This paper addresses the now entrenched historiography of the Australian Settlement and New Zealand variations thereof. Against the central premise of this historiography, that a particular regime of domestic insulation and external orientation to the British market constrained development and persisted unchanged until the neo-liberal restructuring of the 1980s, it is argued here that the political economy of the beginning of the 20th century was profoundly destabilized by the Depression. As a result, a new, Keynesian regime was established in New (...) Zealand from the late 1930s and in Australia a few years later. The entrenchment of this regime depended upon adoption by remade conservative parties by the end of the 1940s. (shrink)

Let P 0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P 0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M ' of M which is a model of T such that the complete diagram of M ' (...) is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive. (shrink)

Background Escalating levels of obesity place enormous and growing demands on Health care provision in the (U.K.) United Kingdom. Resources are limited with increasing and competing demands upon them. Ethical considerations underpin clinical decision making generally, but there is limited evidence regarding the relationship between these variables particularly in terms of treating individuals with obesity. Research aim To investigate the views of National Health Service (NHS) clinicians on navigating the ethical challenges and decision making associated with obesity management in adults (...) with chronic illness. Research design A cross-sectional, multi-site survey distributed electronically. Participants A consensus sample of nurses, doctors, dietitians and final year students in two NHS Trusts and two Universities. Ethical considerations Ethical and governance approvals obtained from a National Ethics Committee (11NIR035), two universities and two teaching hospitals. Results Of the total ( n = 395) participants, the majority were nurses (48%), female (79%) and qualified clinicians (59%). Participants strongly considered the individual to have primary responsibility for a healthy weight and an obligation to attempt to maintain that healthy weight if they wish to access NHS care. Yet two thirds would not withhold treatment for patients with obesity. Discussion While clinicians were clear about patient responsibility and obligations, the majority prioritised their duty of care and would not invoke a utilitarian approach to decision making. This may reflect awareness of obesity as a multi-faceted entity, with responsibility for support and management shared amongst society in general. Conclusions The attitudes of this sample of clinicians complemented the concept of the health service as being built on a principle of community, with each treated according to their need. However limited resources challenge the concept of needs-based decisions consequently societal engagement is necessary to agree a pragmatic way forward. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...) are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (shrink)

For a function $f$ with domain $[X]^{n}$, where $X\subseteq\mathbb{N}$, we say that $H\subseteq X$ is canonical for $f$ if there is a $\upsilon\subseteq n$ such that for any $x_{0},\ldots,x_{n-1}$ and $y_{0},\ldots,y_{n-1}$ in $H$, $f=f$ iff $x_{i}=y_{i}$ for all $i\in\upsilon$. The canonical Ramsey theorem is the statement that for any $n\in\mathbb{N}$, if $f:[\mathbb{N}]^{n}\rightarrow\mathbb{N}$, then there is an infinite $H\subseteq\mathbb{N}$ canonical for $f$. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness (...) condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying $\operatorname{PA}$. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

Paul Erdős was a mathematicianpar excellencewhose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. Hismodus operandiwas to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressing and ever reaching, and hismodus vivendiwas to be itinerant (...) in the world, stimulating and interacting about mathematics at every port and capital. (shrink)

Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership (...) distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. (shrink)

We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing.

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. (...) Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions (...) and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was. (shrink)

On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.

(2005). The Problem of Vitalism Revisited. Angelaki: Vol. 10, continental philosophy and the sciences the french tradition issue editor: andrew aitken, pp. 13-26.

Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse. This article address the historical emergence of the infinite in set theory, and how we are to take the (...) infinite in and out of mathematics.The first section discusses the emergence of the infinite as a matter of method in set theory. The second section discusses the infinite in and out of mathematics, and how it is to be taken. (shrink)

Analytic philosophy, a dominant tradition of twentieth-century philosophy, can be informatively cast as the outgrowth of the investigations of logic and language of Gottlob Frege, Bertrand Russell, and Ludwig Wittgenstein, and in the next generation, of Rudolf Carnap and W.V. Quine. As such, it is a specific historical development, one that featured subtle dialectical interactions among its propounders, interactions that have been reflected or reenacted in later developments. Whatever its heritage, contemporary analytic philosophy continues to use investigations of language and (...) thought to get at fundamental issues at the heart of philosophy: truth, meaning, and knowledge. (shrink)

We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory.

Azriel Levy did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions (...) to set theory. (shrink)