In this article, Savage’s theory of decision-making under uncertainty is extended from a classical environment into a non-classical one. The Boolean lattice of events is replaced by an arbitrary ortho-complemented poset. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility. Then, we discuss the issue of beliefs updating and investigate a transition probability model. An application to a simple game context is proposed.

Quantum cognition in decision making is a recent and rapidly growing field. In this paper, we develop an expected utility theory in a context of non-classical uncertainty. We replace the classical state space with a Hilbert space which allows introducing the concept of quantum lottery. Within that framework, we formulate axioms on preferences over quantum lotteries to establish a representation theorem. We show that demanding the consistency of choice behavior conditional on new information is equivalent to the von Neumann–Lüders postulate (...) applied to beliefs. A dynamically consistent quantum-like agent may violate dynamic recursive consistency, however. This feature suggests interesting applications in behavioral economics as we illustrate in an example of persuasion. (shrink)

Attention has been drawn, particularly since Kant, to propositions which can not have negative instances. They used to be called a priori, axioms, first principles. Today, they are usually called postulates—C. I. Lewis uses both the old and new terminology—because there is a growing recognition of the fact that at least some of them are not “necessary” in the traditional sense. Kant placed a limitation on the apriorism of the continental rationalists. Current epistemologists and logicians have outstripped Kant in the (...) direction signalled by him. Both he and they, however, concur in the opinion that apriori propositions are subject to negation of a sort. The Kantian dinge-an-sich, though they can not contradict these propositions by being negative instances of them, do nevertheless stand to them in the relation of insubordination, thereby exerting on them a “negative action” of a sort. And, according to current views, there are instances which, though they fall outside the range of significance of a proposition or system and hence can not contradict it, do nevertheless tend to bring about a revision of it by a negative action of some sort. The purpose of this brief essay is to coin a convenient name for such instances and to describe certain properties of their peculiar negative action. (shrink)

Hilbert's conference lectures during the year 1922, Neuebegründung der Mathematik. Erste Mitteilung and Die logischen Grundlagen der Mathematik (both are published in (Hilbert [1935] 1965) pp. 157-195), contain his first public presentation of an axiom system for propositional logic, or at least for a fragment of propositional logic, which is largely influenced by the study on logical woks of Frege and Russell during the previous years.The year 1922 is at the beginning of Hilbert's foundational program in its definitive form. (...) The final public presentation by Hilbert of an axiom system for propositional logic is contained in the 1934 book Grundlagen der Mathematik, I (Hilbert and Bernays [1934] 1968). Sequent .. (shrink)

Social cognition can be based on two contrary axioms that answer the question of whether the society can provide for the universal survival of all of its members. Negative answer (relativist model of society) is more productive methodologically. The key notion here is the technosocial formula of society, the physical meaning of which is that the society as an aggregate of people needs bigger vital space than it can create. The growth of man in nature was the result not only (...) of the supplanting of other species but also of the intraspecific struggle which became the main factor of the development process. The criterion of social progress is reduced in the end to only one index – the ability of society’s survival and it doesn’t have any absolute value in space and time. (shrink)

The debates now in progress about the interactions of science and art compel one involuntarily to recall that such discussions have been held more than once and were, a long time ago, perhaps no less heated. It suffices to cite virtually at random certain statements of Pisarev, for example , for us to see, as in a cloudy mirror, both today's advocates of scientism and the romantics of art. Does this mean that all we need is to bear in mind (...) the wise words of the past in order to advance, as is believed by Iu. I. Kagarlitskii ? Obviously, that is not the case. The problem of the interactions of science and art is a philosophical one, not one of natural science; and the results achieved in debates of the past do not eliminate the questions of world view that constantly arise anew. Philosophy has no axioms such as exist in mathematics. It is therefore difficult to imagine that subsequent history has added nothing whatever to the understanding of past debates. Yet reference to these debates would appear to be essential today to clarify at least one of the factors of the problem we have posed. It might appear that this factor has been pointed to, but it has constantly been beclouded by the explanations of how science influences artists, and how art influences scientists who are at the same time artists, by references to the unity of the creative character of science and art, and so forth. In my opinion, it is essential, however, to emphasize that the very problematics of the interactions between science and art depend on the social and cultural environment within which they exist and are discussed. Turning to the past, we can see that science and art are not realms of activity handed down to humankind "since the creation of the world." They develop in the "body," if one may put it thus, of a specific socium, a specific culture. It would, of course, be absurd to conceive of any direct contact between science and art without the mediation of human beings, of society, of culture. Moreover, the debates themselves are determined by the level at which science and art are found within a given sociocultural space, by virtue of which analysis of these debates may provide data to the researcher for an understanding of the direction of social development. The RST influences all spheres of human activity, including the esthetic. Today, however, particularly in the West, there is a tendency to elevate the significance of the RST to an absolute, to explain many processes by its direct influence on social consciousness independently of the social conditions in which it occurs. Therefore, it would seem to be methodologically important for today's debates to demonstrate, on the basis of some concrete historical material, the connection between those shifts in social consciousness that sometimes explain the direct influence of progress in science and technology, on the one hand, and certain sociocultural conditions, on the other. (shrink)

In applied problems mathematics is used as language or as a metalanguage on which metatheories are built, E.G., Mathematical theory of experiment. The structure of pure mathematics is grammar of the language. As opposed to pure mathematics, In applied problems we must keep in mind what underlies the sign system. Optimality criteria-Axioms of applied mathematics-Prove mutually incompatible, They form a mosaic and not mathematical structures which, According to bourbaki, Make mathematics a unified science. One of the peculiarities of applied mathematical (...) language is a variety of dialects: a problem can be presented in terms of various mathematical notions. Another peculiarity is polysemy: a problem can be presented in the framework of one dialect by a set of various models with equal right to exist. The pragmatic sense of distinction between applied and pure mathematics must lead to specific training in each case. (shrink)

The article is devoted to the distinction between real-unreal content hidden under the name. The relations of mark with the concept and denotation are studied. These relations are manifested in such significant acts as characterization, attribution, interpretation, aimed at establishing the substantive values of the concepts. It is highlighted that ideological process identifies super-phrasal information about reality that lies at the intersection of ‘culture‘ and ‘nature‘. In formalized languages, the problem of connecting ‘meaningless‘ syntactic expressions with object-semantic structures is solved (...) by appeal to interpretation - a special procedure, which makes it possible to compare the names of objects and relations between them. In the circle of the interpretive procedure, the ‘empty‘ axiom, symbolic entries become statements about objects, areas of the world. Any suggestion corresponds to a statement about objects in a particular subject system. In natural languages, the relationship of signs with the object basis is set a) genealogically by rules of usage; b) metaphorically by tropic unification, equating with the introduction of harmony; c) didactically - by rules of definition and selection; d) metaphysically by expressing of philosophical indefinable. (shrink)

This chapter contains sections titled: Aristotle's Declaration of a General Science of Being qua Being A Problem for the Science of Being The Content of the General Science of Being Including Axioms in the General Science of Being The Notion of the Firmest Principle Proving Something about an Axiom: the Indubitability Proof of PNC PNC as the Ultimate Principle Defending an Axiom: the Elenctic Proof of PNC Theology and the General Science of Being Notes Bibliography.

Research on bounded rationality has two cultures, which I call ‘idealistic’ and ‘pragmatic’. Technically, the cultures differ on whether they build models based on normative axioms or empirical facts, assume that people's goal is to optimize or to satisfice, do not or do model psychological processes, let parameters vary freely or fix them, aim at explanation or prediction and test models from one or both cultures. Each culture tells a story about people's rationality. The story of the idealistic culture is (...) frustrating, with people in principle being able to know what they should do, but in practice systematically failing to do it. This story makes one hide in books for intellectual solace or surrender to the designs of someone smarter. The story of the pragmatic culture is empowering: If people are educated to use the right tool in the right situation, they do well. (shrink)

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms (...) of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructed. (shrink)

The logical structure of Quantum Mechanics (QM) and its relation to other fundamental principles of Nature has been for decades a subject of intensive research. In particular, the question whether the dynamical axiom of QM can be derived from other principles has been often considered. In this contribution, we show that unitary evolutions arise as a consequences of demanding preservation of entropy in the evolution of a single pure quantum system, and preservation of entanglement in the evolution of composite (...) quantum systems. 6. (shrink)

RECENT attempts to explain and justify Aristotle's principle of non-contradiction have focused to a great extent on the dialectical dimension of Aristotle's account. For example, T. Irwin maintains that Aristotle justifies the PNC by arguing that there is a sub-set of dialectical opinions which no one can rationally give up. J. Lear supports the importance of the dialectical dimension by summarizing Aristotle's defense of the PNC as follows: The opponent of the PNC tries to argue dialectically that one should not (...) accept it. "Aristotle's point is that there is no conceptual space in which such a rational discussion can occur." By appealing to the limits of rationally, externally, expressible discourse as the basis of the PNC, Irwin and Lear fail to take seriously the more explicitly psychological and metaphysical bases of the PNC. C. Kirwan, and most recently, R. Smith also either argue against or ignore the role of the psychological and metaphysical bases. Smith goes so far as to argue that common axioms like the PNC are simply empty statement schemata of universal validity, i.e., general verbal forms or patterns which no one can deny with intelligibility. (shrink)

This study examines BDI logics in the context of Gabbay's fibring semantics. We show that dovetailing can be adopted as a semantic methodology to combine BDI logics. We develop a set of interaction axioms that can capture static as well as dynamic aspects of the mental states in BDI systems, using Catach's incestual schema G^[a, b, c, d]. Further we exemplify the constraints required on fibring function to capture the semantics of interactions among modalities. The advantages of having a fibred (...) approach is discussed in the final section. (shrink)

Weber’s concept of “vocation” in science implies “anti-monumentalism”: research can always be continued, and the results obtained can be used in various ways. The scientist cannot be completely aware of the final impact of their work, so they are faced with a paradox of consequences. This paradox is based on value polytheism, a concept put forward by Weber. There are two ideas central to polytheism: first, one must recognize the internal logic of value spheres and, second, one must consider their (...) fundamental incommensurability. But how does this idea emerge in Weber’s theory? Interpretations of value polytheism as a “fact” of a cultural situation and as the logical foundation of science do not allow one to answer the question of its origin. The conceptual bridge is found in Weber’s sociology of religion. Tenbruck’s, Schluchter’s, and Hennis’s models are examined to identify variations of value polytheism. However, their macro-orientation does not demonstrate the internal structure and functioning of polytheism. The present paper explicates the logical-methodological foundations of Weber’s scientific programme to clarify these points. Primarily, it investigates the problem of the consequences of an action carried out in a “vocation” mode and the boundaries of “adequate” causal explanations as presented in Weber’s works. It makes it possible to consider Weber’s value polytheism and concepts associated with it not as value metaphysics or unreasonable axioms,but as a methodologically based conceptual apparatus. (shrink)

David Hilbert’s distinction between mathematics and metamathematics assumes mathematics is not metamathematics, cardinality of mathematics is less than cardinality of metamathematics, and metamathematics contains mathematics. Only by abandoning the last renders these characteristics consistent. Every set identifiable only in a metaset, following Kurt Gödel, the metaset is convertible into the set by translation of its constituents into constituents of the set, rendering the set indistinguishable from the metaset. Reversing Kurt Gödel, the set is convertible into the metaset by translation of (...) its constituents into constituents of themetaset, rendering the set indistinguishable from the metaset. Set being indistinguishable from metaset, the set of mathematics is unidentifiable as constituent of the set of metamathematics. Only inductively by exclusive resolution of all constituents of the set of all disjunctives of the sets of mathematics and metamathematics is the set of mathematics identifiable. Generated is endless arbitrary qualification of mathematical identity exhibited in continual proofsof its axioms. Understood as clarification, when everything is unique, no concealed contradiction is contained. Constituted is the endless non-repeating digit to the left of the decimal of an algebraic unknown number. Manifest is casuistry, now neither objective nor specious identity, but instead necessary subjective identity distinguishing sets. (shrink)

In the middle of myMathematical logicI defined a certain class of formulae as “stratified,” and conjectured that exclusion from this class is a feature “shared, presumably, by all the untenable statements(p. 157). This ushered in a set of axioms of class-membership which Rosser has since shown to be inconsistent. Accordingly, inElement and numberI dropped the principle*200, in which had been assembled axioms to the effect, roughly, that “stratified functions of elements are elements.” In lieu of*200 I set forth alternatives in (...) which no appeal is made to stratification. The system ofMathematical logicexclusive of*200 carries over as an unchanging framework; and this framework admits, we know, of a simple consistency proof. My concern in the present paper is to draw attention to certain relationships between this framework and earlier theories. (shrink)

The comprehensive nature of information and insight available in the Upanishads, the Indian philosophical systems like the Advaita Philosophy, Sabdabrahma Siddhanta, Sphota Vaada and the Shaddarsanas, in relation to the idea of human consciousness, mind and its functions, cognitive science and scheme of human cognition and communication are presented. All this is highlighted with vivid classification of conscious-, cognitive-, functional- states of mind; by differentiating cognition as a combination of cognitive agent, cognizing element, cognized element; formation; form and structure of (...) cognition, instruments and means of cognition, validity of cognition and the nature of energy/matter which facilitates and also is the medium of cognition- cognizing process; and also as the container and content of cognition. The human communication process which is just the reverse of cognizing process is also presented with necessary description and detail. The sameness of cognitive / communicative process during language acquisition and communication processes and the modes of language acquisition and communication are also given. The hardware and software of human cognition, language acquisition and communication as envisaged in Indian spiritual and philosophical expressions are given. In the light of the information and insight obtained as above, the axioms for human cognition / communication are formed, framed and presented. A brain-wave frequency modulation / demodulation model of human cognition, communication and language acquisition and communication process based on Upanishadic expressions, Shaddarsanas and Sabdabrahma Siddhhanta is defined and discussed here. The use of these theories and models and axioms in mind-machine modelling and natural language comprehension branch of artificial intelligence will be put forward and highlighted. (shrink)

One of the characteristic features of the dynamic development of science and technology in recent decades is the constantly rising significance of probabilistic, statistical and information-theory methods in research, both theoretical and applied. Nor is the mathematical theory of probability standing still. The internal logic of its development is leading steadily to enrichment of the traditional study of probability with new axioms and constructive formal calculi.

Quantum mechanics is currently being tried to be used as a probe to unravel the mysteries of consciousness. Present paper deals with this probe, quantum mechanics and its usefulness in getting an insight of working of human consciousness. The formation of quantum mechanics based on certain axioms, its development to study the dynamical behavior and motions of fundamental particles and quantum energy particles moving with the velocity of light, its insistence on wave functions, its probability approach, its dependence on uncertainty (...) principles will all be critically discussed. The result of this discussion will be presented. Its limitations in unraveling the form, function and biological nature of consciousness will be presented. The alternative probe available and being used – the translation of Upanishadic insight and Advaita philosophy into cognitive science elements in delineating the definition, form and function of Consciousness will be given. The usefulness of this modeling and analysis in understanding the consciousness, mind and its functions in cognitive science and theory of human language acquisition and communication will also be presented. The physic-chemical nature of ideas, senses, thoughts, feelings, utterances will be grossly dealt with. The functions of consciousness and mind, in knowledge and language acquisition and communication will be dealt with. (shrink)

Peirce and Frege both distinguished between the propositional content of an assertion and the assertion of a propositional content, but with different notational means. We present a modification of Peirce’s graphical method of logic that can be used to reason about assertions in a manner similar to Peirce’s original method. We propose a new system of Assertive Graphs, which unlike the tradition that follows Frege involves no ad hoc sign of assertion. We show that axioms of intuitionistic logic can be (...) derived from AGs, and argue that AGs analyse and represent assertions and illocutionary content in a way which is motivated both by its logical properties and its historical connection with the ideas that led to the development of the graphical method. (shrink)

Purpose. The aim of the article is to clarify the content of the concept of culture as an explication of vitality within the philosophy of life and its further modifications in current problems of contemporary. The analysis performed standing from the point, that contrasting of nature and culture is irrelevant, since culture does not contradict natural determinants and patterns, but rather qualitatively alters them. So, are justified the idea of culture as a phenomenon that exist accordingly and in proportion to (...) nature, need to form its potential and content and not contradict the axioms and values of life. Theoretical basis. In the theoretical field of philosophy of life, the local development of the problem of culture as an explication of vitality produces grounds for analytical and prognostic activity concerning meaningful transformations in a separate historical and social horizon. The fundamental categories of culture: spirit, value, symbol, freedom, justice and harmony receive the requested content and meaning. The idea of the constancy and super-naturality of cultural universals is illusory and dangerous. The consequences of such a "non-cosmological" justification of freedom and will, and the assertion of values, that contradict the logic of life, are the global environmental, economic and social crisis of our time. Originality. The originality of the authors’ thought lies in the interpretation of the essence of culture as an explication of vitality, as a logical and natural extension of life. In this formulation of the problem of culture, the possibility of reconciling the natural, social and value determinants of human life is formed. Theorists of the philosophy of life substantiated the primacy and supremacy of the values of life over the values and meanings of culture. The position of authors position consists in the need to understand culture as an environmentally appropriate and dimensional phenomenon, the content and strategies of which are determined by a single ontology. Conclusions. The analysis let authors understand the voluntarily chaotic element of life. Culture in its philosophical analysis took on a clearer anthropomorphic dimension: the immanent logic of being in substantiating the essence and purpose of man and the value of his being localized the universe of transcendence in the concept of "living world", "inhabited space", "human, too human". Accordingly, the range of cultural evaluations has been polarized: from the approving statement of its vital essence, to the disparaging calls for its reform. The chaotic state of voluntarily acts is transformed into cultural codes and stereotypes by rationalization. The modern global nature of crisis phenomena, both in the worldview, in the social, and in the ecological dimension, requires reformatting the understanding of culture as a continuation of nature, and not its antipode. (shrink)

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts (...) are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts (...) are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French (...) philosopher’s fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory. (shrink)

The paradox of biological diversity is the key problem of theoretical ecology. The paradox consists in the contradiction between the competitive exclusion principle and the observed biodiversity. The principle is important as the basis for ecological theory. On a relatively simple model we show a mechanism of indefinite coexistence of complete competitors which violates the known formulations of the competitive exclusion principle. This mechanism is based on timely recovery of limiting resources and their spatio-temporal allocation between competitors. Because of limitations (...) of the black-box modeling there was a problem to formulate the exclusion principle correctly. Our white-box multiscale model of two-species competition is based on logical deterministic individual-based cellular automata. This approach provides an automatic deductive inference on the basis of a system of axioms, and gives a direct insight into mechanisms of the studied system. It is one of the most promising methods of artificial intelligence. We reformulate and generalize the competitive exclusion principle and explain why this formulation provides a solution of the biodiversity paradox. In addition, we propose a principle of competitive coexistence. (shrink)

In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik Frege hardly could have construed Hume's Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

In a well-known passage in the last chapter of Frege: Philosophy of Mathematics Michael Dummett suggests that Frege’s major “mistake”—the key to the collapse of the project of Grundgesetze—consisted in “his supposing there to be a totality containing the extension of every concept defined over it; more generally [the mistake] lay in his not having the glimmering of a suspicion of the existence of indefinitely extensible concepts” (Dummett [1991, 317]). Now, claims of the form, Frege fell into paradox because……. are (...) notoriously difficult to assess even when what replaces the dots is relatively straightforward. Offerings have included, for instance, that — (A) Unrestricted quantification: Frege fell into paradox because he allowed himself to quantify over a single, all-inclusive domain of objects (Russell, Dummett). (shrink)

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought (...) to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity. (shrink)

The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language $\{\in,j\}$ , and that asserts the existence of a nontrivial elementary embedding $j:V\to V$ . The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an $I_1$ embedding. This answers a question (...) about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an $I_1$ -embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.”. (shrink)

If the Wholeness Axiom wa $_0$ is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa $_0$ is finitely axiomatizable.

Say that the property Φ of a cardinal λ strongly implies the property Ψ. If and only if for every λ,Φ implies that Ψ and that for some λ′<λ,Ψ. Frequently in the hierarchy of large cardinal axioms, stronger axioms strongly imply weaker ones. Some strong implications are proved between axioms of the form “there is an elementary embedding j:Lα[Vλ+1]→Lα[Vλ+1] with ”.

We prove the following Main Theorem: $ZF + AD + V = L(R) \Rightarrow DC$ . As a corollary we have that $\operatorname{Con}(ZF + AD) \Rightarrow \operatorname{Con}(ZF + AD + DC)$ . Combined with the result of Woodin that $\operatorname{Con}(ZF + AD) \Rightarrow \operatorname{Con}(ZF + AD + \neg AC^\omega)$ it follows that DC (as well as AC ω ) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬ DC (...) R , where DC R is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R # (i.e. the sharp of L(R)) exists. (shrink)

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