The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

The axiom of symmetry (A ℵ 0 ) asserts that for every function F: ω 2 → ω 2 there is a pair of reals x and y in ω 2 so that y is not in the countable set $\{(F(x))_n:n coded by F(x) and x is not in the set coded by F(y). A(Γ) denotes axiom A ℵ 0 with the restriction that graph(F) belongs to the pointclass Γ. In § 2 we prove A(Σ 1 1 ). In § (...) 3 we show A(Π 1 1 ), A(Σ 1 2 ) and $^\omega 2 \nsubseteq L$ are equivalent. In § 4 several effective versions of A(REC) are examined. (shrink)

A nucleus is an operation on the collection of truth values which, like double negation in intuitionistic logic, is monotone, inflationary, idempotent and commutes with conjunction. Any nucleus determines a proof-theoretic translation of intuitionistic logic into itself by applying it to atomic formulas, disjunctions and existentially quantified subformulas, as in the Gödel–Gentzen negative translation. Here we show that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda’s negative translation. The key is (...) to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well. (shrink)

Since Turing proposed the first test of intelligence, several modifications have been proposed with the aim of making Turing’s proposal more realistic and applicable in the search for artificial intelligence. In the modern context, it turns out that some of these definitions of intelligence and the corresponding tests merely measure computational power. Furthermore, in the framework of the original Turing test, for a system to prove itself to be intelligent, a certain amount of deceit is implicitly required which can have (...) serious security implications for future human societies. In this article, we propose a unified framework for developing intelligence tests which takes care of important ethical and practical issues. Our proposed framework has several important consequences. Firstly, it results in the suggestion that it is not possible to construct a single, context independent, intelligence test. Secondly, any measure of intelligence must have access to the process by which a problem is solved by the system under consideration and not merely the final solution. Finally, it requires an intelligent agent to be evolutionary in nature with the flexibility to explore new algorithms on its own. (shrink)

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\scr{D}_{h}$ . In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of $\scr{D}_{h}$ . Corollaries include the decidability of the two quantifier theory of $\scr{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a (...) new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$ . Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω₁. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of $\scr{D}_{h}$. (shrink)

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.

En este artículo se examinan algunas implicaciones del naturalismo matemático de P. Maddy como una concepción filosófica que permite superar las dificultades del ficcionalismo y el realismo fisicalista en matemáticas. Aparte de esto, la mayor virtud de tal concepción parece ser que resuelve el problema que plantea para la aplicabilidad de la matemática el no asumir la tesis de indispensabilidad de Quine sin comprometerse con su holismo confirmacional. A continuación, sobre la base de dificultades intrínsecas al programa de Maddy, exploramos (...) un camino naturalista mejor motivado, el enfoque cognitivista corporeizado, y sugerimos que él permite explicar de manera adecuada la postulación de ciertos cardinales grandes, en particular, la aceptación de conjuntos no-construibles. (shrink)

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the (...) theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals. (shrink)

In this essay I consider what is meant by the description ‘great’ philosophy and then offer some broadly applicable criteria by which to assess candidate thinkers or works. On the one hand are philosophers in whose case the epithet, even if contested, is not grossly misconceived or merely the product of doctrinal adherence on the part of those who apply it. On the other are those – however gifted, acute, or technically adroit – to whom its application is inappropriate because (...) their work cannot justifiably be held to rise to a level of creative-exploratory thought where the description would have any meaningful purchase. I develop this contrast with reference to Thomas Kuhn’s distinction between ‘revolutionary’ and ‘normal’ science, and also in light of J.L. Austin’s anecdotal remark – à propos Leibniz – that it was the mark of truly great thinkers to make great mistakes, or to risk falling into certain kinds of significant or consequential error. My essay goes on to put the case that great philosophy should be thought of as involving a constant readiness to venture and pursue speculative hypotheses beyond any limits typically imposed by a culture of ‘safe’, well-established, or academically sanctioned debate. At the same time – and just as crucially – it must be conceived as subject to the strictest, most demanding standards of formal assessment, i.e., with respect to basic requirements of logical rigour and conceptual precision. Focusing mainly on the work of Jacques Derrida and Alain Badiou I suggest that these criteria are more often met by philosophers in the broadly ‘continental’ rather than the mainstream ‘analytic’ line of descent. However – as should be clear – the very possibility of meeting them, and of their being jointly met by any one thinker, is itself sufficient indication that this is a false and pernicious dichotomy. (shrink)

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...) talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched. (shrink)

Though many working mathematicians embrace a rough and ready form of Platonism, that venerable position has suffered a checkered philosophical career. Indeed the three schools of thought with which most of us began our official philosophizing about mathematics—Intuitionism, Formalism, and Logicism—all stand in fundamental disagreement with Platonism. Nevertheless, various versions of Platonistic thinking survive in contemporary philosophical circles. The aim of this paper is to describe these views, and, as my title suggests, to trace their roots.I'll begin with some preliminary (...) remarks about the big three schools. This seems a reasonable approach to the issues both because most observers are familiar, at least in a general way, with the tenets of Intuitionism, Formalism, and Logicism, and because it is in reaction to these that contemporary Platonism has taken shape. (shrink)

In this article my primary intention is to engage in a discussion on the inherent constraints of models, taken as models of theories, that reaches beyond the epistemological level. Naturally the paper takes into account the ongoing debate between proponents of the syntactic and the semantic view of theories and that between proponents of the various versions of scientific realism, reaching down to the most fundamental, subjective level of discourse. In this approach, while allowing for a limited discussion of physical (...) and positive science models, I am primarily focused on the structure and ontology of mathematical models, in particular Cohen’s forcing models and to a lesser extent Gödel’s constructible universe, to the extent that these were designed to answer questions bearing on the scope, the capacity and ultimately the ontology of models themselves, therefore influencing in one or the other way the status of models in general. This status, it is argued, is largely defined by the way models subsume a set-theoretical structure whose constraints, reducible to an extra-linguistic level of discourse, may implicitly condition the epistemic status of models as representations of axiomatic theories. In the last section I deal extensively with the inner constraints of theories as subjectively originated in a less technical philosophically oriented discussion with certain prompts from phenomenology. (shrink)

Based on formal arguments from Zermelo–Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo–Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing (...) such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of \. The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth \ from which one determines the realistic, agreeing with observation, small value of the vacuum energy density. (shrink)

The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.

In [3] F. P. Ramsey proved as a theorem of Zermelo-Fraenkel set theory (ZF) with the Axiom of Choice (AC) the following result:(1) Theorem. Let A be an infinite class. For each integer n and partition {X, Y} of the size n subsets of A, there exists an infinite subclass of A all of whose size n subsets are contained in only one of X or Y.

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.

Where $\underline{a}$ is a Turing degree and ξ is an ordinal $ , the result of performing ξ jumps on $\underline{a},\underline{a}^{(\xi)}$ , is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? (...) What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski. In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

In spite of the work of Gödel and Cohen, which showed the undecidability of the Generalized Continuum Hypothesis from the axioms of set theory, the problem still remains to decide GCH on the basis of new axioms. It is almost 100 years since Cantor first conjectured the Continuum Hypothesis, yet we seem to be no closer to determining its truth. Nevertheless, it is a sound methodological principle that given any undecidable set-theoretical statement, we should search for “other axioms of set (...) theory which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts”. (shrink)

We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...) why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy--if you reject CH you are only two steps away from rejecting the axiom of choice (AC)--we will point out along the way some extensions of our intuition which contradict AC. (shrink)

In this paper, we strengthen Hardy’s [1995] and Ketland’s [2005] arguments on the issues surrounding the self-referential nature of Yablo’s paradox [1993]. We first begin by observing that Priest’s [1997] construction of the binary satisfaction relation in revealing a fixed point relies on impredicative definitions. We then show that Yablo’s paradox is ‘ω-circular’, based on ω-inconsistent theories, by arguing that the paradox is not self-referential in the classical sense but rather admits circularity at the least transfinite countable ordinal. Hence, we (...) both strengthen arguments for the ω-inconsistency of Yablo’s paradox and present a compromise solution of the problem emerged from Yablo’s and Priest’s conflicting theses. (shrink)

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK∞+. We show that in these conditions, we can interprete the Kelley‐Morse theory (KM) in GPK∞+ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo‐Fraenkel set theory (ZF) in GPK∞+. We also see what happens in (...) the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections. (shrink)

A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF, T has a Leibnizian model if and only if T proves LM. Here we prove: THEOREM A. Every complete theory T extending ZF + LM has $2^{\aleph_{0}}$ nonisomorphic countable Leibnizian models. THEOREM B. If $\kappa$ is aprescribed definable infinite cardinal of a (...) complete theory T extending ZF + V = OD. then there are $2^{\aleph_{1}}$ nonisomorphic Leibnizian models $\mathfrak{M}$ of T of power $\aleph_{1}$ such that $(\kappa^{+})^\mathfrak{M}$ is $\aleph_{1}-like$ . THEOREM C. Every complete theory T extending ZF + V = OD has $2^{\aleph_{1}}$ nonisomorphic \aleph_{1}-like$ Leibnizian models. (shrink)

There is a fairly simple algebraic property that distinguishes isols from cosimple isols.

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some remarks on standard models (...) are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55. (shrink)

This is an examination, a commentary, of links between some philosophical views ascribed to Gödel and general proof theory. In these views deduction is of central concern not only in predicate logic, but in set theory too, understood from an infinitistic ideal perspective. It is inquired whether this centrality of deduction could also be kept in the intensional logic of concepts whose building Gödel seems to have taken as the main task of logic for the future.