We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+ (...) class='Hi'>Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf (...) {MV}_T$ (where T is $\mathsf {ZFC}$ +Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework. (shrink)

“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such (...) disaparate and relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

A defense of Kant’s moral philosophy. The author seeks to counteract those interpretations of Kant that restrict their focus to the Groundwork of the Metaphysics of Morals. He argues that one must look at the whole of Kant’s writings, the earlier and later ethical writings as well as the theoretical works. This makes it possible for him to challenge the popular misconceptions of Kant’s teaching: the overemphasis on the correct motive of an action, the mistaken impression that consequences are of (...) no consequence when evaluating the moral worth of an act, and the overattention paid to logical contradiction as the test of a maxim’s morality. The Groundwork is not simply a conclusion of Kant’s early reflections on morality, but it also represents a new departure whose full significance is not seen until the later works. With the Groundwork Kant realizes that morality cannot be established on the grounds set out by the Critique of Pure Reason but requires an independent foundation. His new concept of practical necessity represents a revolution in morality. It is a mistake, then, to read the practical philosophy solely in the light of the theoretical critique. Though founded on a blind "fact of reason," Kant’s teaching is yet not abstract, overly formalistic, or rigoristic. "One must go through all the different formulas [of the categorical imperative] to understand the concept of moral judgment". Rossvaer’s rethinking brings him to stress that "the categorical imperative has no validity beyond being the form of... sensuous striving". This contention that the intellectual and sensuous faculties are not to be radically divorced leads to the ultimate intent of Rossvaer’s work. He argues that the categorical imperative is a principle of execution and only assumes its fullest form in a social philosophy. Consideration of Kant’s notion of the highest good, especially as it is developed in the later works on religion and history, allows Rossvaer to argue for the social and historical relevance of Kant’s teaching. "Since the highest good cannot be justified within an individualistic perspective, but only within a philosophy which takes up society as a problem, morality is meaningless from a human point of view unless we consider ourselves obligated to realize a universal, ethical community". With this, the kingdom of ends is transformed into an historical idealism, practical philosophy becomes a theory of action, and Kant turns out to be not so different from his detractors, Fichte, Hegel, Marx, Habermas, and others. Yet the historicization of the moral agent is yet not complete in Kant. The moral agent retains his critical standpoint, that is the distinction between rationality and the social system. Yet one is brought to ask how this critical distance can be maintained. To what extent does Kant’s overstepping the bounds of theoretical reason also entail the loss of that critical validity won by the first Critique? And, if one is forced in the end to justify one’s practical philosophy only by a social philosophy which does not find its foundation in a theoretical critique, then is the original fear of the first Critique confirmed, that morality may be meaningless after all? (shrink)

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend on the context in (...) which they are embedded. The context here will be the theory CZF− of constructive Zermelo–Fraenkel set theory but without -Induction . Let INAC be the statement that for every set there is an inaccessible set containing it. CZF−+INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF−+INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF−+INAC is a small ordinal known as the Feferman–Schütte ordinal Γ0. (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 prove two general results about the preservation of extendible and $C^{(n)}$ -extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{(n)}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{(n)}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$ -definable behaviour of the (...) power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving $C^{(n)}$ -extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings–Foreman–Magidor for forcing $\diamondsuit _{\kappa ^+}^+$ at every $\kappa $ [10] preserves $C^{(n)}$ -extendible cardinals. We give an optimal result on the consistency of weak square principles and $C^{(n)}$ -extendible cardinals. In the last section prove another preservation result for $C^{(n)}$ -extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations. As applications we prove the consistency of $C^{(n)}$ -extendible cardinals with $\mathrm {{V}}=\mathrm {{HOD}}$, and also with $\mathrm {GA}$ (the Ground Axiom) plus $\mathrm {V}\neq \mathrm {HOD}$, the latter being a strengthening of a result from [14]. (shrink)

We define a potentialist system of ‐structures, i.e., a collection of possible worlds in the language of connected by a binary accessibility relation, achieving a potentialist account of the full background set‐theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those (...) in the modal theory. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory, both for assertions in the language of and for assertions in the full potentialist language. (shrink)

A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, the mantle, the intersection of all grounds, must be a model of ZFC. V has only set many grounds if and only if the mantle is a ground. We also show that (...) if the universe has some very large cardinal, then the mantle must be a ground. (shrink)

With historical hindsight, it can be little questioned that the view of protozoa as unicellular organisms was important for the development of the discipline of protozoology. In the early years of this century, the assumption of unicellularity provided a sound justification for the study of protists: it linked them to the metazoa and supported the claim that the study of these “simple” unicellular organisms could shed light on the organization of the metazoan cell. This prospect was significant, given the state (...) of cytology circa 1910. In the wake of the major gains made in understanding nuclear division in the last two decades of the nineteenth century, cytology was suddenly confronted with many, seemingly less penetrable, problems. Several aspects of nuclear organization still remained unexplained, and recent research had revealed the presence in the cytoplasm of structures whose functions in cell life were unknown. Classical methods of cytology, relying on descriptive, morphological analysis, seemed ill equipped to resolve these questions.Hertwig's program for protozoology, grounded in the assumption of the fundamental unity of organization in protozoa and metazoa, offered a potential means for investigating these and other problems of cell theory. Linked to mainstream cytology, protozoology was advocated as a means of experimentally investigating key biolobical processes — reproduction, metabolism, and organelle morphology and physiology — less accessible in higher organisms. Protozoa were hailed as prime experimental organisms, in which cell structure and function could be more easily studied. Unlike the metazoan cell, they could be subjected to controlled experiments in which the external environment was modified and the effects monitored. Protozoa offered, in other words, a promising experimental means by which to investigate the cell — its structures and its processes.The success of this program within Germany was soon apparent. In contrast to the rather neglected state of the discipline in 1900, protozoology began to be recognized as more than a somewhat obscure area of study for specialists. In practical terms, this translated into greater numbers of students attracted to the field, increased institutional support for the discipline, and its elevation in status within the biological sciences as new developments, particularly in connection with medical applications, began to draw attention to the field.64The unicellular hypothesis also promoted the internal development of the science. It provided a rationale for introducing the various techniques used in cytology, embryology, physiology, and the new field of biochemistry as suitable research tools for protozoology as well. This vastly extended the research possibilities and facilitated the understanding of, among other things, protozoan organization, modes of reproduction, and evolutionary relationships. The new experimental grounding of the discipline in turn fitted in well with developments in biology at large: in contrast to the descriptive methodology that had predominated in the nineteenth century, this new experimental program placed protozoology at the forefront of the early twentieth-century movement to make biology an experimental science comparable to physics and chemistry.Yet the criticism of the unicellular hypothesis can also be seen as having served a valuable function within the development of the discipline. It focused attention on the study of protists as organisms in their own right, not simply as models for metazoan cells. More generally, it helped to remind biologists of the particular evolutionary assumptions that supported this conception. Dobell and other British critics pointed out, among other things, the association between the theory of recapitulation and the interpretation of protozoa as unicellular organisms. At the time when the recapitulation theory and the germ-layer doctrine were in decline, it was important to stress how these evolutionary ideas also entered into the contemporary concepts of subsidiary specialties. This was as true for protozoology as it was for cell theory itself. The chromidial theory, in its various guises, and the binuclearity hypothesis did in fact contain elements of recapitulationist reasoning, and they were open to criticism for the same kind of overly speculative theorizing that characterized this evolution theory. Dobell's critique forced protozoologists and cell theorists alike to review the theoretical postulates guiding their investigations.In the absence of further historical studies, it is hard to evaluate the consequences of this debate in later years. The issue was not whether protozoa were the precursors of metazoa — both sides accepted this. They disagreed over which particular protozoon had served as the ancestral form, and this, in turn, influenced their stance on the question of unicellularity. The situation is little changed today. Because the former question is still an open one, the latter remains so too. Both of the models for the origin of multicellular organisms — colonies of ciliates versus multinucleate protists — are still presented as possible mechanisms in modern textbooks of evolution. Unable to judge the dispute in terms of the ultimate validity of the competing conceptions, the historian requires other criteria. It perhaps becomes more important to evaluate the issues in the context of the internal and external stimulus they provided the discipline.65 In these terms, and from the present historical vantage point, Hertwig's research program for protozoology, based upon the unicellular hypothesis, appears to have been a successful one. (shrink)

Literary criticism at the present moment seems ready to open its doors once again to the outside world, even if that world is only a series of other academic disciplines, each cloistered in its own way. For the reader of black African literature in French, the opening comes none too soon. The program for reading Camara Laye, Ahmadou Kourouma, and Yambo Ouologuem should never have been the program prescribed for Rousseau, Wordsworth, or Blanchot. If one is willing to read a (...) literature that might not be a rewriting of Hegel , and if the negative knowledge of recent theoretical criticism is questioned in the universality of its applications, then what is really open to a Western reader of non-Western literature? Claiming a break with his/her own culture and critical upbringing, can he/she the Other, the African, as if from an authentically African point of view, interpreting Africa in African terms, perceiving rather than projecting?The goal of breaking through the nets of Western criticism, of reading African literature in a nonethnocentric, nonprojective fashion, will remain both indisputably desirable and ultimately unattainable. No matter how many languages I learn or ethnologies I study, I cannot make myself into an African. The Western scholar’s claim to mastery of things African, albeit motivated by xenophilia rather than xenophobia, risks subjugation of the object to a new set of Western models. J. P. Makouta-M’Boukou rightly scolds Western critics who refuse to take into account the distance between themselves and African culture, and who read African literature only in function of their own cultural context.1 Wole Soyinka, more forbiddingly, complains: “We black Africans have been blandly invited to submit ourselves to a second epoch of colonisation—this time by a universal-humanoid abstraction defined and conducted by individuals whose theories and prescriptions are derived from the apprehension of their world and their history, their social neuroses and their value systems.”2 1. See J. P. Makouta-M’Boukou, Introduction à l’étude du roman négro-africain de langue française , p. 9.2. Wole Soyinka, Myth, Literature, and the African World , p. x. Christopher L. Miller, Charles B. G. Murphy Assistant Professor of French and of African and Afro-American Studies at Yale University, is author of Blank Darkness: Africanist Discourse in French . He is working at present on a study of francophone black African literature, for which he will have a Fulbright Africa Research grant. (shrink)

Early 20th century philosopher Henri Bergson posited an initial push that propelled the diversity of life forward into a varied, novel future: The élan vital, a necessary force or impulse that animated life’s progress and development. His idea had largely been abandoned by mid-century. Even so, much of the conceptual and explanatory work this impulse targeted is yet in want of an explanation. In particular, Bergson’s derelict ideas on evolution addressed three areas that have once again become relevant in the (...) effort to unite evolutionary genetics, biological development, and ecological context : the purposeful nature of individual organisms and their parts; the integrative, holistic, non-linear emergent dynamics seen in evolutionary processes; and how genuine novelty emerges into the universe :422, 2016; Simondon et al. in On the mode of existence of technical objects. Univocal series, Univocal Publishing LLC, Minneapolis, 2017; Bang, in: Winther-Lindqvist, Bang, Valsiner Nothingness: philosophical insights into psychology, Transaction Publishers, Somerset, 2016; Moreno and Mossio in Biological autonomy: a philosophical and theoretical enquiry. History, philosophy and theory of the life sciences, Springer, Dordrecht, 2015). In this paper I argue that Bergson’s ideas may yet be relevant to these questions, and his work warrants a reexamination in light of current problems in evolutionary biology. This is not a call to ‘return’ to Bergson, nevertheless his notions about complexity suggest ways of looking at current biological problems in ways that offer a heuristic insight worth entertaining. Bergson’s Nobel Prize-winning book, Creative Evolution, provided a strikingly prescient early 20th century framework for understanding how Darwinian evolution acts as an engine for generating new forms, University Press of America, Lanham, Bergson 1911). (shrink)

This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of (...) the set-theoretic community---set-theoretic practitioners with a pluralist view and set-theoretic practitioners who aim for ZFC-proofs. By describing this, the paper gives a more complete picture of new axioms in set-theoretic practice. These observations, for instance, show that set-theoretic practitioners interested in ZFC-proofs use tools that go beyond ZFC. The analysis is based on empirical data that was collected in an extensive interview study with set-theoretic practitioners. (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)

We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...) analytic quasi-orders on View the MathML source. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal [17] and Friedman and Motto Ros [6], and it also partially extends a result from Baumgartner [3] concerning the structure of the embeddability relation on linear orders of size κ. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to (...) the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis. It was Gödel who first suggested that perhaps “strong axioms of infinity” could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH—one can show that virtually all large cardinals defined so far do not affect the status of CH. It seems to be an inherent feature of large cardinals that they do (...) not determine properties of sets low in the cumulative hierarchy if such properties can be forced to hold or fail by small forcings.The paper can also be used as an introductory text on large cardinals as it defines all relevant concepts. (shrink)

We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong , hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of (...) the partial order; we refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author’s outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals. (shrink)

Is there a key for ‘translating' some set-theoretical paradoxes into counterpart semantical paradoxes and vice-versa? There is, and this encourages the hope of a unified solution. The solution turns not on inventing new axioms that do not entail contradiction, but on imposing a completely intuitive restriction on the comprehension axiom of naive set theory in order to avoid illegitimate (circular) stipulation.

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since Cantor’s introduction of the theory of transfinite numbers in the late nineteenth century. A special focus has been given to this theory’s interrelation with the forcing theory, introduced by P. Cohen in his lectures of 1963 and further extended and deepened since then, which (...) leads to a development and further refinement of the theory of large cardinals ultimately touching, especially in view of the discussion in the last section, on the metatheoretical nature of infinity. The whole undertaking, which takes into account major stages of the research in large cardinals theory, tries to present a defensible argumentation against an ontology of infinity actually rooted in the notion of subjectivity within the world. This means that rather than talking of a general ontology of infinity in the ideal platonic or in the aristotelian sense of potentiality, even in the alternative sense of an ontology of the event in A. Badiou’s sense, one can argue from a subjective point of view about the impossibility of defining cardinalities greater than the first uncountable one \ that would correspond to a distinct existence in real world terms or would be supported by a mathematical intuition in terms of reciprocity with experience. The argumentation from the particular standpoint includes also certain comments on the delimitative character of Gödel’s constructive universe L and the influence of the constructive approach in narrowing the breadth of an ‘ontology’ of infinity. (shrink)

After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey (...) of results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter. (shrink)

The dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle. In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics.We commence Part I by investigating the Identity Crisis phenomenon in (...) the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $^{\mathrm {HOD}}<\lambda ^+$, for every regular cardinal $\lambda $.” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $^{\textrm {{HOD}}}=\lambda ^+$, there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$, whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously. (shrink)

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

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, (...) asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. (shrink)

The independence results in set theory invite the search for new and justified axioms. In Chapter 1 I set the stage by examining three approaches to justifying the axioms of standard set theory and argue that the approach via reflection principles is the most successful. In Chapter 2 I analyse the limitations of ZF and use this analysis to set up a mathematically precise minimal hurdle which any set of new axioms must overcome if it is to effect a significant (...) reduction in incompleteness. In Chapter 3 I examine the standard method of justifying new axioms---reflection principles---and prove a result which shows that no reflection principle can overcome the minimal hurdle and yield a significant reduction in incompleteness. In Chapter 4 I introduce a new approach to justifying new axioms---extension principles---and show that such principles can overcome the minimal hurdle and much more, in particular, such principles imply PD and that the theory of second-order arithmetic cannot be altered by set size forcing. I show that in a sense these principles are inevitable. In Chapter 5 I close with a brief discussion of meta-mathematical justifications stemming from the work of Woodin. These touch on the continuum hypothesis and other questions which are beyond the reach of standard large cardinals. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large (...) cardinal axioms are restrictive. I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles. (shrink)

The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author asked the first author about the development of forcing, the use of new axioms and set-theoretic intuition on (...) independence. Parts of these conversations are directly presented in this article. They are supplemented by important mathematical results as well as discussion sections. Finally, we present three hypotheses about set-theoretic practice: First, that most set theorists were surprised by the introduction of the forcing method. Second, that most set theorists think that forcing is a natural part of contemporary set theory. Third, that most set theorists prefer an answer to a problem with the help of a new axiom of lowest possible consistency strength, and that for most set theorists, a difference in consistency strength weighs much more than the difference between Forcing Axiom and Large Cardinal Axiom. (shrink)

In this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω 1 . Define $\operatorname{cof}(H)$ as (...) the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that $\operatorname{card}(\lfloor\log_2(H)\rfloor) \leq \operatorname{cof}(H) \leq \operatorname{card}(2^H)$ , where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes $M \preccurlyeq N$ and hyperfinite integer H ∈ M such that H is not enlarged by N, 2 H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe. (shrink)

In deciding whether an event was caused by chance or agency, it is incorrect to attribute the event to chance on the grounds that there have been enough broadly similar situations in the universe to provide opportunities for the event to occur by chance somewhere or other. In order to include a set of instances in an inference as opportunities for the event to occur by chance, we must calculate the impact of the other proposed opportunities upon the prior probabilities (...) of chance and design and argue that the absence of a result confirming design in the other instances directly drives down the prior for design in the instance at hand. This approach forces us to narrow our focus to those instances which are directly relevant to a causal inference about the case at issue, and it avoids confusions generated by treating all broadly similar instances as a lottery that is nearly guaranteed to have a winner if the set is large enough. (shrink)

Interpreting Figurative Meaning critically evaluates the recent empirical work from psycholinguistics and neuroscience examining the successes and difficulties associated with interpreting figurative language. There is now a huge, often contradictory literature on how people understand figures of speech. Gibbs and Colston argue that there may not be a single theory or model that adequately explains both the processes and products of figurative meaning experience. Experimental research may ultimately be unable to simply adjudicate between current models in psychology, linguistics and philosophy (...) of how figurative meaning is interpreted. Alternatively, the authors advance a broad theoretical framework, motivated by ideas from 'dynamical systems theory', that describes the multiple, interacting influences which shape people's experiences of figurative meaning in discourse. This book details past research and theory, offers a critical assessment of this work and sets the stage for a new vision of figurative experience in human life. (shrink)

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent (...) formulation of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of [Formula: see text] for all [Formula: see text]. (shrink)

We begin with a discussion of the Borde-Guth-Vilenkin past-incompleteness theorem for inflationary universes and discuss its significance for various pre-big-bang inflationary scenarios in string cosmology, including landscape and cyclic ekpyrotic models. We then undertake a general critique of inflationary cosmology in respect of its stated goals and conclude with a critcal discussion of the string-theoretic multiverse as a "solution" to the problem of cosmological fine-tuning.

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.

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to (...) the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni-Friedman :77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

Cognitive science is transforming our understanding of the mind. New discoveries are changing how we comprehend not just language, but thought itself. Yet, surprisingly little of the new learning has penetrated discussions and analysis of the most important social institution affecting our lives-the law. Drawing on work in philosophy, psychology, anthropology, linguistics, and literary theory, Steven L. Winter has created nothing less than a tour de force of interdisciplinary analysis. A Clearing in the Forest rests on the simple notion that (...) the better we understand the workings of the mind, the better we will understand all its products-especially law. Legal studies today focus on analytic skills and grand normative theories. But, to understand how real-world, legal actors reason and decide, we need a different set of tools. Cognitive science provides those tools, opening a window on the imaginative, yet orderly mental processes that animate thinking and decisionmaking among lawyers, judges, and lay persons alike. Recent findings about how humans actually categorize and reason make it possible to explain legal reasoning in new, more cogent, more productive ways. A Clearing in the Forest is a compelling meditation on both how the law works and what it all means. In uncovering the irrepressibly imaginative, creative quality of human reason, Winter shows how what we are learning about the mind changes not only our understanding of law, but ultimately of ourselves. He charts a unique course to understanding the world we inhabit, showing us the way to the clearing in the forest. (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 essay aims to identify a semiotic consciousness found in New England Transcendentalism, consisting of the worldview that signs are pervasively present throughout nature and society. It finds that this worldview exists as a historical strand of thought stretching through the 19th century and, ultimately, further beyond, thereby making up an early movement in American semiotics. In this context, I furthermore see Transcendentalist thought informing the backdrop of Charles Peirce’s groundbreaking theory of signs later in the century, especially his metaphysical (...) claims about a “universe... perfused with signs”1 (1906: EP 2.394). In order to bring into full view the presence of a semiotic consciousness in Transcendentalism, I first address the intellectual history and genealogical roots that helped shape the minds of the Transcendentalists. Relevant influences include those both local to New England and imported from abroad, in particular the theologies of Jonathan Edwards and of Emanuel Swedenborg. Next, I directly examine the ideas of the figurehead of the Transcendentalist movement, Ralph Waldo Emerson, together with his Concord peers, Amos Bronson Alcott and Henry David Thoreau. Taking them as my case studies, I track the different ways a like-minded interest in signs takes shape within their individual projects. Focus-ing on their descriptions of nature, we find it in their experiences of the natural environment, their understanding of phenomena as representational and poetic, and their belief in the dialogical sharing of ideas across minds and species. Along the way, I further work out some of the aspects of a general theory of signs identifi-able within the Transcendentalist perspective, as well as distinguish it from other theoretical alternatives. Ultimately, I contend that the Transcendentalists held a similar idea of nature existing as a sign representing deep and varied meanings. (shrink)

In this paper I apply the concept of _inter-Model Inconsistency in Set Theory_ (MIST), introduced by Carolin Antos (this volume), to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s _Ultimate L_, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these positions on the (...) spectrum from inconsistency avoidance to inconsistency toleration. By doing so, I connect a debate in philosophy of set theory with a debate in philosophy of science about the role of inconsistencies in the natural sciences. While there are important differences, like the lack of threatening explosive inferences, I show how specific philosophical positions in the philosophy of set theory can be interpreted as reactions to a state of inconsistency similar to analogous reactions studied in the philosophy of science literature. My hope is that this transfer operation from philosophy of science to mathematics sheds a new light on the current discussion in philosophy of set theory; and that it can help to bring philosophy of mathematics and philosophy of science closer together. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue (...) that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory. (shrink)

Purpose. The paper considers the fear of social alienation of love. It is within the limits of psychoanalytic epistemology, the analysis of which will be presented in the article, the tendencies to monotony and universal solutions with an emphasis on ensuring the objectivity of the problem of gender alienation, to be more exact, the fear of love, which causes the gender process, are viewed most reliably. In view of the above the purpose of the paper is to investigate the conceptual (...) contradictions associated with the reconstruction of basic measurements of the fear of alienation of love within different ideological approaches; to reveal the most significant contradictions that manifest themselves in the differences between secular and religious frames; to determine a philosophical and fundamental foundation for the types of fear: metaphysical, transcendental and social, which to some extent have influenced the development of a gender tradition. The paper investigates the fear of alienation of love, not as an instinct or psychological experience, or as a social factor, but as a correlation between the functional types of fear and the peculiarities of social and cultural processes, and it acts as a dependence of the European societies on the various functional forms of fear of love that influence the processes of social transformations. Theoretical basis. The authors determine that in the gender approach the subject, who is cognized, has specific desires and interests. The process of cognation is local, contextual, in which physiologically and spiritually the fear of alienation of love is closely linked to the conditions of existence of people, with the arrangement of social and individual life. In modern civilization, true love with some exceptions, is fleeting. It is now hindered by the burdens of life, and ideological settings, which go only from the "I", and not from the common "I and You", contributing to frustration and alienation. Originality. The emphasis is placed on the fact that the fear of alienation of love, as a way of identifying the essential characteristics of "I", acts as an existential, which concentrates on the sense-life range of problems. The range of problem in a single topic is in the "point I", surrounded by gender boundaries, it contains as a condition for its manifestation the personality distancing – free or forced consolidation and assertion of the autonomy of the individual. Alienation of love as an existential admits in the register of the intimate, intentional for self-perception, internal state to organize and identify with the "authentic I". Conclusions. The article establishes that gender identity under the conditions of modern socio-psychological crisis is associated with forming a special "picture of fear, anxiety, loneliness" as a part of that internal and external world, generated by alienation from society. The fear of alienated love acts as the essential existential of human being, which is a necessary existential condition of the existential tension that occurs as an antipode and a characteristic of courage, asceticism, and the "beginning of wisdom". The considered phenomenon of the fear of alienation of love is interpreted as the result of the ontological and gender alienation of a person who has lost a sense of security and plunged into a continuous interindividual rivalry. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these (...) higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [math]. We also show the existence of many classes [math] with [math] giving rise to pairwise incompatible theories for [math]. (shrink)

The INBIOSA project brings together a group of experts across many disciplines who believe that science requires a revolutionary transformative step in order to address many of the vexing challenges presented by the world. It is INBIOSA’s purpose to enable the focused collaboration of an interdisciplinary community of original thinkers. This paper sets out the case for support for this effort. The focus of the transformative research program proposal is biology-centric. We admit that biology to date has been more fact-oriented (...) and less theoretical than physics. However, the key leverageable idea is that careful extension of the science of living systems can be more effectively applied to some of our most vexing modern problems than the prevailing scheme, derived from abstractions in physics. While these have some universal application and demonstrate computational advantages, they are not theoretically mandated for the living. A new set of mathematical abstractions derived from biology can now be similarly extended. This is made possible by leveraging new formal tools to understand abstraction and enable computability. [The latter has a much expanded meaning in our context from the one known and used in computer science and biology today, that is "by rote algorithmic means", since it is not known if a living system is computable in this sense (Mossio et al., 2009).] Two major challenges constitute the effort. The first challenge is to design an original general system of abstractions within the biological domain. The initial issue is descriptive leading to the explanatory. There has not yet been a serious formal examination of the abstractions of the biological domain. What is used today is an amalgam; much is inherited from physics (via the bridging abstractions of chemistry) and there are many new abstractions from advances in mathematics (incentivized by the need for more capable computational analyses). Interspersed are abstractions, concepts and underlying assumptions “native” to biology and distinct from the mechanical language of physics and computation as we know them. A pressing agenda should be to single out the most concrete and at the same time the most fundamental process-units in biology and to recruit them into the descriptive domain. Therefore, the first challenge is to build a coherent formal system of abstractions and operations that is truly native to living systems. Nothing will be thrown away, but many common methods will be philosophically recast, just as in physics relativity subsumed and reinterpreted Newtonian mechanics. -/- This step is required because we need a comprehensible, formal system to apply in many domains. Emphasis should be placed on the distinction between multi-perspective analysis and synthesis and on what could be the basic terms or tools needed. The second challenge is relatively simple: the actual application of this set of biology-centric ways and means to cross-disciplinary problems. In its early stages, this will seem to be a “new science”. This White Paper sets out the case of continuing support of Information and Communication Technology (ICT) for transformative research in biology and information processing centered on paradigm changes in the epistemological, ontological, mathematical and computational bases of the science of living systems. Today, curiously, living systems cannot be said to be anything more than dissipative structures organized internally by genetic information. There is not anything substantially different from abiotic systems other than the empirical nature of their robustness. We believe that there are other new and unique properties and patterns comprehensible at this bio-logical level. The report lays out a fundamental set of approaches to articulate these properties and patterns, and is composed as follows. -/- Sections 1 through 4 (preamble, introduction, motivation and major biomathematical problems) are incipient. Section 5 describes the issues affecting Integral Biomathics and Section 6 -- the aspects of the Grand Challenge we face with this project. Section 7 contemplates the effort to formalize a General Theory of Living Systems (GTLS) from what we have today. The goal is to have a formal system, equivalent to that which exists in the physics community. Here we define how to perceive the role of time in biology. Section 8 describes the initial efforts to apply this general theory of living systems in many domains, with special emphasis on crossdisciplinary problems and multiple domains spanning both “hard” and “soft” sciences. The expected result is a coherent collection of integrated mathematical techniques. Section 9 discusses the first two test cases, project proposals, of our approach. They are designed to demonstrate the ability of our approach to address “wicked problems” which span across physics, chemistry, biology, societies and societal dynamics. The solutions require integrated measurable results at multiple levels known as “grand challenges” to existing methods. Finally, Section 10 adheres to an appeal for action, advocating the necessity for further long-term support of the INBIOSA program. -/- The report is concluded with preliminary non-exclusive list of challenging research themes to address, as well as required administrative actions. The efforts described in the ten sections of this White Paper will proceed concurrently. Collectively, they describe a program that can be managed and measured as it progresses. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Spatializing Culture: The Ethnography of Space and Place by Setha LowCarlos J. L. BalsasSpatializing Culture: The Ethnography of Space and Placeby setha low London: Routledge, 2017Spatializing Culture: The Ethnography of Space and Place adds clarity to our understanding of the value of ethnographic scholarship in the study of socio-economic, cultural, and developmental transformations. The book is a thorough review of two established conceptual frames of analysis—the social production (...) of space and the social construction of space—coupled with an in-depth discussion of four newer ways to make sense of the world around us: embodied space, language and discourse, emotion and affect, and translocal space. As the world is transformed by a myriad of forces and events occurring simultaneously, its complexity intensifies insurmountably and we require clear and robust concepts, theories, methods, and tools to not only interpret it, but above all to attempt to change it in ways that are more beneficial to our collective co-existence while enabling the flourishing of those most at risk.Readers of Spatializing Culture will find plentiful theoretical constructs, lineages, diverse viewpoints, examples, and overall opportunities for reflecting on how the (built) environment, spaces, and places have been changing mostly due to our behavior. Inspired by Foucault’s take on genealogy, in chapter 2, Setha Low traces the history of space and place according to distinct interpretative schools of thought ranging from philosophical and mathematical, French social theory, geographical, architectural, environmental psychology, to anthropological, and even archeological. Five Venn diagrams are utilized to illustrate the main conceptual relationships between space and place. Most of the book’s in-depth case studies were researched by the author with only a small number investigated and written up by fellow scholars, which Low utilizes in the book with gratitude. The interrelated thematic and distinct geographical scope of the cases is one of the book’s hallmarks. The illustrative case studies come from such faraway places as Latin America (San José, Costa Rica), Asia (Taipei, Taiwan), North America (New York City; Washington, D.C.; Philadelphia, Pennsylvania; and San [End Page 151] Antonio, Texas), Southern Europe (Smederevska Palanka, Serbia), and the Middle East (Beirut, Lebanon; Cairo, Egypt; and Tel Aviv, Israel).Spatializing Culture was first mentioned to me years ago by a former colleague at the University at Albany, SUNY. His ongoing work in Latin America, coupled with his ethnographic training and prior work in a region of the world also extensively examined by Low in other scholarly works, was likely the reason for bringing it up in a conversation about the latest developments in the field and potential research opportunities. I acquired the book when I was teaching Global Urbanism and Culture and International Urban Planning. I was pleasantly surprised with the book’s thorough conceptual lineage, historic characterizations, attention to detail, comprehensive ethnographic investigations, and implications for transforming reality as well as institutional processes. Spatializing Culture is a welcome addition to urban planning textbooks, which tend to be relatively less critical, are often covered with normative constructs and statistics, and are more institutional in their characterization of problems and recommendations for further action.For the most part, the ethnographic method is qualitative, participa-tory, detailed, and quite comprehensive in characterizing key variables as the researcher found them on the ground. This differs slightly from urban planning’s preoccupation with public policy, policy analysis, rules and regulations, institutionalized procedures, the public good, and overall positivist and rationalist concerns with documenting reality and helping to lead transformative planning processes, whether in the realm of land use, transportation, housing, economic development, infrastructure, or the environment. Although International Urban Planning was to be taught in a rather orthodox manner with recourse to understanding planning cultures, best practices, lessons learned, and implications for similar situations, the Global Urbanism and Culture course was slightly more aligned with the discussions put forward by Low. These included attention to postmodern ideals, the use of creativity in community planning, distinctions between popular and highbrow culture, and the building of iconic structures in contexts of urban regeneration and mega-event planning. As such, my students were somewhat already exposed to issues of embodied spaces and to ideas of... (shrink)

In this paper we argue that to adequately capture the complicated relationship between Descartes' work and late medieval thought, philosophers need to think not only about his ideas but also about his presentation and choice of genre. Reading the Meditations as a mere discursive treatise containing a progressive and consistent set of arguments intended to establish a particular philosophical position fails to appreciate the eponymous genre that Descartes explicitly chose to employ in writing them. Instead, we argue that reading the (...) text in light of the long tradition of contemplative meditative literature opens up a radically new way of interpreting the Meditations as a “mystagogical” project of spiritual edification and epistemic and volitional transformation. During such a transformation, the meditator undergoes a series of revelatory experiences structurally similar to those we find in various so-called “mystical” or “mystagogical”literature, in the Christian West. We show how paying attention to these mystagogical aspects of the Meditations can open up new avenues for understanding the connection between the theoretical, practical, and theological aspects of Descartes' thought. Such a reading resolves a number of seemingly glaring problems and peculiarities in the Meditations including the Cartesian circle, and the tension in the Cartesian notion of the freedom of the will. We also argue that the odd detours the meditator takes in the Sixth Meditation through physiology are not mere digressions but rather mark an often overlooked aspect of the mystagogical story – the return to embodiment as a transformed self. Indeed, the overly intellectualist emphasis of contemporary scholars on both the contemplative aspects of mystagogical literature and the epistemological focus of Descartes' project in the Meditations have caused them to largely overlook the ways in which the aim of both medieval and Cartesian meditation involves a transformative re-orientation of the whole person and their understanding of the world – one which has practical, moral, and social consequences that have gone underexplored in discussions of these texts. (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)

This book is essential for those interested in Russell, or in the philosophy of logic and mathematics. Only two of the fifteen selections are readily available elsewhere: "On Denoting" and "Mr Strawson on Referring." Amazingly, three substantial selections have never been published previously: "On the Substitutional Theory of Classes and Relations," "The Regressive Method of Discovering the Premisses of Mathematics," and "Is Mathematics Purely Linguistic?". Three other papers have appeared previously only in French : one of these is a better (...) introduction to the theory of types than the classic "Mathematical Logic as based on the Theory of Types." Finally, of the five remaining papers, which are not easily available, three are commentaries on Meinong, one is a note on the axiom of infinity, and the most substantial piece is a discussion of three notions of sets—in effect, Zermelo-Frankel, New Foundations, and Russell’s own "no-classes" theory, which required a theory of types applied to propositional functions. What is striking throughout these essays—all but one from the first decade of the century-is Russell’s persistent sympathy with some sort of realism in foundation theory, and his perception that the theory of definite descriptions, the "no-classes" proposals, and the theory of types, were very closely interrelated. Also included are three bibliographies. The chronological bibliography includes several still unpublished mss in the McMaster University Russell Archives.—J.L. (shrink)

Our paper explores how the affective energies and cultural expectations set in motion by best-selling American sentimental novels like Hannah Foster's The Coquette and Susanna Rowson's Charlotte Temple informed the notorious mid-nineteenth-century American trial of Amelia Norman, who attempted to kill the man who seduced her. Once newspapers, defense lawyers, and reformers such as Lydia Maria Child recast the defendant as a sentimental heroine, the trial became about seduction, and Norman was acquitted against the weight of the evidence. Sentimental novels (...) turn on the contrast between the passive victim status of the heroine and the active libidinal quest of the rake-villain. As Cathy Davidson points out, this fiction is "about silence, subservience, stasis (the accepted attributes of women as traditionally defined) in contradistinction to conflicting impulses toward independence, action, and self expression (the ideals of the new American nation)." The seduction plot diverts attention from this disparity by establishing an affective solidarity between heroine and reader and elevating the ruined woman to a tragic status. The sentimental emplotment of Norman's life marshaled a powerful set of emotional responses and moral judgements on her behalf. Norman claimed insanity. And since sentimental heroines are supposed to go mad when they are seduced and abandoned, the jury was prepared to interpret her symptoms according to her lawyers' very strategy for establishing her innocence. Ultimately, however, Norman embodied the plight of the sentimental heroine at the same time that she contested her fictional counterpart's fate. In this way, her trial spectacularized the disparity which the sentimental novel conjures up and displaces but never resolves. The common law theory of coverture, which severely limited the legal personhood of married women, has received a great deal of scholarly attention. Cases like Norman's remind us that unmarried women were also subject to draconian constraints on their legal personhood. The tort of seduction is a key example. Legal historians trace the development of the seduction tort from its common-law origins, when men's property interest in women's bodies formed the basis of the cause of action, to 1851, when Field Code authors (including Norman's lawyer David Graham) persuaded several states to grant seduced women standing to bring their own cause of action. Consequently, courts were forced to reckon with the seduced woman as a moral agent capable of consenting to sex. As trials like Norman's demonstrate, sentimental novels helped lay the groundwork for this shift in the law by elucidating a subjectivity for the seduced woman. Yet the doctrinal implications of Norman's precedent-setting trial had a second, more ambiguous strain. Norman's sentimental strategy proved so powerful that men on trial for killing their wive's seducers appropriated it to bring their own stories before juries and to reinforce male sexual norms. In the end, then, Norman's trial fostered legal reform, but it also suggested - as Child's fictionalization of the case in "Rosenglory" recognized - that only sustained and multifaceted efforts to change cultural as well as legal norms could improve the sexual status of women. Scholars such as Wai Chee Dimock have argued for a focus "on the historical and historically shifting relations between law and literature" - a view we endorse. Where we differ from Dimock is in our diversion of attention away from abstract ideas of law laid out by treatise writers and philosophers in favor of law experienced and manipulated by individuals. So, too, we are interested less in representations of concepts such as justice in legal and literary texts than we are in the ways in which literature (broadly conceived) can create provisional and fragile opportunities for concrete instantiations of justice and even generate legal change (for good or ill). Going further, we would argue that to the extent legal change motivates rather than simply mirrors cultural change, it needs literature to be effective. Thus, we see no need to characterize - and indeed reject the depiction of - law as "a limited arena" in which ideas of morality have little place. This project, then, responds to Gregg Crane's call for attention to "the complex and slippery historical interactions of law and literature that shape and are shaped by an ever changing cultural idiom of justice." The extended story of Amelia Norman not only constitutes a case study in the inescapable interaction between the overlapping and interdependent discourses of law and literature, but also reveals the literary and legal consequences of that interaction. (shrink)