We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of (...) reals in models of ZF. In particular, we show that ZFC + is equiconsistent with ZF + \\) where \ is the pointclass of all \-Suslin sets of reals, and also with ZF + \\) + \\) where \ is the least ordinal that is not a surjective image of the reals. (shrink)

We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of (...) reals in models of ZF. In particular, we show that ZFC + is equiconsistent with ZF + \\) where \ is the pointclass of all \-Suslin sets of reals, and also with ZF + \\) + \\) where \ is the least ordinal that is not a surjective image of the reals. (shrink)

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by (...) defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal. (shrink)

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal princi...

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + (...) “Ord is not ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. (shrink)

Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad (...) class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders. (shrink)

We study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the existence (...) of a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $. (shrink)

For each natural number n, let C(n) be the closed and unbounded proper class of ordinals α such that Vα is a Σn elementary substructure of V. We say that κ is a C(n)-cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C(n). By analyzing the notion of C(n)-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong (...) cardinals and up to the level of rank-into-rank embeddings, C(n)-cardinals form a much finer hierarchy. The naturalness of the notion of C(n)-cardinal is exemplified by showing that the existence of C(n)-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of Bagaria et al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C(n)-extendible cardinals. (shrink)

We generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopěnka-Hrbáček theorem from ZF + DC only (the proof (...) of Vopěnka and Hrbáček used the full axiom of choice): if there exists a strongly compact cardinal, then the universe is not constructible from a set. The same method shows that, in L[ 2 ω ], there cannot exist a θ-compact cardinal less than θ (where θ is the least cardinal onto which the continuum cannot be mapped); a similar result can be proven for other models of the form L[ A ]. The result for L[ 2 ω ] is of particular interest in connection with the axiom of determinacy. The extended ultrapower construction of this paper is an improved version of the author's earlier pseudo-ultrapower method, making use of forcing rather than the omitting types theorem. (shrink)

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding j:V→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j: V \to M}$$\end{document} such that M is closed under sequences of length sup{j|f:κ→κ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sup\{{j\,|\,f: \kappa \to \kappa}\}}$$\end{document}. Some of (...) the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopěnka cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Among the results, I highlight the following. Vopěnka cardinals are the same as Woodin-for-supercompactness cardinals.There are no excessively hypercompact cardinals.Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. (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 consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our (...) investigation of this helical structure, we have new characterizations of extendible cardinals and of Vopěnkan cardinals in terms of elementary embeddings from the universe V, a new large large cardinal property which looks like Shelahness properly between Vopěnkaness and almost hugeness, and a more direct relation between Vopěnkaness and Woodinness than already known.We also consider limits of the helices and relations with the axioms I1–I3. (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)

Karol Wojtyła found phenomenology very helpful for the analysis of concrete human experience and for overcoming the ethical formalism ofKant. Phenomenology, he believed, could also enrich classical Thomism by exploring the lived experience of freedom, interiority, and self-governance. But phenomenology, in his opinion, needed to be supplemented by metaphysics in order to ground experiences such as the sense of duty in the real order. He criticized much modern philosophy for abandoning metaphysics and thus neglecting the sapiential dimension. Since his career (...) as a professor was very short, he did not have time to complete his project of a personalist Thomism in which phenomenology and metaphysics would be harmoniously combined. (shrink)

Cardinal Mercier’s Manual of Modern Scholastic Philosophy is a standard work, prepared at the Higher Institute of Philosophy, Louvain, mainly for the use of clerical students in Catholic Seminaries. Though undoubtedly elementary, it contains a clear, simple, and methodological exposition of the principles and problems of every department of philosophy, and its appeal is not to any particular class, but broadly human and universal. Volume I includes a general introduction to philosophy and sections on cosmology, psychology, criteriology, and metaphysics or (...) ontology. (shrink)

Rousseau’s political project consists in ensuring that the citizens of the social contract, in uniting with each other, preserve their ability to self-legislate, or be autonomous. For this to work, however, members of the social contract would need to feel intrinsically linked to the political whole. This essay investigates what that feeling might be and how it can be grown. I argue that Rousseau develops a model of the energy or character of the being capable of autonomy, capable of experiencing (...) themselves as part of the whole. That energy is a pathos of vigour, a strong sentiment and way of being that I develop from Rousseau’s educational precepts in Émile, which makes the citizen feel free and robust in dependence and boundedness. Autonomy, then, comes from the active exercise of oneself, physically and mentally, in an environment bounded by things, and this results in a sentiment of vigour, a vitality that produces confidence and poise, which then entices further activity. (shrink)

Real-life stories from teachers who share their passion for shaping the lives of young people today.

'ground-breaking anthology... wide array of perspectives on WW1, from both sides of the fighting' -B. Adler, Choice 'a very fine anthology' -Times Literary SupplementThe First World War inspired a huge outpouring of writing that, until recently, was thought to be almost the exclusive preserve of men. Yet the war also acted as a catalyst which enabled women writers to find a literary and political voice. This anthology bears witness to the great variety and scope of women's writing about the war. (...) Covering the years 1914-1930, the book brings together little-known writing from across the world, drawing on a wide spectrum of genres from letters and diary entries to essays, novels, and short stories. Among the celebrated writers featured in this anthology are: Radclyffe Hall, Sylvia Pankhurst, Maud Gonne, Rebecca West, May Sinclair, Olive Schreiner, Colette, Vera Brittain, Gertrude Stein, Katherine Mansfield, HD, and Virginia Woolf. (shrink)

The proposed heavy dependence on Pavlovian conditioning to account for social behavior confounds phylogenically and ontogenically selected behavior patterns and ignores the extension of the principle of selection by consequences from biological to learning theory. Instead of acknowledging operant relations, Domjan et al. construct vaguely specified mechanisms based upon anticipatory cost-benefit considerations that are not supported by the Pavlovian conditioning literature.

In Balcony in the Forest, Julien Gracq composes a soundscape in four dimensions: he establishes an undulating background by involving spatial events; he forms temporal figures by involving material affects—spatial events and material affects extracted from depths before being enveloped by a resonant place. Each of the four dimensions has a particular relationship to the sounds of power and the power of sound, and it is up to the reader to decide whether the soundscape composed in and by the writing (...) makes this double... (shrink)