In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of \((\in, \in)\) and \((\in, \in\! \vee \, {q})\)-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of \((\alpha,\beta)\)-fuzzy subalgebras and \((\alpha,\beta)\)-fuzzy ideals on equality algebras. (...) Also, by using the notion of \((\in, \in)\)-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of \(X\), and we proved that the set of all classes of \(X\) by these order is a poset. (shrink)

The falling shadow theory is applied to subhoops and filters in hoops. The notions of falling fuzzy subhoops and falling fuzzy filters in hoops are introduced, and several properties are investigated. Relationship between falling fuzzy subhoops and falling fuzzy filters are discussed, and conditions for a falling fuzzy subhoop to be a falling fuzzy filter are provided. Also conditions for a falling shadow of a random set to be a falling fuzzy filer are displayed.

The falling shadow theory is applied to subhoops and filters in hoops. The notions of falling fuzzy subhoops and falling fuzzy filters in hoops are introduced, and several properties are investigated. Relationship between falling fuzzy subhoops and falling fuzzy filters are discussed, and conditions for a falling fuzzy subhoop to be a falling fuzzy filter are provided. Also conditions for a falling shadow of a random set to be a falling fuzzy filter are displayed.

When it comes to making decisions in vague problems, rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations- lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of (...) all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop. (shrink)

The notions of soju filters in a hoop algebra are introduced, and related properties are investigated. Relations between a soju sub-hoop, a soju filter and an implicative soju filter are discussed. Conditions for a soju filter to be implicative are displayed, and characterizations of an implicative soju filters are considered. The extension property of an implicative soju filter is established.

The notions of soju filters in a hoop algebra are introduced, and related properties are investigated. Relations between a soju sub-hoop, a soju filter and an implicative soju filter are discussed. Conditions for a soju filter to be implicative are displayed, and characterizations of an implicative soju filters are considered. The extension property of an implicative soju filter is established.

The notion of positive implicative soju ideal in BCK-algebra is introduced, and several properties are investigated. Relations between soju ideal and positive implicative soju ideal are considered, and characterizations of positive implicative soju ideal are established. Finally, extension property for positive implicative soju ideal is constructed.

In this paper, considering \(L\)-algebras, which include a significant number of other algebraic structures, we present a definition of modules on \(L\)-algebras (\(L\)-modules). Then we provide some examples and obtain some results on \(L\)-modules. Also, we present definitions of prime ideals of \(L\)-algebras and \(L\)-submodules (prime \(L\)-submodules) of \(L\)-modules, and investigate the relationship between them. Finally, by proving a number of theorems, we provide some conditions for having prime \(L\)-submodules.

In this paper, we apply the notion of \(n\)-fold filters to the \(EQ\)-algebras and introduce the concepts of \(n\)-fold pseudo implicative, \(n\)-fold implicative, \(n\)-fold obstinate, \(n\)-fold fantastic prefilters and filters on an \(EQ\)-algebra \(\mathcal{E}\). Then we investigate some properties and relations among them. We prove that the quotient algebra \(\mathcal{E}/F\) modulo an 1-fold pseudo implicative filter of an \(EQ\)-algebra \(\mathcal{E}\) is a good \(EQ\)-algebra and the quotient algebra \(\mathcal{E}/F\) modulo an 1-fold fantastic filter of a good \(EQ\)-algebra \(\mathcal{E}\) is an (...) \(IEQ\)-algebra. (shrink)

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition is (...) a distributive BCK-algebra. (shrink)

In this paper, by considering the notions of effect algebra and product effect algebra, we define the concept of effect module. Then we investigate some properties of effect modules, and we present some examples on them. Finally, we introduce some topologies on effect modules.

Rough set theory is an excellent mathematical tool for the analysis of a vague description of actions in decision problems. Now, in this paper by considering the notion of an equality algebra, the notion of the lower and the upper approximations are introduced and some properties of them are given. Moreover, it is proved that the lower and the upper approximations define an interior operator and a closure operator, respectively. Also, using D-lower and D-upper approximation, conditions for a nonempty subset (...) to be definable are provided and investigated that under which condition D-lower and D-upper approximation can be filter. (shrink)

More general form of -neutrosophic ideal is introduced, and their properties are investigated. Relations between -neutrosophic ideal and )-neutrosophic ideal are discussed. Characterizations of )-neutrosophic ideal are discussed, and conditions for a neutrosophic set to be an )-neutrosophic ideal are displayed.

In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a special case. Also, we constructed a free EQ-algebra on a singleton.

In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a special case. Also, we constructed a free EQ-algebra on a singleton.

Rajabʻalī Tabrīzī, -1669 or 1670 -- Philosophical thoughts -- Islamic philosophy - Iran.

This fourteenth volume of Collected Papers is an eclectic tome of 87 papers in Neutrosophics and other fields, such as mathematics, fuzzy sets, intuitionistic fuzzy sets, picture fuzzy sets, information fusion, robotics, statistics, or extenics, comprising 936 pages, published between 2008-2022 in different scientific journals or currently in press, by the author alone or in collaboration with the following 99 co-authors (alphabetically ordered) from 26 countries: Ahmed B. Al-Nafee, Adesina Abdul Akeem Agboola, Akbar Rezaei, Shariful Alam, Marina Alonso, Fran Andujar, (...) Toshinori Asai, Assia Bakali, Azmat Hussain, Daniela Baran, Bijan Davvaz, Bilal Hadjadji, Carlos Díaz Bohorquez, Robert N. Boyd, M. Caldas, Cenap Özel, Pankaj Chauhan, Victor Christianto, Salvador Coll, Shyamal Dalapati, Irfan Deli, Balasubramanian Elavarasan, Fahad Alsharari, Yonfei Feng, Daniela Gîfu, Rafael Rojas Gualdrón, Haipeng Wang, Hemant Kumar Gianey, Noel Batista Hernández, Abdel-Nasser Hussein, Ibrahim M. Hezam, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Muthusamy Karthika, Nour Eldeen M. Khalifa, Madad Khan, Kifayat Ullah, Valeri Kroumov, Tapan Kumar Roy, Deepesh Kunwar, Le Thi Nhung, Pedro López, Mai Mohamed, Manh Van Vu, Miguel A. Quiroz-Martínez, Marcel Migdalovici, Kritika Mishra, Mohamed Abdel-Basset, Mohamed Talea, Mohammad Hamidi, Mohammed Alshumrani, Mohamed Loey, Muhammad Akram, Muhammad Shabir, Mumtaz Ali, Nassim Abbas, Munazza Naz, Ngan Thi Roan, Nguyen Xuan Thao, Rishwanth Mani Parimala, Ion Pătrașcu, Surapati Pramanik, Quek Shio Gai, Qiang Guo, Rajab Ali Borzooei, Nimitha Rajesh, Jesús Estupiñan Ricardo, Juan Miguel Martínez Rubio, Saeed Mirvakili, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, Ahmed A. Salama, Nirmala Sawan, Gheorghe Săvoiu, Ganeshsree Selvachandran, Seok-Zun Song, Shahzaib Ashraf, Jayant Singh, Rajesh Singh, Son Hoang Le, Tahir Mahmood, Kenta Takaya, Mirela Teodorescu, Ramalingam Udhayakumar, Maikel Y. Leyva Vázquez, V. Venkateswara Rao, Luige Vlădăreanu, Victor Vlădăreanu, Gabriela Vlădeanu, Michael Voskoglou, Yaser Saber, Yong Deng, You He, Youcef Chibani, Young Bae Jun, Wadei F. Al-Omeri, Hongbo Wang, Zayen Azzouz Omar. (shrink)

This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad (...) Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang. (shrink)

Edition and study of P.Vindob. A.Ch. 36616, a literary paper fragment from 3rd/9th-century Egypt. The fragment contains a tradition that depicts ʿAlī b. Abī Ṭālib (d. 40/661) performing a four-rakʿa prayer and reciting a special supplication on the 15th of the month of Rajab. The tradition is only known from noncanonical Shīʿī ḥadīth collections. Situating it in a broader historical context, the paper provides a glimpse on the Shīʿī presence in Egypt as well as the sanctity of the month (...) of Rajab in Sunnī and Shīʿī literatures. This tradition, in conclusion, presents an early evidence for the observances of the month of Rajab in general and the middle of the month in particular in the local Shīʿī community in pre-Fatimid Egypt. (shrink)

To a modern observer of Western culture, Romanticism might appear conflicted about size. On the one hand, the likes of Chopin and Scriabin best expressed themselves through small-scale compositions, while, on the other, there were those who, like Wagner and Mahler, produced colossal works. The aim of the present article is to explore the phenomenon of miniaturization in Western culture and to examine how miniature works (e.g., literary fragments, preludes) competed with their much larger counterparts. My central claims are threefold: (...) (1) that miniaturization, contrary to common belief, is not a contemporary phenomenon; (2) that the attitude of Romanticism to size was not paradoxical but rather an attempt to harmonize the fragment and the colossal; and (3) that in contemporary culture, the fragment and the colossal are no longer in harmony with each other. (shrink)

Luck (2009) argues that gamers face a dilemma when it comes to performing certain virtual acts. Most gamers regularly commit acts of virtual murder, and take these acts to be morally permissible. They are permissible because unlike real murder, no one is harmed in performing them; their only victims are computer-controlled characters, and such characters are not moral patients. What Luck points out is that this justification equally applies to virtual pedophelia, but gamers intuitively think that such acts are not (...) morally permissible. The result is a dilemma: either gamers must reject the intuition that virtual pedophelic acts are impermissible and so accept partaking in such acts, or they must reject the intuition that virtual murder acts are permissible, and so abstain from many (if not most) extant games. While the prevailing solution to this dilemma has been to try and find a morally relevant feature to distinguish the two cases, I argue that a different route should be pursued. It is neither the case that all acts of virtual murder are morally permissible, nor are all acts of virtual pedophelia impermissible. Our intuitions falter and produce this dilemma because they are not sensitive to the different contexts in which games present virtual acts. (shrink)

In the Buddhist tradition, there is an expansive collection of texts that explore the topic of ethics, addressing moral questions concerning the right and wrong behaviors, virtues, vices, and so forth. However, when examining the main texts of this tradition, we find an absence of a structured moral philosophy that systematically and critically analyzes moral values and principles. Therefore, Buddhist scholars have responded in different ways to the perplexing situation in which Buddhism largely lacks an explicit theory in moral philosophy. (...) Some scholars argue that we should read Buddhist moral teachings as one of the contemporary ethical theories, such as consequentialism or virtue ethics. Damien Keown is one of the scholars who claims that “virtue ethics” is the best way to understand Buddhist ethics. This paper analyzes and critiques Damien Kewon’s reading of Buddhist moral teachings as Virtue ethics. I argue that such interpretation poses problems, primarily because it may overlook key aspects of Buddhist beliefs deemed inconsequential to contemporary ethical debates, and secondly, it could result in the imposition of non-authentic Buddhist ideas on Buddhist ethics. (shrink)

The gamer’s dilemma offers three plausible but jointly inconsistent premises: (1) Virtual murder in video games is morally permissible. (2) Virtual paedophelia in video games is not morally permissible. (3) There is no morally relevant difference between virtual murder and virtual paedophelia in video games. In this paper I argue that the gamer’s dilemma can be understood as one of three distinct dilemmas, depending on how we understand two key ideas in Morgan Luck’s (2009) original formulation. The two ideas are (...) those of (1) occurring in a video game and (2) being a virtual instance of murder or paedophelia. Depending on the weight placed on the gaming context, the dilemma is either about in-game acts or virtual acts. And depending on the type of virtual acts we have in mind, the dilemma is either about virtual representations or virtual partial reproductions of murder and paedophelia. This gives us three dilemmas worth resolving: a gaming dilemma, a representation dilemma, and a simulation dilemma. I argue that these dilemmas are about different issues, apply to different cases, and are susceptible to different solutions. I also consider how different participants in the debate have interpreted the dilemma in one or more of these three ways. (shrink)

How do we assign values to virtual items, which include virtual objects, properties, events, subjects, worlds, environments, and experiences? In this article, I offer a framework for answering this question. After considering different value theses in the literature, I argue that whether we think these theses mutually exclusive or not turns on our view about the number of value-salient kinds virtual items belong to. Virtual monism is the view that virtual Xs belong to only one value-salient kind in relation to (...) X. Virtual pluralism is the view that they belong to more than one value-salient kind. I argue for two claims. First, virtual monism is mistaken. Minimally some virtual Xs are Xs, while others are not. Second, dualistic virtual pluralism is also mistaken because it is too coarse grained. Instead, I argue for fourfold pluralism: virtual items either represent an original's properties or reproduce essential properties and do so to lesser or greater extents. This gives us four value-salient kinds of virtual X: virtual reproductions, simulations, representations, and simulacra of X. I apply this view to various debates in the literature and conclude with a discussion of less basic hybrid kinds. (shrink)

In this paper, by considering the notion of MV-module, which is the structure that naturally correspond to lu-modules over lu-rings, we study injective MV-modules and we investigate some conditions for constructing injective MV-modules. Then we define the notions of essential A-homomorphisms and essential extension of A-homomorphisms, where A is a product MV-algebra, and we get some of there properties. Finally, we prove that a maximal essential extension of any A-ideal of an injective MV-module is an injective A-module, too.

A central debate in the philosophy of virtual reality (VR) focuses on the reality of virtual items. Broadly, there are two main disagreements. Some views accept a metaphysical orientation to VR, and disagree on the reality of virtual items. For instance, David Chalmers (Disputatio 9(46):309-352, 2017, Disputatio 11(55):453- 486, 2019, 2022) defends digitalism, the view that virtual items are real digital items. Neil McDonnell & Nathan Wildman (Disputatio 11(55):371-397, 2019), by contrast, defend fictionalism, which maintains that virtual items are unreal (...) fictions. Other views, like Grant Tavinor’s (2021), reject the metaphysical orientation, argu- ing that focusing on the reality of virtual items is a mistake. In what follows I evalu- ate these two disagreements. I argue that experiences of virtual items depend on episodes of picture perception. Recognizing this reveals that the current debates are either easily resolved in favor of one view, or are not disagreeing about the reality of virtual items, but about what virtual items are. (shrink)

Aristotle (b. 322 BC) was the first philosopher in the history of thought to examine all modes and types of belief acquisition such as knowledge, supposition, error and indirectly imagination. In his _Prior Analytics_, which he wrote primarily to clarify his theory of demonstration, Aristotle examined in detail the syllogism, which he saw as the most important form of reasoning, and his analysis was subject to interpretation by different traditions of thought for centuries. Aristotle’s _Prior Analytics_ was translated into Arabic (...) in the classical period and later interpreted by many Islamic logicians. This article aims to show how Aristotle’s definition of the syllogism in the _Prior Analytics_ was taken up and interpreted in the Peripatetic Islamic logical tradition. These interpretations of the definition of the syllogism will provide us with an idea about the unique features of the commentaries on the _Prior Analytics_ in the Islamic world. Here, introductory information about the translation of the _Prior Analytics_ into Arabic and its commentators in Arabic will be given, Aristotle’s definition of the syllogism and its versions in Peripatetic Islamic logicians will be indicated, and the interpretation of the basic terms in the definition by Peripatetic Islamic logicians will be revealed. (shrink)

Scientists cannot devise theories, construct models, propose explanations, make predictions, or even carry out observations, without first classifying their subject matter. The goal of scientific taxonomy is to come up with classification schemes that conform to nature's own. Another way of putting this is that science aims to devise categories that correspond to 'natural kinds.' The interest in ascertaining the real kinds of things in nature is as old as philosophy itself, but it takes on a different guise when one (...) adopts a naturalist stance in philosophy, that is when one looks closely at scientific practice and takes it as a guide for identifying natural kinds and investigating their general features. This Element surveys existing philosophical accounts of natural kinds, defends a naturalist alternative, and applies it to case studies in a diverse set of sciences. This title is also available as Open Access on Cambridge Core. (shrink)

An action plan for a globally equitable sociology.

Recently there is an increasing technological development in intelligent tutoring systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching information security. This intelligent tutoring systems target the students enrolled in Advanced Topics in Information Security in the faculty of Engineering and Information Technology at Al-Azhar University in Gaza. Through which the student will be able to study the course and solve related problems. An evaluation of the intelligent tutoring systems (...) was carried out and the results were promising. (shrink)

This article draws on the ecological model of teacher agency and elaborates on how teacher agency is achieved, its components and how it could be strengthened. This model highlights professional competence, structural and cultural context, and professional purpose as the main elements of achieving agency. In this paper, we specify some elements of the ecological model and elaborate on how three types of reflection could be used to strengthen conditions for achieving teacher agency. These include, first, procedures aimed at articulating (...) the theory of action, including effective strategies, rules and principles for practice; second, procedures of practical reflection, aimed at fostering the conceptualization of practice and the understanding of oneself as a teacher; and third, procedures of critical reflection, which help to address the inequalities and malpractices of education and society. The main importance of this paper is to extend the practical relevance of the ecological model for teacher learning settings. (shrink)

In “Compassionate Phenomenal Conservatism” (2007), “Phenomenal Conservatism and the Internalist Intuition” (2006), and Skepticism and the Veil of Perception (2001), Michael Huemer endorses the principle of phenomenal conservatism, according to which appearances or seemings constitute a fundamental source of (defeasible) justification for belief. He claims that those who deny phenomenal conservatism, including classical foundationalists, are in a self-defeating position, for their views cannot be both true and justified; that classical foundationalists have difficulty accommodating false introspective beliefs; and that phenomenal conservatism (...) is most faithful to the central internalist intuition. I argue that Huemer’s self-defeat argument fails, that classical foundationalism is able to accommodate fallible introspective beliefs, and that classical foundationalism captures a relatively clear internalist intuition. I also show that the motivation for phenomenal conservatism is less than clear. (shrink)

BackgroundProfessional ethics refers to the use of logical and consistent communication, knowledge, clinical skills, emotions and values in nursing practice. This study aimed to explore and describe factors that affect professional ethics in nursing practice in Iran.MethodsThis qualitative study was conducted using conventional content analysis approach. Thirty nurses with at least 5 years of experience participated in the study; they were selected using purposive sampling. Data were collected through semi-structured interviews and analyzed using thematic analysis.ResultsAfter encoding and classifying the data, (...) five major categories were identified: individual character and responsibility, communication challenges, organizational preconditions, support systems, educational and cultural development.ConclusionsAwareness of professional ethics and its contributing factors could help nurses and healthcare professionals provide better services for patients. At the same time, such understanding would be valuable for educational administrators for effective planning and management. (shrink)

Most refugees residing in the European Union (EU) do not retain their voting rights in states of origin or lack the means to exercise them effectively. Most member states of the EU do not extend voting rights to refugees. This leaves a large population of refugees residing within the borders of the EU in a unique state of disenfranchisement. In this article, I consider this problem from a democratic perspective. Should refugees in the EU have voting rights? My answer turns (...) on three aspects characterizing the circumstances of refugees: First, refugees reside in member states for an indeterminate, and typically protracted, amount of time without any exit options. Second, the political governance of the EU and its member states significantly shapes the lives of refugees. And third, refugees lack effective voting rights, and consequently, a meaningful political membership, in any political community across the world. In turn, based on an all-subjected principle of democratic inclusion, I argue that refugees should be secured state-level voting rights in member states, albeit with a condition of the period of residency. Moreover, they should have immediate voting rights at the supranational institutions of the EU such as the European Parliament. (shrink)

According to Wright’s Judgement-Dependent account of intention, facts about a subject’s intentions can be taken to be constituted by facts about the subject’s best opinions about them formed under certain optimal conditions. This paper aims to defend this account against three main objections which have been made to it by Boghossian, Miller and implicitly by Wright himself. It will be argued that Miller’s objection is implausible because it fails to take into account the partial-determination claim in this account. Boghossian’s objection (...) also fails because it is based on an unjustified reductionist reading of Wright’s account. However, Wright’s own attempt to resist Boghossian’s objection seems to display a shift from his Judgement-Dependent account to an Interpretationist account of self-knowledge, in which case Wright’s new account would face the same problem which he himself has previously put forward in the case of Davidson’s Interpretationist account of self-knowledge. Nonetheless, I will argue that Wright does not need to make such a move because Boghossian’s objection is not applicable to his account. (shrink)

The “problem of memory” in epistemology is concerned with whether and how we could have knowledge, or at least justification, for trusting our apparent memories. I defend an inductive solution—more precisely, an abductive solution—to the problem. A natural worry is that any such solution would be circular, for it would have to depend on memory. I argue that belief in the reliability of memory can be justified from the armchair, without relying on memory. The justification is, roughly, that my having (...) the sort of experience that my apparent memory should lead me to expect is best explained by the hypothesis that my memories are reliable. My solution is inspired by Harrod’s (1942) inductive solution. Coburn (1960) argued that Harrod’s solution contains a fatal flaw. I show that my solution is not vulnerable to Coburn’s objection, and respond to a number of other, recent and likely objections. (shrink)

A model M = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be θ-like, where E interprets ∈ and θ is an uncountable cardinal, if |M| = θ but $|\{b \in M: bEa\}| for each a ∈ M. An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ 1 -like model. Coupled with Chang's two cardinal theorem this implies (...) that if θ is a regular cardinal θ such that $2^{ then every consistent extension of ZF also has a θ + -like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ 2 -like model. Here we prove: THEOREM A. If θ has the tree property then the following are equivalent for any completion T of ZFC: (i) T has a θ-like model. (ii) $\Phi \subseteq T$ , where Φ is the recursive set of axioms {∃ κ(κ is n-Mahlo and "V κ is a Σ n -elementary submodel of the universe"): n ∈ ω}. (iii) T has a λ-like model for every uncountable cardinal λ. THEOREM B. The following are equiconsistent over ZFC: (i) "There exists an ω-Mahlo cardinal". (ii) "For every finite language L, all ℵ 2 -like models of ZFC(L) satisfy the scheme Φ(L). (shrink)