Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph (...) conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

Conceptions of Set and the Foundations of Mathematics By IncurvatiLucaCambridge University Press, 2020. xvi + 238 pp.

Many philosophers are aware of the paradoxes of set theory (e.g. Russell's paradox). For many people, these were solved by the iterative conception of set which holds that sets are formed in stages by collecting sets available at previous stages. This Element will examine possibilities for articulating this solution. In particular, the author argues that there are different kinds of iterative conception, and it's open which of them (if any) is the best. Along the way, the author hopes to make (...) some of the underlying mathematical and philosophical ideas behind tricky bits of the philosophy of set theory clear for philosophers more widely and make their relationships to some other questions in philosophy perspicuous. (shrink)

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative conception of set is a good (...) one, for all the usual reasons. However, the cumulative hierarchy is merely one way among many of working out this conception, and arguments in favour of an iterative conception have been mistaken for arguments in favour of this one special instance of it. (This may be the point to get out of the way the observation that although philosophers of mathematics write of the iterative conception of set, what they really mean – in the terminology of modern computer science at least – is the recursive conception of sets. Nevertheless, having got that quibble off my chest, I shall continue to write of the iterative conception like everyone else.). (shrink)

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...) is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...) the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine. (shrink)

In explaining his concept of set Cantor intimates a connection with the metaphysical scheme put forward in Plato’s Philebus to determine the place of pleasure. We argue that these determinations capture key ideas of Cantorian set theory and, moreover, extend to intuitions which continue to play a central role in the modern mathematics of infinity.

Surveying and criticising attitudes towards the role and strength of the iterative conception of set—widely seen as the justificatory basis of Zermelo-Fraenkel set theory with Choice—this paper highlights a tension in both contemporary and historic accounts of the iterative conception’s justificatory role: on the one hand its advocates wish to claim that it justifies ZFC, but on the other hand they abstain from stating whether the preconditions for such justification exists. Expanding the number of axioms that the conception is standardly (...) charged with failing to justify, in the forms of the Emptyset and Powerset, this paper aims to extend the critique that the iterative conception does not justify ZFC to its subsystems. Exploring the historic and contemporary relationship between the iterative conception and set theory, the paper then attempts to defuse strategies that avoid the problems of intrinsic justification by weakening the iterative conception’s role to the ‘motivational’ or ‘heuristic’ by showing that what the iterative conception has been seen to motivate has changed over time. It is suggested that the conjunction of these arguments seriously weakens the programme of reasoning on an ‘atheoretic’ notion of set. (shrink)

Recenze knihy:Luca Incurvati, Conceptions of Set and the Foundations of Mathematics. Cambridge University Press, 2020, 238 s.

It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...) pictures a monadology already via its form. And secondly, because Gödel wanted to establish in his Philosophical Remarks a modern monadology in adapting the philosophical principles of Leibniz’s monadology to modern science. This article will only deal with the first aspect that has been mentioned namely that a single remark by Gödel (very often) mirrors his whole philosophy (at the time) for example: set theory, perception and intuition of mathematical objects and axioms, the nature of the human mind, the concept and existence of God, and so on. This renders it extremely difficult to interpret Gödel’s philosophical remarks but interpreting it provides also a deep insight in Gödel’s philosophical thoughts. (shrink)

이 글의 주된 목적은 반복적 집합 개념을 검토하여 그 개념이 지니는 내부적인 문제들을 드러내고 그 개념에 대한 구성주의적 비판을 제시하는 것이다. 내부적인 문제는 고전수학자들에게 반복적 집합 개념이 불분명한 데에서 기인한다. 이 점은 불로스의 단계이론을 반복적 집합 개념에 관한 다른 설명과 비교하고 치환공리와 선택공리에 관한 논란을 검토하여 드러낼 것이다. 이보다 훨씬 심각한 문제는 구성주의적 비판이다. 특히, 고전적인 반복적 집합 개념에 내재한 무한에 관한 실재론적 이해에 대한 비판과, 집합론에서 고전논리의 사용에 대한 비판 및 비서술성에 관한 비판이 제기될 것이며, 반복적 집합 개념이 구성적으로 (...) 이해될 때, 어떤 원칙들이 정당화될 수 있는지 검토될 것이다. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

The main concern of this paper is the justification of the axioms of Zermelo-Fraenkel set theory, either as true statements about a concept of set or, alternatively, as true statements about abstract objects . I want to argue here that, in either case, set theory can be seen as a body of knowledge largely built on intuitive foundations . I call this inquiry “phenomenological” for it approaches its subject from the perspective of the intentional acts that originate sets as (...) doubly dependent objects . Such an inquiry, I believe, brings to light the essential characters of sets as objects or, alternatively, the concept of set, which the axioms of the theory express. (shrink)

The main concern of this paper is the justification of the axioms of Zermelo-Fraenkel set theory, either as true statements about a concept of set or, alternatively, as true statements about abstract objects. I want to argue here that, in either case, set theory can be seen as a body of knowledge largely built on intuitive foundations. I call this inquiry “phenomenological” for it approaches its subject from the perspective of the intentional acts that originate sets as doubly dependent (...) objects. Such an inquiry, I believe, brings to light the essential characters of sets as objects or, alternatively, the concept of set, which the axioms of the theory express. (shrink)

Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we (...) first need to provide a full-fledged function theory. I supply such a theory: it axiomatizes the iterative notion of function in exactly the same sense that ZF axiomatizes the iterative notion of set. Indeed, this function theory is synonymous with ZF. It might seem that set theory and function theory present us with rival foundations for mathematics, since they postulate different ontologies. But appearances are deceptive. Set theory and function theory provide the very same judicial foundation for mathematics. They do not supply rival metaphysical foundations; indeed, if they supply metaphysical foundations at all, then they supply the very same metaphysical foundations. (shrink)

Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative (...) conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles. (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)

Individuals with high self-concept will likely have high life satisfaction, they easily get adjusted to life, and they communicate their feeling more appropriately. However, it was not certain whether self-concept would decline or improve as individuals age, or whether self-concept would vary between genders and ethnic groups. To prove, a study was carried out to compare the self-concept of college students in an Asian context. The inquiry utilized the cross-sectional design in finding out significant differences in (...) the self-concept of participants in terms of age, gender, and ethnicity. A 22-item questionnaire was adapted and administered to 222 Bachelor of Public Administration and Bachelor of Science in Business Administration students from the satellite campus of Bukidnon State University in the Philippines. Initially, a sample was randomly drawn from the population. During the actual data collection, the researchers had difficulty getting the responses from the randomly selected individuals due to internet connection and it was done amidst a pandemic. Instead, it took all responses from those who were available, have access to the internet, and could accomplish the google forms. The data were analyzed using Mean, Standard Deviation, T-test for independent sample, ANOVA and Post Hoc test. The results revealed that college students at the locale have a high level of self-concept in self-fulfilment, emotional adjustment, and honesty. Yet, they only had a moderate level of self-concept in autonomy. Further, there were significant differences in college students’ autonomy and honesty in terms of age and gender. Furthermore, there were significant differences in their emotional adjustment and self-fulfilment as to their ethnicity. The results have implications for instruction, administration, guidance services, and future research. (shrink)

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal (...) and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃ x) [ x is a strongly inaccessible cardinal]. (shrink)

The aim of this analysis was to examine the concept of time to rejuvenate and extend existing narratives of time within the nursing literature. In particular, we hope to promote a new trajectory in nursing research and practice which focuses on time and person‐centred care, specifically of older people with cognitive impairment hospitalized in the acute care setting. We consider the explanatory power of concepts such as clock time, process time, fast care, slow care and time debt for elucidating (...) the relationship between ‘good care’ and ‘time use’. We conclude by offering two additional concepts of time, plurotemporality and person‐centred time (PCT) which we propose will help advance of nursing knowledge and practice. Nurse clinicians and researchers can use these alternative concepts of time to explore and describe different temporal structures that honour the patient's values and preferences using experiential, observation‐based nursing research approaches. (shrink)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

The work of Anthony J. Steinbock on emotions―particularly moral emotions―and on religious experience is closely related to a methodological claim. This claim is that the concepts of “experience” and “manifestation” should be understood in a broader manner than that of classical phenomenology, particularly Edmund Husserl’s phenomenology. In this paper, I examine the way in which Steinbock understands and conceptualizes the kind of givenness to which he refers with the notion of “vertical experience”. I focus on his claim that vertical experiences (...) are irreducible to the kind of experiences that can be described in terms of what he calls “provocation”, “presentation” and the “noesis-noema structure”. Even though I make a criticism of his assertion that the latter implies that they should not be understood as forms of givenness founded on the above-mentioned structure, I agree with some major implications that he draws from them. In the last part of the paper, I discuss his suggestion that the Husserlian conceptualization of emotional givenness should be revised to set forward their structure in terms of what he calls “evocation” and try to give additional reasons, drawn from Husserl himself, to support this claim. The paper comes concludes by stressing the relevance of Steinbock’s analyses concerning what he calls “idolatry”. I argue that his analyses of attitudes that negate the vertical dimension of experience have far reaching implications that go beyond the field of philosophy of religion and open new, promising paths for phenomenological research on social and moral problems. (shrink)

Although a large body of literature exists propounding the importance of space in aged care and care of the older person with dementia, there is, however, only limited exploration of the ‘acute care space’ as a particular type of space with archetypal constraints that maybe unfavourable to older people with cognitive impairment and nurses wanting to provide care that is person‐centred. In this article, we explore concepts of space and examine the implications of these for the delivery of care to (...) older people who are cognitively impaired. Our exploration is grounded in theorisations of space offered by key geographers and phenomenologists, but also draws on how space has been constructed within the nursing literature that refers specifically to acute care. We argue that space, once created, can be created and that nursing has a significant role to play in the process of its recreation in the pursuit of care that is person‐centred. We conclude by introducing an alternative logic of space aimed at promoting the creation of more salutogenic spaces that invokes a sense of sanctuary, safeness, and inclusion, all of which are essential if the care provided to the older person with cognitive impairment is apposite to their needs. The concept of ‘person‐centred space’ helps to crystallize the relationship between space and person‐centred care and implies more intentional manipulation of space that is more conducive to caring and healing. Significantly, it marks a return to Nightingale's wisdom, that is, to put the person in the best possible conditions for nature to act upon them. (shrink)

In the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic, and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen, was not a major motivation for operative (...) logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

_The Concept of Analytic Contact_ presents practitioners with new ways to assist the often severely disturbed patients that come to see them in both private and institutional settings. In this book Robert Waska outlines the use of psychoanalysis as a method of engagement that can be utilised with or without the addition of multiple weekly visits and the analytic couch. The chapters in this book follow a wide spectrum of cases and clinical situations where hard to reach patients are (...) provided with the best opportunity for health and healing through the establishment of analytic contact. Divided into four parts, this book covers: the concept of analytic contact caution and reluctance concerning psychological engagement drugs, mutilation, and psychic fragmentation clinical reality, psychoanalysis and the utility of analytic contact. Analytic contact is demonstrated to be a valuable clinical approach to working analytically with a complicated group of patients in a successful manner. It will be of great interest to all practitioners in the field of psychoanalysis and psychotherapy. (shrink)

This article aims to shed some light on the concepts embedded in the expressions ‘services of general interest’ (‘SGI’), ‘services of general economic interest’ (‘SGEI’), ‘non-economic services of general interest’ (‘NSGI’) and ‘social services of general interest’ (‘SSGI’). It is submitted that the expression ‘SGI’ conveys a general concept which comprises both SGEI and NSGI. SGEI may be distinguished from NSGI in that only the former involve an economic activity. In contrast to SGI, SGEI and NSGI, the expression ‘SSGI’ (...) is nowhere to be found in primary EU law. This means that it is for the EU legislator and, as the case may be, for the Member States to define such expression. Furthermore, this article supports the contention that a definition of the principles and conditions underpinning the operation of SGI must be capable of adapting to changing times and social perceptions, whilst being respectful of the vertical and horizontal allocation of powers set out in the Treaties. Vertically, a definition of SGI must not impinge upon the powers retained by the Member States. In the absence of harmonisation, it is for the Member States to define the services they consider to be of general interest, unless they commit a manifest error of assessment. In the presence of EU harmonising measures, the margin of discretion enjoyed by the Member States is, if still existent, narrowed down, given that national authorities are required to comply with the objectives pursued by the EU legislator. Horizontally, a definition of SGI must not encroach upon the prerogatives of the Commission in the realm of competition law. Moreover, in light of Article 9 TFEU and secondary EU legislation, the specific features of SSGI must be taken into consideration when determining the compatibility with EU State aid rules of public service compensation awarded to the providers of those services. An EU conceptual framework for the SGI must thus be the result of a constructive dialogue between the different levels of governance, as well as of a balanced solution among different policy areas in relation to which the EU enjoys competences. (shrink)

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I (...) conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation. (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)

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...) existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)