THE MATHEMATICS OF DEFLATIONARY TRUTH

Analytic conceptions of truth can be broadly classified along two lines, depending on how they answer the following two questions: 1. Has truth a substantial nature? 2. How many properties of truth are there? Traditional views, like correspondence, coherence, pragmatist views all answer “yes” to both questions. The second half of the last century and the last decades in particular, however, has witnessed the rapid growth in popularity of conceptions of truth answering “no” to at least one. Pluralist views hold that there is more than one property of truth, whereas deflationary views hold that truth lacks a substantial nature. Pluralist and deflationary views are today the main rivals in the field. This book focuses on deflationary views of truth and contributes to the general contemporary debate on truth by addressing an issue emerging from an appealing way to clarify what the insubstantiality o f (deflationary) t ruth might amount to: conservativeness

Envelopes, indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that BΣn +1 (the fragment of Arithmetic given by the collection scheme restricted to Σn +1‐formulas) is a Πn +2‐conservative extension of IΣn (the fragment given by the induction scheme restricted to Σn ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δn +1(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + BΣn +1 is a Πn +2‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Πn ‐envelopes and Πn ‐indicators for T . The analysis of Σn +1‐collection we develop here is also applied to Σn +1‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): BΣn +1 and IΣn +1 are Σn +3‐conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1.Junta de Andalucía TIC-13

Expansions, omitting types, and standard systems

Recursive saturation and resplendence are two important notions in models of arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic saturation and argued that recursive saturation might not be as rigid as first assumed. In this thesis we give further examples of variations of recursive saturation, all of which are connected with expandability properties similar to resplendence. However, the expandability properties are stronger than resplendence and implies, in one way or another, that the expansion not only satisfies a theory, but also omits a type. We conjecture that a special version of this expandability is in fact equivalent to arithmetic saturation. We prove that another of these properties is equivalent to \beta-saturation. We also introduce a variant on recursive saturation which makes sense in the context of a standard predicate, and which is equivalent to a certain amount of ordinary saturation. The theory of all models which omit a certain type p(x) is also investigated. We define a proof system, which proves a sentence if and only if it is true in all models omitting the type p(x). The complexity of such proof systems are discussed and some explicit examples of theories and types with high complexity, in a special sense, are given. We end the thesis by a small comment on Scott's problem. We prove that, under the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0} closed under arithmetic comprehension which has the countable chain condition is the standard system of some model of PA. However, we do not know if there exists any such uncountable Scott sets.Comment: Doctoral thesi

Countable Short Recursively Saturated Models of Arithmetic

Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated

Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s 'Completeness Paper' (1930)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined

Infinitary cut-elimination via finite approximations

We investigate non-wellfounded proof systems based on parsimonious logic, a weaker variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. Logical consistency is maintained at a global level by adapting a standard progressing criterion. We present an infinitary version of cut-elimination based on finite approximations, and we prove that, in presence of the progressing criterion, it returns well-defined non-wellfounded proofs at its limit. Furthermore, we show that cut-elimination preserves the progressive criterion and various regularity conditions internalizing degrees of proof-theoretical uniformity. Finally, we provide a denotational semantics for our systems based on the relational model

Teaching Multiplication with Lesson Study

This open access book is intended to assist teachers, teacher trainers, curriculum designers, editors and authors of textbooks in developing strategies to teach the multiplication of natural numbers based on the experience of the Lesson Study in Japan. This approach to mathematics education dates back to the 1870s and reconciles the emphasis on problem solving with the treatment of the curricular contents. It has gained international recognition since the 1990s and thanks to it mathematics education in Japan has been recognized as one of the most efficient and innovative in the world. This growing international awareness has led to an effort to apply the principles of Lesson Study to other parts of the world and this book shows how experienced authors from Brazil, Chile, Mexico, Spain and Portugal have worked to adapt some of these methods and techniques to the Portuguese and Spanish speaking countries of Ibero-America. Drawing on the impact of Lesson Study on government curriculum decisions and teacher behavior in Japanese classrooms; offering examples of lessons, lesson plans and suggestions for teaching; and presenting examples of the good reception of the principles of Lesson Study in Ibero-America, Teaching Multiplication with Lesson Study – Japanese and Ibero-American Theories for Mathematics Education shows how an efficient and cutting-edge experience in mathematics education can travel the world and help teachers in many different countries. ; Shows how teachers can apply the principles of Japanese Lesson Study to teach the multiplication of natural numbers Presents an approach to mathematics education that reconciles the emphasis on problem solving with the treatment of the curricular contents Provides examples of how a method developed in Japan can be adapted to different cultural contexts, such as the Portuguese and Spanish speaking countries of Ibero-Americ