Life history theory is often invoked to make universal predictions about phenotypic evolution. For example, it is conventional wisdom that organisms should evolve older ages at first reproduction if they have longer lifespans. We clarify that life history theory does not currently provide such universal predictions about phenotypic diversity. Using the classic Euler–Lotka model of adaptive life history evolution, we demonstrate how predictions about optimal age at first reproduction depend on rarely acknowledged, prior theoretical assumptions (i.e., axioms) about organismal (...) development. These developmental axioms include the rates, forms, and tradeoffs that relate to growth or differentiation. Developmental innovations transform the biology underlying these axioms. Consequently, Euler–Lotka and related life history models do not make coherent predictions at macroevolutionary scales, where developmental innovations occur (e.g., across mammals, birds, or insects). By focusing on historical innovations instead of universal rules, life history theory can reconnect with flourishing research in evolutionary developmental biology. (shrink)

We study the theories I∇n, L∇n and overspill principles for ∇n formulas. We show that IEn ⇒ L∇n ⇒ I∇n, but we do not know if I∇n L∇n. We introduce a new scheme, the growth scheme Crγ, and we prove that L∇n ⇒ Cr∇n⇒ I∇n. Also, we analyse the utility of bounded collection axioms for the study of the above theories.

The axiom of extensionality of set theory states that any two classes that have identical members are identical. Yet the class of persons age i at time t and the class of persons age i + 1 at t + l, both including same persons, possess different demographic attributes, and thus appear to be two different classes. The contradiction could be resolved by making a clear distinction between age groups and cohorts. Cohort is a multitude of individuals, which is constituted (...) within a time interval, and endures throughout part of the time continuum. Age group, on the other hand, is only a reference term to which empirical measurement relates, as in birth or death rates. Accordingly, the two concepts, age group i at t, and age group i + 1 at t + 1, are different. The standard population growth model of Leslie and Lotka, however, does not support such a distinction in age groups. An alternative model, proposed recently, implies precisely such a distinction. (shrink)

Yet this is precisely what I intend to do. As a way of conferring some initial legitimacy Perhaps the most fundamental axiom of upon this enterprise, I would like to start out modern economic science is that there simply by appealing to the “no free lunch” is no such thing as a free lunch. It is principle. To adopt productivity growth as a this axiom that gives us the concept of opporsocial priority is to set aside other objectives tunity (...) cost, an idea that has led to enormous that we might like to pursue. Therefore, one gains in the clarity of our understanding of cannot maintain rational adherence to a proindividual and societal choice. But while congram of increased productivity without a clear tinuing to endorse this fundamental axiom, sense of what the benefits of such a program many economists have also been extremely are likely to be, compared to the other things attracted by the appeal of increased economic that we might choose to do. The bulk of this efficiency. Efficiency gains are often treated, paper consists of an attempt to evaluate the if not exactly as a free lunch, certainly about size and scope of these benefits. More controas close to a free lunch as one can get in this versially, I will try to show that these benesublunar realm. As a result, economists often fits are often exaggerated. react with a certain incredulity when some- The conclusion that I draw on the basis one questions the need for such gains. It is of this survey will strike many as complacent. tempting to suppose that the critic simply But it is a principled complacency. I argue misunderstands the relevant concepts, or else that we have good reason, as a society, to strive fails to grasp the full set of constraints under for productivity gains, but that there is nothwhich economic activity occurs. ing urgent about this objective. Furthermore, Productivity growth is widely regarded we should not strive to maximize productivity as a subset of the efficiency gains that can be gains, nor should we be overly concerned realized in our economy. (shrink)

Dieser Artikel konzentriert sich auf zwei der frühesten Schriften aus Leibniz' Pariser Zeit, die „Accessio ad Arithmeticam Infinitorum“ und die „De Minimo et Maximo. De corporibus et mentibus“ , und er beurteilt die Beziehung zwischen den in der AAI angegebenen mathematischen Ergebnissen und dem Wandel in Leibniz' Metaphysik, der in MM und in den damit in Verbindung stehenden Schriften und Briefen zum Ausdruck gebracht wird. Sowohl Leibniz' mathematische Ergebnisse als auch der Wandel seiner Haltung in der Metaphysik können als Resultat (...) einer stärkeren Verpflichtung gegenüber dem Euklidischen Axiom betrachtet werden, daß das Ganze größer ist als der Teil. Indem das Euklidische Axiom eine bedeutendere Rolle in Leibniz' Metaphysik und mathematischer Arbeit einnimmt, bewegt sich Leibniz von der Vorstellung der Zusammensetzung des Kontinuums aus unteilbaren Punkten zu einer Aufgliederung des Kontinuums in unvergleichbar kleine Teile, die in sich selbst kontinuierlich sind. Dieser Wandel führt Leibniz dazu, die Rolle der Identität und eines Kalküls von Teil und Ganzem zu betonen und sich in Richtung einer Vorstellung vom Unbestimmten als eines Unendlichen zu bewegen. (shrink)

We introduce Broad Infinity, a new set-theoretic axiom scheme based on the slogan “Every time we construct a new element, we gain a new arity.” It says that three-dimensional trees whose growth is controlled by a specified class function form a set. Such trees are called broad numbers. Assuming AC (the axiom of choice) or at least the weak version known as WISC (weakly initial set of covers), we show that Broad Infinity is equivalent to Mahlo’s principle, which says (...) that the class of all regular limit ordinals is stationary. Assuming AC or WISC, Broad Infinity also yields a convenient principle for generating a subset of a class using a “rubric” (family of rules); this directly gives the existence of Grothendieck universes, without requiring a detour via ordinals. In the absence of choice, Broad Infinity implies that the derivations of elements from a rubric form a set; this yields the existence of Tarski-style universes. Additionally, we reveal a pattern of resemblance between “wide” principles, that are provable in ZFC, and “broad” principles, that go beyond ZFC. Note: this paper uses a base theory that is weaker than ZF but includes classical first-order logic and Replacement. (shrink)

We explain how the field of logarithmic-exponential series constructed in 20 and 21 embeds as an exponential field in any field of exponential-logarithmic series constructed in 9, 6, and 13. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Thequation image; the elementary theory of the ordered field of real numbers, with the (...) exponential function and restricted analytic functions. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (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)

We show that Matijasevič's Theorem on the diophantine representation of r.e. predicates is provable in the subsystem I ∃ - 1 of Peano Arithmetic formed by restricting the induction scheme to diophantine formulas with no parameters. More specifically, I ∃ - 1 ⊢ IE - 1 + E ⊢ Matijasevič's Theorem where IE - 1 is the scheme of parameter-free bounded existential induction and E is an ∀∃ axiom expressing the existence of a function of exponential growth. We conclude (...) by examining the consequences of these results to the structure of countable nonstandard models of IE - 1. (shrink)

This article pursues the question whether and inasmuch theories of corporate responsibility are dependent on conceptions of managerial freedom. I argue that neglect of the idea of freedom in economic theory has led to an inadequate conceptualization of the ethical responsibilities of corporations within management theory. In a critical review of the history of economic ideas, I investigate why and how the idea of freedom was gradually removed from the canon of economics. This reconstruction aims at a deconstruction of certain (...) axioms of neoclassical economics that hamper contemporary efforts in integrating ethics firmly into management education. I intend to show by these deconstructive endeavors that a constructive use of the idea of responsible freedom could correct and complement the current quantitative focus in business theory through qualitative orientations. I argue that with the help of qualitative success criteria, the strategic integration of the tenets of both business ethics and Corporate Social Responsibility into business practice can be advanced by further research. (shrink)

Are the employees working in different countries and enjoying healthy compensation truly loyal to their organization? Our study attempts to answer this question by examining the role of perception of politics on the affective commitment of expatriates in the Sultanate of Oman. Following the axiom of equity theory, the relationships between the perception of politics (POP) to affective commitment (AC) and Islamic work ethics (IWE) to affective commitment (AC) was tested. The moderating role of Islamic work ethics was also examined (...) in the POP-AC relationship. Responses from 146 employees were collected following a simple random sampling technique. Employees working in different organizations at various managerial levels were approached for data collection. Descriptive statistics, correlation, and regression tests were applied to test hypotheses using SPSS-22.0 and SmartPLS-4. Results confirmed the positive relationship between Islamic work ethics to affective commitment and the negative relationship between perception of politics to affective commitment. Moreover, the negative relationship between the perception of politics with affective commitment was weaker when Islamic work ethics were high. Practical implications of the study are presented based on the results. (shrink)

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 725–731] , Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. (...) of the LMS 14 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory .In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.For these independence principles we classify their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles.As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results.This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem. (shrink)

Modern science developed in the interflow of culture between west and east. Combing of pratice technology with philosophic thoughts formed experimental method. Holistic views contacting atomism produced system theory. System thoughts are applicated in the science and engineering of biosystems, and the cencepts of system biomedicine (Kamada T.1992), systems biology (Zieglgansberger W, Tolle TR.1993), system bioengineering and system genetics (Zeng BJ. 1994) were established. From positive to synthetic thoughts, philosophy have been developed ontology, cosmology, organism theories. Structurity is structure logic (...) system founded on entity, develop, exist axioms, tolerance, adaptation, fluctuate, interweave, transform theorems and integrate, adaptation, construct laws. Structurity be discussed on the cosmos, life, culture system, creation, hologram theories of structure ontology, construct mutation, simi-structure organism and entity emergence, symbol implication. From the relation of structure, function and development, the structurity put forward the cycle, spiral, triangle structure stability patterns of self-organization in structure complement each other and stratification, functional couple and interflow bounds growth, coordinativetransformation and holographic symmetry. (shrink)

Since the critique of psychologism initiated by Gottlob Frege and championed by Edmund Husserl, logicians and psychologists alike have adhered to a strict division of labour. This has created a gap between reasoning as a psychological phenomenon and logic. However, reasoning involves logic, and logic is the benchmark of rationality; intuitively at least, reasoning and logic are connected. Recently, attempts have been made to bridge the gap, but the strict division of labour is often eroded. Jean Piaget conceived genetic epistemology (...) as a science of the growth of knowledge, and since logical knowledge is in its purview the accusation of psychologism was omnipresent. In this paper, I outline Piaget's psychological account of the development of logic and argue that he successfully navigated the hazards of psychologism. In essence, he minds the gap by preserving the division of labour; however, by making logic the mirror of thought, he also intimated at a bridge. Concrete correspondences between logic and psychological structures of thought lie at the heart of Piaget's metaphor, and, by building on these foundations, I illustrate how an axiom schema for sentential logic can be derived from the interpropositional grouping, Piaget's psychological model of the operational structure characterising hypothetico-deductive reasoning. (shrink)

One cannot really speak of a school of personalistic economists. Moreover, there is a wide gulf between the economic philosophy of the personalists and the mathematical context of economic science. Since the thirties, philosophers such as Alexandre Marc, Jacques Maritain, Emmanuel Mounier and many others have been searching, on the basis of a personalistic view of man, for a `third way' between individualistic capitalism and statist socialism , but there was seldom interest from the side of the scientific economists.Fortunately, there (...) are some notable exceptions. François Perroux, Kenneth Boulding, Ernst Schumacher, Serge Christophe Kolm, and Amartya Sen are economists who, in various ways, have attempted to bridge the gulf between the personalistic view of man and economic rationality. By carrying out an immanent critique of some of the standard economic assumptions, they opened a path for a normative economics that was clearly distinct from a purely positive economics as well as from Pareto's welfare economics. One can affix various labels to these attempts: humanistic economics , socio-economics , moral economics . In what follows, I shall interpret these attempts as initiatives in the direction of a personalistic economics. I would also situate the current research in `economics and ethics', which is becoming part of the mainstream of economic thought, and which has received official recognition with the recent Nobel prize award to Sen, within this tendency towards a more personalistic economics.Attention to moral feelings, human rights, altruistic motives, justice and solidarity, non-profit organizations, etc., contribute to situating the economic problem in a field where economics, politics and ethics no longer function as three distinct domains but where their interaction is central.Yet does this interactive economics fit into a personalistic way of thinking? Is a generalized system theory not more attractive? System theory has a different focus than personalistic economics. System theory considers the axioms of system equilibrium and sustainable growth as the ultimate horizon, whereas a personalist sees them as merely important conditions for a good society and economy. However refined it may be, system theory views the person from an external point of view, while the personalist refers to the internal experience of human freedom and meaning as something that transcends and regulates the system's functionality.One could object that neo-classical economists do not use system theory as their point of departure. In their eyes, system equilibrium is the result of the utility-maximizing behaviour of individuals. These individuals subjectively determine what their preferences are and then act accordingly. Is this openness for a free, utility-maximizing actor — where there is room for both material and immaterial, egoistic and altruistic preferences — not sufficient to speak of a personalistic economics ? If this were true, then the discourse of personalism would be more or less superfluous.I take the term `personalistic economics' to refer to a normative economics that does not simply reconstruct the problem of efficient allocation as an individual or social problem of utility, but in the first instance as a problem of human dignity and social justice. To reduce the problem of human choice to a problem of subjective utility is in itself a far-reaching normative standpoint. As soon as one realizes this, there is no reason to exclude other normative assumptions, such as personalistic ones. (shrink)

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal (...) analysis. (shrink)

We study the extension of public announcement logic PAL by public assignments, which we call PALA. Just as in the case of PAL, the standard procedure for deciding PALA validity, i.e. the use of so-called reduction axioms to translate PALA formulae into formulae in epistemic logic EL, may lead to exponential growth. In this paper, we show that such a price is not mandatory, for we provide a polynomial translation of PALA into EL. This is based on abbreviations (...) of subformulae by new propositional letters. Such optimal translation also enables us to show the computational complexity of the problem of deciding PALA validity, which turns out to be coNP-complete in the single-agent case and PSPACE-complete in the multiagent case. (shrink)

The role of divine power in Aquinas’s spiritual doctrine has often been neglected in favor of a focus on the primacy of charity, the controlling virtue of spiritual progress. The tendency among some thinkers (e.g. Polkinghorne) to juxtapose divine love and power stems from the stress on divine immanence at the cost of divine transcendence, and from an evolutionary (vs. classical) view of God with its ‘kenotic’ theodicy. A study of the ways in which divine power grounds and directs the (...) spiritual life highlights the robust role that metaphysics plays in spiritual ascent for Aquinas, and offers a philosophical entry point to his doctrine. Themes in his doctrine of the spiritual life incorporate Platonic transcendent causal plenitude and Aristotelian causal axioms and motifs of growth and unity. From the side of theology, divine power is analyzed through several lenses, including power through weakness in Christ, the sin of Lucifer against the gift of being in contrast to the counsel of obedience, and the role of Christ’s human nature in the Church. Taken together, these themes combine to characterize divine power as redemptive medicine, as opposed to a distant, arbitrary force, and to reveal the ways in which Aquinas applies metaphysical insights to the supernatural order. (shrink)

The article examines the concept of “revolving ownership” as a metamodel of property, whose structure allows for the interchange (rotation) of diverse components included in it, each capable of achieving different objectives through varied means. It is noted that the metamodel of revolving ownership can be utilized in the creation and structuring of property models over specific entities, necessitating consideration of specific social and environmental factors. In this context, a distinction is made between yang-ownership (“classic” private ownership used as an (...) instrument for growth through extraction and accumulation, analogous to so-called “extractive” ownership) and yin-ownership (engaged possession involving a healthful balance between rights and obligations, i.e., flexible distribution of various components aimed at achieving sustainable equilibrium, analogous to the so-called “generative” ownership). The thesis is presented that the sooner the “proprietary” nature of yang-ownership is overcome, yielding to its dominance through the integration of balancing yin-components, the more easily and effectively legal professionals can propose novel and diverse ownership models. The recognition and acknowledgment of the complexity of nature imply greater intricacy in contemporary property law. It is time for property law to reconsider its unquestionable axioms and seek an appropriate solution to the challenges humanity faces today. (shrink)

I consider the general question of whether a study of the life of a philosopher can help us to understand his/her philosophical principles. This topic is narrowed to the consideration of principles of moral and political philosophy, especially in instances in which the philosopher deliberately uses the experiences of his/her own life in formulating his/her views. Such use raises the problem of justification of the self as sampIe. As part of my general defense of the merits of studying a life (...) to grasp a philosophy, I argue that the meaning of a principle in the philosophy of X can be illuminated by trying to recognize what that principIe meant to X. All of these general issues are deployed in my study of the philosophy of John Stuart Mill. In particular I focus on five points, the understanding of which I feel is deepened by studying how they involved his own life. These five are: a change in tone in ch. III of On Liberty, his conviction about free will, his commitment to a principle of vigorous action, his principle of balanced growth, and his fundamental axiom of Individualism. (shrink)

Welfare biology is the study of living things and their environment with respect to their welfare. Despite difficulties of ascertaining and measuring welfare and relevancy to normative issues, welfare biology is a positive science. Evolutionary economics and population dynamics are used to help answer basic questions in welfare biology : Which species are affective sentients capable of welfare? Do they enjoy positive or negative welfare? Can their welfare be dramatically increased? Under plausible axioms, all conscious species are plastic and (...) all plastic species are conscious. More complex niches favour the evolution of more rational species. Evolutionary economics also supports the common-sense view that individual sentients failing to survive to mate suffer negative welfare. A kind of God-made fairness between species is also unexpectedly found. The contrast between growth maximization, average welfare, and total welfare maximization is discussed. It is shown that welfare could be increased without even sacrificing numbers. Since the long-term reduction in animal suffering depends on scientific advances, strict restrictions on animal experimentation may be counter-productive to animal welfare. (shrink)

This paper examines whether philosophers of argument, in spite of their disavowing ‘timeless principles’, nevertheless embrace a set of principles, or axioms, to underlie argumentation theory. First, it reviews the thinking of some prominent philosophers of argument; second, it extracts some principles common to their philosophies; and third, it draws out possible consequences for argumentation theory and asks whether such theory has an underlying political posture.

Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.

Asperó and Schindler have completely solved the Axiom [Formula: see text] vs. [Formula: see text] problem. They have proved that if [Formula: see text] holds then Axiom [Formula: see text] holds, with no additional assumptions. The key question now concerns the relationship between [Formula: see text] and Axiom [Formula: see text]. This is because the foundational issues raised by the problem of Axiom [Formula: see text] vs. [Formula: see text] arguably persist in the problem of Axiom [Formula: see text] vs. (...) [Formula: see text]. The first of our two main theorems is that Axiom [Formula: see text] is equivalent to Axiom [Formula: see text], and as a corollary we show that Axiom [Formula: see text] fails in all the known models of [Formula: see text]. This suggests that [Formula: see text] actually refutes Axiom [Formula: see text]. Our second main theorem is that the [Formula: see text] Conjecture holds assuming [Formula: see text]. This is the strongest partial result known on this conjecture which is one of the central open problems of [Formula: see text]-theory and [Formula: see text]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [Formula: see text]-expressible and which implies [Formula: see text], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [Formula: see text]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [Formula: see text] for which some property involving only quantification over [Formula: see text] holds. (shrink)

No detailed description available for "From Axiom to Dialogue".

In epistemic logic, some axioms dealing with the notion of knowledge are rather convoluted and difficult to interpret intuitively, even though some of them, such as the axioms.2 and.3, are considered to be key axioms by some epistemic logicians. We show that they can be characterized in terms of understandable interaction axioms relating knowledge and belief or knowledge and conditional belief. In order to show it, we first sketch a theory dealing with the characterization of (...) class='Hi'>axioms in terms of interaction axioms in modal logic. We then apply the main results and methods of this theory to obtain specific results related to epistemic and doxastic logics. (shrink)

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

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)

We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.

We compare two well-known set-theoretical statements, namely the axiom of choice and the continuum hypothesis, with regard to their historical development and formulation, as well as their consequences in mathematics. It is known that both statements are independent from the other axioms of set theory (if they are consistent). The axiom of choice – despite initial controversies – is today almost universally accepted as an axiom. However, the status of the continuum hypothesis is more complex and no agreement has (...) been found so far: both the continuum hypothesis and its negation (often as consequences of stronger statements) decide several mathematical problems differently, but in contrast with the axiom of choice it is not clear which of the two solutions should be the “correct” one (in the sense of an agreement within the community). (shrink)

The search for axioms like those of set theories for dealing with collections of indistinguishable elementary particles was posed by Yu. I. Manin, in 1974, as one of the important problems of present day researches on the foundations of mathematics. In this paper we presented a quasi-set theory which stands for a mathematical framework for dealing with collections of indistinguishable objects, whose ’intended interpretation’ is precisely the behaviour of elementary particles as described by non-relativistic quantum mechanics. A sketch of (...) the proof that this theory and ZFC are equiconsistent is also presented and the relationship with the case of quantum particles is mentioned throughout the paper. (shrink)

We analyze axioms and postulates as speech acts. After a brief historical appraisal of the concept of axiom in Euclid, Frege, and Hilbert, we evaluate contemporary axiomatics from a linguistic perspective. Our reading is inspired by Hilbert and is meant to account for the assertive, directive, and declarative components of modern axiomatics. We will do this by describing the constitutive and regulative roles that axioms possess with respect to the linguistic practice of mathematics.

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable (...) property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC00m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${AB_{1/2,0}^{2^{\mathbb{n}}}}$$\end{document}. We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd. (shrink)

The present article deals with the power of the axiom of choice within the second-order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one-sorted second-order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well-ordering theorem for unary (...) predicates is independent from AC for binary predicates and from the trichotomy law for unary predicates. Moreover, we show that the AC for binary predicates follows neither from the trichotomy law for unary predicates nor from Zorn's lemma for unary predicates nor from the formalization of the axiom of choice for disjoint families of sets for binary predicates, and that the trichotomy law for unary predicates does not follow from AC for binary predicates. (shrink)

In this paper we give axiom systems for classical and intuitionistic hybrid logic. Our axiom systems can be extended with additional rules corresponding to conditions on the accessibility relation expressed by so-called geometric theories. In the classical case other axiomatisations than ours can be found in the literature but in the intuitionistic case no axiomatisations have been published. We consider plain intuitionistic hybrid logic as well as a hybridized version of the constructive and paraconsistent logic N4.

Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added. In this paper we propose new versions that extend the underlying static epistemic language in such a way that dynamic completeness proofs can be obtained by perspicuous reduction axioms.

In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, but the (...) reversal is provable even in ACA 0 . These results illuminate the substantial differences among lightface theorems which can not be observed in boldface. (shrink)

The axiom of reducibility plays an important role in the logic of Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. If we understand (...) propositional functions as constructions of the mind, it is clear that the axiom is clearly not a logical axiom and in fact makes an implausible claim. I look at two other ways of understanding propositional functions, a nominalist interpretation along the lines of Landini and a realist interpretation along the lines of Linsky and Mares. I argue that while on either of these interpretations it is not easy to see the axiom as a non-logical claim about the world, there are also appear to be difficulties in accepting it as a purely logical axiom. (shrink)

In this paper rejection systems for the “nonsense-logic” W and the k-valued implicational-negational sentential calculi of Sobociński are given. Considered systems consist of computable sets of rejected axioms and only one rejection rule: the rejection version of detachment rule.

We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.

We consider forcing axioms for suitable families of μ‐complete ‐c.c. forcing notions. We show that some form of the condition “ have a in ” is necessary. We also show some versions are really stronger than others.

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. (...) The distinctions introduced aim at clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions (...) to geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.