This paper looks at philosophical questions that arise in the context of AI alignment. It defends three propositions. First, normative and technical aspects of the AI alignment problem are interrelated, creating space for productive engagement between people working in both domains. Second, it is important to be clear about the goal of alignment. There are significant differences between AI that aligns with instructions, intentions, revealed preferences, ideal preferences, interests and values. A principle-based approach to AI alignment, which combines these elements (...) in a systematic way, has considerable advantages in this context. Third, the central challenge for theorists is not to identify ‘true’ moral principles for AI; rather, it is to identify fair principles for alignment that receive reflective endorsement despite widespread variation in people’s moral beliefs. The final part of the paper explores three ways in which fair principles for AI alignment could potentially be identified. (shrink)

What is it for a picture to depict a scene? The most orthodox philosophical theory of pictorial representation holds that depiction is grounded in resemblance. A picture represents a scene in virtue of being similar to that scene in certain ways. This essay presents evidence against this claim: curvilinear perspective is one common style of depiction in which successful pictorial representation depends as much on a picture's systematic differences with the scene depicted as on the similarities; it cannot be analyzed (...) in terms of similarity alone. The same problem arises for many other kinds of depiction. The essay concludes that depiction in general is not grounded in resemblance but geometrical transformation. (shrink)

A semantics of pictorial representation should provide an account of how pictorial signs are associated with the contents they express. Unlike the familiar semantics of spoken languages, this problem has a distinctively spatial cast for depiction. Pictures themselves are two-dimensional artifacts, and their contents take the form of pictorial spaces, perspectival arrangements of objects and properties in three dimensions. A basic challenge is to explain how pictures are associated with the particular pictorial spaces they express. Inspiration here comes from recent (...) proposals that analyze depiction in terms of geometrical projection. In this essay, I will argue that, for a central class of pictures, the projection-based theory of depiction provides the best explanation for how pictures express pictorial spaces, while rival perceptual and resemblance theories fall short. Since the composition of pictorial space is itself the basis for all other aspects of pictorial content, the proposal provides a natural foundation for further pictorial semantics. (shrink)

Because humans cannot know one another’s minds directly, every form of communication is a solution to the same basic problem: how can privately held information be made publicly accessible through manipulations of the physical environment? Language is by far the best studied response to this challenge. But there are a diversity of non-linguistic strategies for representation with external signs as well, from facial expressions and fog horns to chronological graphs and architectural renderings. The general thesis of this dissertation is that (...) there is an impressively wide spectrum of conventional systems of representation, corresponding to the many ways that the problem of communication can be solved, and that these systems can be described and explained using the tools of contemporary mathematical semantics. As a partial corrective to the countervailing norm, this work concentrates on the class of systems arguably most different from language— those governing the interpretation of pictorial images. Such representations dominate practical communication: witness the proliferation of maps, road signs, newspaper photographs, scientific illustrations, television shows, engineering drawings, and even the fleeting imagery of manual gesture. I argue that systems of depiction and languages embody a parallel technologies of communication. Both are based on semantics: systematic and conventional mappings from signs to representational content. But I also provide evidence that these semantics are profoundly divergent. Whereas the semantics of languages are based on arbitrary associations of signs and denotations, the semantics of systems of depiction are based on rules of geometrical transformation. Drawing on recent research in computer graphics and computational vision, I go on to develop a precise theory of pictorial semantics. This in turn facilitates a detailed comparison of iconic, image-based representation, and symbolic, language-based representation. A consequence of these conclusions is that the traditional, language-centric conception of semantics must be overhauled to allow for a more general semantic theory, one which countenances the wide variety of interpretive mechanisms actually at work in human communication. (shrink)

We look at recent accounts of the indefinite extensibility of the concept set and compare them with a certain linguistic model of indefinite extensibility. We suggest that the linguistic model has much to recommend over alternative accounts of indefinite extensibility, and we defend it against three prima facie objections.

The aim of this paper is to assess the relative merits of two accounts of the semantics of proper names. The enterprise is of particular interest because the theories are very similar in fundamental respects. In particular, they can agree on three major features of names: names are rigid designators; different co-extensive names can have different cognitive significance; empty proper names can be meaningful. Neither theory by itself offers complete explanations of all three features. But each theory is consistent with (...) them and goes some way towards explaining them. (shrink)

Bertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell's paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the (...) uses to which modern descendente of Russell's paradox of propositions have been put in recent literature. (shrink)

Where do we come from? Are we merely a cluster of elementary particles in a gigantic world receptacle? And what does it all mean? In this highly original new book, the philosopher Markus Gabriel challenges our notion of what exists and what it means to exist. He questions the idea that there is a world that encompasses everything like a container life, the universe, and everything else. This all-inclusive being does not exist and cannot exist. For the world itself (...) is not found in the world. And even when we think about the world, the world about which we think is obviously not identical with the world in which we think. For, as we are thinking about the world, this is only a very small event in the world. Besides this, there are still innumerable other objects and events: rain showers, toothaches and the World Cup. Drawing on the recent history of philosophy, Gabriel asserts that the world cannot exist at all, because it is not found in the world. Yet with the exception of the world, everything else exists; even unicorns on the far side of the moon wearing police uniforms. Revelling in witty thought experiments, word play, and the courage of provocation, Markus Gabriel demonstrates the necessity of a questioning mind and the role that humour can play in coming to terms with the abyss of human existence. (shrink)

We criticize attempts to define hope in terms of other psychological states and argue that hope is a primitive mental state whose nature can be illuminated by specifying key aspects of its functional profile.

I would like to discuss the claim that the resources of plural reference and plural quantification are sufficient for the purpose of paraphrasing all ordinary statements apparently concerned with composite material objects into plural statements concerned exclusively with simples.

Several philosophers have argued recently that semantic properties do play a causal role. 1 It is our view that none of these arguments are satisfactory. Our aim is to reveal some of the deficiencies of these arguments, and to reassess the question in our own way. In section 1, we shall explain in more detail what is involved in the pretheoretical idea of a causally efficacious property and so provide a fuller sense of the issue. In section 2 we shall (...) discuss Fodor's and Kim's arguments that the semantic properties of mental events are causally efficacious. Their tactic is to articulate sufficient conditions for the causal efficacy of a property, and then apply these conditions to the psychological properties under consideration. We shall show that both formulations are inadequate. However, each account can remedy some of the defects of the other. We shall argue that by putting the two together, and modifying this combined account in certain ways, we can arrive at something that might just pass as a general sufficient condition for causal efficacy. We believe that the question of causal efficacy of content is best approached as a question in the philosophy of science. If one wants to know whether a given species of property is efficacious, the most productive strategy is to find a science that investigates the properties and examine the role they are assigned. In keeping with this attitude we shall restrict our discussion to the notion of content deployed within cognitive science. In section 3 we examine the role of content in cognitive science and discuss whether this role suits content to the conditions on efficacy articulated in section two. Our conclusion will be conditional and hedged: on certain views of cognitive science, content will meet the sufficient condition for efficacy. If other views are correct, the question remains open. (shrink)

In recent years, many disciplines have become interested in the scientific study of morality. However, a conceptual framework for this work is still lacking. In The Moral Background, Gabriel Abend develops just such a framework and uses it to investigate the history of business ethics in the United States from the 1850s to the 1930s. According to Abend, morality consists of three levels: moral and immoral behavior, or the behavioral level; moral understandings and norms, or the normative level; and (...) the moral background, which includes what moral concepts exist in a society, what moral methods can be used, what reasons can be given, and what objects can be morally evaluated at all. This background underlies the behavioral and normative levels; it supports, facilitates, and enables them. Through this perspective, Abend historically examines the work of numerous business ethicists and organizations--such as Protestant ministers, business associations, and business schools--and identifies two types of moral background. "Standards of Practice" is characterized by its scientific worldview, moral relativism, and emphasis on individuals' actions and decisions. The "Christian Merchant" type is characterized by its Christian worldview, moral objectivism, and conception of a person's life as a unity. The Moral Background offers both an original account of the history of business ethics and a novel framework for understanding and investigating morality in general. (shrink)

This paper takes a close look at the thought that mereological relations on material objects mirror, and are mirrored by, parallel mereological relations on their exact locations. This hypothesis is made more precise by means of a battery of principles from which more substantive consequences are derived. Mereological harmony turns out to entail, for example, that atomistic space is an inhospitable environment for material gunk or that Whiteheadian space is not a hospitable environment for unextended material atoms.

ABSTRACT This essay has three goals. The first is to introduce the notion of fundamentality and to argue that physicalism can usefully be conceived of as a thesis about fundamentality. The second is to argue for the advantages of fundamentality physicalism over modal formulations and that fundamentality physicalism is what many who endorse modal formulations of physicalism had in mind all along. Third, I describe what I take to be the main obstacle for a fundamentality-oriented formulation of physicalism: ‘the problem (...) of abstracta’, which asks how physical can accommodate phenomena such as mathematics and universals, and which modal formulations do not face. I canvas three solutions: the inapt for ground solution, the concrete restriction, and the contingency restriction. (shrink)

Two semantic theories of proper names are explained and assessed. The theories are Burge’s treatment of proper names as complex demonstratives and Larson and Segal’s quasi-descriptivist account of names. The two theories are evaluated for empirical plausibility. Data from deficits, processing models, developmental studies and syntax are all discussed. It is concluded that neither theory is fully confirmed or refuted by the data, but that Larson and Segal’s theory has more empirical plausibility.

Ethical issues at the workplace have once again become topical and important due to considerable adverse publicity surrounding reports of unethical business practices by corporate managers. Accordingly, this paper re-visits the question of whether gender, age and work experience do have an effect on ethical judgement, using 655 business students as respondents. This is necessary as business students are likely to become managers during their career and will face complex ethical concerns and dilemmas in their daily, routine affairs. The findings (...) currently demonstrate that females are more ethically aware than their male counterparts – that is, there are differences between males and females regarding ethical judgement. There is also evidence to suggest that age is a factor that does impact on ethical judgement. However, we also found evidence to suggest that in some cases, the age factor does not necessarily have a significant impact on ethical awareness. The results further indicate that there is a difference in ethical judgement related to work experience. However, at the same time, in one question there is no evidence to suggest the claim that work experience does impact significantly on ethical judgement. Overall, it appears that ethical awareness does increase alongside work experience. (shrink)

Hubert Dreyfus has defended a novel view of agency, most notably in his debate with John McDowell. Dreyfus argues that expert actions are primarily unreflective and do not involve conceptual activity. In unreflective action, embodied know-how plays the role reflection and conceptuality play in the actions of novices. Dreyfus employs two arguments to support his conclusion: the argument from speed and the phenomenological argument. I argue that Dreyfus's argumentative strategies are not successful, since he relies on a dubious assumption about (...) concepts and reflection. I suggest that Dreyfus is committed to a minimal view of conceptuality in action. (shrink)

This article seeks to contribute to a vision for leadership in business based on a recovery of virtue. The vision for leadership articulated here draws principally on the writings of the classical philosopher Aristotle and of the contemporary philosopher Josef Pieper. Without discounting the ever-increasing complexity of modern business, this essay will attempt to reconstruct Aristotle’s emphasis on virtue and moral character, and argues for the philosopher’s relevance to modern management and corporate leadership. The paper concludes that the message of (...) virtue is a message of hope and attempts to find plain language to articulate its value to those engaged in business or concerned with the formulation of government policy. (shrink)

This essay argues for a model of pictorial representation which aims to explain the relationship between pictorial content and pictorial accuracy. Focusing on cases where pictures are intended to convey accurate information, the model distinguishes between two fundamental representational relations: on one hand, a picture expresses a content; on the other, it aims at a target scene. Such a picture is accurate when the content it expresses fits the target scene it aims at. In addition, the model follows the traditional (...) division of content into two aspects: singular content specifies the particular individuals which a picture is of, and attributive content specifies the properties and relations which the picture ascribes to those individuals. For a picture to be accurate, both aspects must be matched in the target. I call this the Three-Part Model because it distinguishes between the triad of factors, singular content, attributive content, and target, which together determine pictorial accuracy. Though previous work on depiction has not recognized the distinctive role played by target, I will argue that it is essential in order to make sense of accuracy judgements across a range of central cases. (shrink)

Thought: A Journal of Philosophy, EarlyView.

Apologizing for Socrates places some of the Platonic and Xenophontic writings in the context of contemporary controversies over Socrates, providing a perspective in which many of the philosophic and literary features of the text can be explained. In addition, it sheds light on the apologetic techniques used by Plato and Xenophon.

Carnap’s seminal ‘Empiricism, Semantics and Ontology’ makes important use of the notion of a framework and the related distinction between internal and external questions. But what exactly is a framework? And what role does the internal/external distinction play in Carnap’s metaontology? In an influential series of papers, Matti Eklund has recently defended a bracingly straightforward interpretation: A Carnapian framework, Eklund says, is just a natural language. To ask an internal question, then, is just to ask a question in, say, English. (...) To try to ask an external question is to try, absurdly, to ask a question in no language at all. Finding that so trivial an I/e distinction can’t help to explain Carnap’s deflationary metaontology, Eklund is led to attribute to Carnap a view he calls ontological pluralism. In this paper, I show that Eklund misreads Carnap, and I argue that this misreading obscures fundamental features of Carnap’s philosophy. I then defend an account of frameworks as what Carnap called semantical systems, and I place this account in the context of Carnap’s philosophical program of explication. Finally, I discuss the role that frameworks and the I/e distinction play in ESO, showing that ESO provides no reason to attribute the doctrine of ontological pluralism to Carnap. (shrink)

In this paper, I first explore Hegel’s own distinctions between various types of idealism, most of which he explicitly rejects. I discuss his notions of subjective, transcendental and absolute idealism and present the outlines of his criticisms of the first two as well as the motivation behind his commitment to a version of absolute idealism. In particular, I argue that the latter does not share the defining features of what is now commonly called ‘idealism’, as Hegel neither denies the existence (...) of an external world nor even holds that we can only somehow indirectly infer the truth of propositions about the external world from the structure of some given mental material. I give a basic account of what Hegel’s concept of ‘the absolute idea’ is about, which lies behind his absolute idealism. In this context, I maintain that it is crucial for our understanding of Hegel and of his potential relevance for contemporary metaphysics and epistemology that the absolute idea is precisely not a mental or ‘spiritual’ (geistig) entity. Rather, it amounts to a set of methodological assumptions designed to guarantee the overall intelligibility of what there is, regardless of its actual natural, social or more broadly normative structure. (shrink)

Warfare is becoming increasingly automated, from automatic missile defense systems to micro-UAVs (WASPs) that can maneuver through urban environments with ease, and each advance brings with it ethical questions in need of resolving. Proponents of lethal autonomous weapons systems (LAWS) provide varied arguments in their favor; robots are capable of better identifying combatants and civilians, thus reducing "collateral damage"; robots need not protect themselves and so can incur more risks to protect innocents or gather more information before using deadly force; (...) robots can assess situations more quickly and do so without emotion, reducing the likelihood of fatal mistakes due to human error; and sending robots to war protects our own soldiers from harm. However, these arguments only point in favor of autonomous weapons systems, failing to demonstrate why such systems need be made *lethal*. In this paper I argue that if one grants all of the proponents' points in favor of LAWS, then, contrary to what might be expected, this leads to the conclusion that it would be both immoral and illegal to deploy *lethal* autonomous weapons, because the many features that speak in favor of them also undermine the need for them to be programmed to take lives. In particular, I argue that such systems, if lethal, would violate the moral and legal principle of necessity, which forbids the use of weapons that impose superfluous injury or unnecessary harm. I conclude by highlighting that the argument is not against autonomous weapons per se, but only against *lethal* autonomous weapons. (shrink)

Agents can think using concepts they do not fully understand. This paper investigates the question “Under what conditions does a thinker fully understand, or have mastery of, a concept?” I lay out a gauntlet of problems and desiderata with which any theory of concept mastery must cope. I use these considerations to argue against three views of concept mastery, according to which mastery is a matter of holding certain beliefs, being disposed to make certain inferences, or having certain intuitions. None (...) of these attitudes is either necessary or sufficient for mastery. I propose and respond to objections to my own recognition view of the conditions under which a thinker has mastery of a concept. (shrink)

Do mereological fusions have their parts necessarily? None of the axioms of non-modal formulations of classical mereology appear to speak directly to this question. And yet a great many philosophers who take the part-whole relation to be governed by classical mereology seem to assume that they do. In addition to this, many philosophers who make allowance for the part-whole relation to obtain merely contingently between a part and a mereological fusion tend to depart from non-modal formulations of classical mereology at (...) least when it comes to the axiom of Unique Fusion, which states that no two different mereological fusions ever fuse exactly the same objects. This is no coincidence. There are reasons of principle why one’s adherence to classical mereology should exert some pull towards the view that mereological fusions have their parts necessarily. There is, however, no direct route from the combination of classical mereology and propositional modal logic to the hypothesis that the part-whole relation obtains necessarily between a part and a mereological fusion. In order to bridge between a modal formulation of classical mereology and the hypothesis that fusions have their parts necessarily, one needs to strengthen the axiom of Unrestricted Fusion in a way that is agreeable to many philosophers on both sides of the debate. (shrink)

The predominant interpretation of Wittgenstein's later remarks on religion takes him to hold that all religious utterances are non-scientific, and to hold that the way to show that religious utterances are non-scientific is to identify and characterise the grammatical rules governing their use. This paper claims that though this does capture one strand of Wittgenstein's later thought on religion, there is an alternative strand of that thought which is quite different and more nuanced. In this alternative strand Wittgenstein stresses that (...) religious utterances and beliefs can come in both scientific and non-scientific varieties. More than that, he claims that the grammar of religious utterances, and the logic of religious beliefs, is often complex – in that individual utterances and beliefs will often be mixed between, indeterminate between, or fluid between being scientific and being non-scientific. This complexity means that it will often be unhelpful to try to pin down one particular grammar or logic for a given utterance or belief. Wittgenstein therefore suggests a new method of grammatical and logical investigation, which is less likely to distort complex grammars or logics by being overly simplistic or rigid. This method is to use simple examples of utterances and beliefs as objects of comparison, so as to illuminate the different aspects of the more complex actual utterances and beliefs under examination. This alternative strand in Wittgenstein's later remarks on religion is a manifestation of a broader strand of Wittgenstein's later thought as a whole, which was first described by Friedrich Waismann, and later developed by Gordon Baker and Oskari Kuusela. The paper concludes by providing examples of religious beliefs which are logically mixed, indeterminate, and fluid, and showing how simple objects of comparison can be used to illuminate them. (shrink)

Part I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung]” of his formal system.

The standard model of precedential constraint holds that a court is equally free to modify a precedent of its own and a precedent of a superior court—overruling aside, it does not differentiate horizontal and vertical precedents. This paper shows that no model can capture the U.S. doctrine of precedent without making that distinction. A precise model is then developed that does just that. This requires situating precedent cases in a formal representation of a hierarchical legal structure, and adjusting the constraint (...) that a precedent imposes based on the relationship of the precedent court and the instant court. The paper closes with suggestions for further improvements of the model. (shrink)

The prelims comprise: To Kalon as Effective Teleological Order The Visibility of the Fine Pleasure and Praise The Value of the Fine Conclusion Acknowledgments Notes References Further reading.

Fichte's Foundations of Natural Right was one of the most influential books in nineteenth-century philosophy. It was read carefully by Schelling, Hegel, and Marx, and initiated a tradition in German philosophy that considers human subjectivity to be relational and intersubjective, thus requiring relations of recognition between subjects. The essays in this volume highlight this little-understood book's most important ideas and innovations. They offer discussions of Fichte's conception of freedom, self-consciousness, coercion, the summons, the body, and human rights, together with new (...) analyses of his deduction of right, his views on the social contract, and his arguments for the separation of right from morality. The essays expand and deepen ongoing debates in the scholarship and chart new avenues of thought about Fichte's most enduring work of political philosophy. They will be essential reading for students and scholars of German Idealism, nineteenth-century philosophy, and the history of political thought. (shrink)

Some political philosophers believe that equality emerges as a moral concern where and because people coerce each other. I shall argue that they are wrong. The idea of coercion as a trigger of equality is neither as plausible nor as powerful as it may initially appear. Those who rely on the idea that coercion is among the conditions that give rise to equality as a moral demand face a threefold challenge. They will have to succeed in jointly (a) offering a (...) convincing account of the wrongness of coercion, (b) rendering cogent the idea that the demand of equality arises in response to the moral problem of coercion, and (c) identifying some types of interaction as relevantly coercive. This challenge, I believe, cannot be met. More precisely, I argue that two important accounts of coercion fail to meet it. (shrink)

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small (...) model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs. (shrink)

We introduce several senses of the principle ofcompositionality. We illustrate the difference between them with thehelp of some recent results obtained by Cameron and Hodges oncompositional semantics for languages of imperfect information.

While much has been written on phronesis, there is a dearth of empirical work on the how the concept can be developed and implemented in practice, particularly in an educational setting. To address this problem, characteristics of phronesis were identified through a review of current literature and an examination of related themes from a special issue of the Philosophy of Management Journal on the philosophy of innovation. The implementation of the concept was investigated using an illustrative study of ongoing work (...) aimed at embedding phronesis in an innovation educational module. The study will contribute to the nascent debate by providing evidence of embedding Aristotle’s phronesis in the curriculum of future managers, in this case Engineering and Business students, working in cross-functional teams. A schema is proposed, based on concepts distilled from the literature and the illustrative study, to examine whether an educational project cultivates phronesis among the students. (shrink)

David Kaplan observed in Kaplan that the principle \\) cannot be verified at a world in a standard possible worlds model for a quantified bimodal propositional language. This raises a puzzle for certain interpretations of the operator Q: it seems that some proposition p is such that is not possible to query p, and p alone. On the other hand, Arthur Prior had observed in Prior that on pain of contradiction, ∀p is Q only if one true proposition is Q (...) and one false proposition is Q. The two observations are related: ∀p is elusive in that it is not possible for the proposition to be uniquely Q. Kaplan based his model-theoretic observation on Cantor’s theorem, but there is a less well-known link between this simple set-theoretic observation and Prior’s remark. We generalize the link to develop a heuristic designed to move from Cantor’s theorem to the observation that a variety of sentences of the bimodal language express propositions that cannot be Q uniquely. We highlight the analogy between some of these results and some set-theoretic antinomies and suggest that the phenomenon is richer than one may have anticipated. (shrink)

We present two studies investigating the impact of causal perceptions and the moral emotions of anger, shame, and guilt on the justification of deviant workplace behavior. Study 1 tests our conceptual framework using a sample of undergraduate business students; Study 2 examines a population of practicing physicians. Results varied significantly between the two samples, suggesting that individual and contextual factors play an important role in shaping the perceptual and emotional processes by which individuals form reactions to undesirable affective workplace events. (...) Implications of these findings for the study of ethics, emotions, and attributions, as well as for promoting ethical behavior, are discussed. (shrink)

We define a formula φ in a first-order language L , to be an equation in a category of L -structures K if for any H in K , and set p = {φ;i ϵI, a i ϵ H} there is a finite set I 0 ⊂ I such that for any f : H → F in K , ▪. We say that an elementary first-order theory T which has the amalgamation property over substructures is equational if every quantifier-free (...) formula is equivalent in T to a boolean combination of equations in Mod, the category of models of T with embeddings for morphisms. Thus, we develop a theory of independence with respect to equations in general categories of structures, which is similar to the one introduced in stability but which, in our context, has an algebraic character. (shrink)

In this highly original book, Markus Gabriel presents 'Neo-Existentialism', an anti-naturalist view that holds that human mindedness consists in an open-ended proliferation of mentalistic vocabularies. Challenged by Charles Taylor, Andrea Kern and Jocelyn Benoist, Gabriel deftly refutes naturalism's metaphysical claim to epistemic exclusiveness.

In this paper we show that first-order languages extended with partially ordered connectives and partially ordered quantifiers define, under a certain interpretation, their own truth-predicate. The interpretation in question is in terms of games of imperfect information. This result is compared with those of Kripke and Feferman.

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of (...) the axioms ofZF. Doubtless, when cast as a first-order theory,ZFdoes not characterize the structures 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-orderZFbe, as usual, the theory that results fromZFwhen the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-orderZFcan only be satisfied in models of the form 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal. (shrink)