or “This can contains less milk than that” are made long before it is possible to say, e.g., “A is twice as large as B” or “A has a volume of 25 cubic feet, B a volume of 20 cubic feet, therefore A is larger than B by 5 cubic feet,” or “This can (...) contains a quart of milk, that one 3 quarts of milk, therefore the former has less milk in it.” In other words, judgments of quantity in terms of units of measure or in terms of number always presuppose, explicitly or implicitly, preliminary judgments of grading. The term four means something only when it is known to refer to a number which is “less than” certain others, say five, six, seven, arranged in an ordered series of relative mores and lesses, and “more than” certain others, say one, two, three, arranged in an ordered series of relative mores and lesses. Similarly, a foot as a unit of linear measure has no meaning whatever unless it is known to be more than some other stretch, say an inch, and less than a third stretch, say a yard. (shrink)

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L.The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and (...) descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL ⇒ HODL ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals. (shrink)

With the deepening crisis of physicalism and the decline in its status as a sustainable research programme, philosophers of mind have begun to investigate the alternative idea—now commonly designated panpsychism—that consciousness is a fundamental feature of nature, and that the mental states, properties, and events exhibited by human beings are metaphysically grounded in the conscious actuality of reality’s most basic entities. Cosmopsychism is the thesis that the cosmos as a whole displays psychological properties, cosmopsychological properties as we might call them, (...) and that the mental states of human beings, and indeed human beings themselves as individual subjects of experience, are metaphysically grounded in the cosmopsychological properties of the cosmos. Richly sophisticated varieties of cosmopsychism can be found in the ancient Sanskrit classics, the Upaniṣads, and more particularly in those Vedānta philosophers who provide interpretations of these texts. The papers in this issue are explorations of the connections and explanatory potential between contemporary work on cosmopsychism and arguments from the Sanskrit tradition. (shrink)

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.

We show that Mitchell order on downward closed extenders below rank-to-rank type is wellfounded.

We provide a new criterion for embedding.

We extend the construction of Mitchell and Steel to produce iterable fine structure models which may contain Woodin limits of Woodin cardinals, and more. The precise level reached is that of a cardinal which is both a Woodin cardinal and a limit of cardinals strong past it.

We prove determinacy for open length ω1 games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω1, with payoff conditions involving the entire run, the club filter on ω1, and a sequence of ω1 disjoint stationary subsets of ω1. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption.

We show that Blackwell determinacy in L(R) implies determinacy in L(R).

This paper engages the extended cognition controversy by advancing a theory which fits nicely into an attractive and surprisingly unoccupied conceptual niche situated comfortably between traditional individualism and the radical externalism espoused by the majority of supporters of the extended mind hypothesis. I call this theory moderate active externalism, or MAE. In alliance with other externalist theories of cognition, MAE is committed to the view that certain cognitive processes extend across brain, body, and world—a conclusion which follows from a theory (...) I develop in “Synergic Coordination: an argument for cognitive process externalism.” Yet, in contradistinction with radical externalism, and in agreement with the internalist orthodoxy, MAE defends the view that mental states are situated invariably inside our heads. This is done, inter alia, by developing a novel hypothesis regarding the vehicles of content (in “Extended cognition without externalized mental states”, and by criticizing arguments in support of mental states externalism (in “Reflections and objections”). The result, I believe, is a coherent theoretical alternative worthy of serious consideration. (shrink)

We present two applications of forcing with finite sequences of models as side conditions, adding objects of size \. The first involves adding a \ sequence and variants of such sequences. The second involves adding partial weak specializing functions for trees of height \.

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary (...) we also see that assuming the existence of a Woodin cardinal δ so that SBHδ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ+ supercompact for all n < ω. We state some results at this level, and indicate how they are proved. (shrink)