Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness

Description

Is mathematics 'entangled' with its various formalisations? Or are the central concepts of mathematics largely insensitive to formalisation, or 'formalism free'? What is the semantic point of view and how is it implemented in foundational practice? Does a given semantic framework always have an implicit syntax? Inspired by what she calls the 'natural language moves' of Gödel and Tarski, Juliette Kennedy considers what roles the concepts of 'entanglement' and 'formalism freeness' play in a range of logical settings, from computability and set theory to model theory and second order logic, to logicality, developing an entirely original philosophy of mathematics along the way. The treatment is historically, logically and set-theoretically rich, and topics such as naturalism and foundations receive their due, but now with a new twist. Cover Half-title page Title page Copyright page Dedication Contents Preface 1 Introduction 1.1 The Syntax/Semantics Distinction 1.2 Our Logical Pluralism 1.3 Formal vs Linguistic Semantics 2 Formalism Freeness and Entanglement: Definitions 2.1 Precedents 2.2 Entanglement and Formalism Freeness: Varieties 2.3 Natural Language Moves 3 Computability: The Primary Example 3.1 On Adequacy 3.2 Different Notions of Computability Emerge in the 1930s 3.3 The "Scope Problem'' 3.4 Turing's Analysis of Computability 3.5 Gödel's Reaction to Turing's Work at the Time 3.6 Coda: A Word about Deviant Encodings 4 Gödel and Formalism Independence 4.1 Gödel on Formalisation 4.2 Episodes of Formalism Independence in Gödel's Writings 4.3 Gödel's Princeton Bicentennial Lecture 4.4 Implementation 4.5 Logical Autonomy? 5 Tarski and "the Mathematical'' 5.1 "The Mathematical'', Definable Sets of Reals, and Naive Set Theory 5.2 Tarski's Naturalism 5.3 Squeezing First Order Definability 5.4 Tarski and Logicality 5.5 In Sum: Parataxis 5.6 Coda: An Improvement of McGee's Theorem 6 Model-Theoretic Aspects 6.1 Abstract Elementary Classes 6.2 Patchwork Foundations, On-Again-Off-Again-Ism and Implicit Syntax 6.3 Implicit Syntax, Implicit Logic 6.4 A Remark on Set Theory 6.5 Symbiosis 6.6 Coda: Symbiosis in Detail 7 On the Side of Natural Language Bibliography Index