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# hunter_1 - Preface My main aim is to make accessible to...

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Preface My main aim is to make accessible to readers without any specialist training in mathematics, and with only an elementary knowledge of modern logic, complete proofs of the fundamental metatheorems of standard (i.e. basically truth-functional) first order logic, including a complete proof of the undecidability of a system of first order predicate logic with identity. Many elementary logic books stop just where the subject gets interesting. This book starts at that point and goes through the interesting parts, as far as and including a proof that it is impossible to program a computer to give the right answer (and no wrong answer) to each question of the form 'Is - a truth of pure logic?' The book is intended for non-mathematicians, and concepts of mathematics and set theory are explained as they are needed. The main contents are: Proofs of the consistency, complete- ness and decidability of a formal system of standard truth- functional propositional logic. The same for first order monadic predicate logic. Proofs of the consistency and completeness of a formal system of first order predicate logic. Proofs of the con- sistency, completeness and undecidability of a formal system of first order predicate logic with identity. A proof of the existence of a non-standard model of a formal system of arithmetic. The reader will be assumed to have an elementary knowledge of truth-functional connectives, truth tables and quantifiers. For the reader with no knowledge of set theory, here very brief explanations of some notations and ideas that will be taken for granted later on: 1 . mr!~ brakt nsteti~;; fii - - '(Fido, Joe)' means 'The set whose sole members are Fido and Joe'. '{3,2, 1,3,2)' means 'The set whose sole members are the numbers 3,2,1,3,2' (and this last set is the same set {1,2,3), i.e. the set whose sole members are the numbers 1.2 and 3).

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Xii METALOGIC 2. The epsilon notation for set-membershb 'n E X' means 'n is a member of the set X'. 3. T;he criterion of identity for sets A set A is the same set as a set B if and only A and B have exactly the same members. Nothing else matters for set identity. 4. empty set, 0 By the criterion of identity for sets [(3) above], if A is a set with no members and B is a set with no members, then A is the same set as B; so if there is a set with no members, there is just one such set. We shall assume that there is such a set. Further introductory material on set theory can be found in, for example, chap. 9 of Suppes (1957) or chap. 1 of Fraenkel (1961). The book deals only with (1) standard (i.e. basically truth- fuctional) logic, and (2) axiomatic systems. (1) Standard first order logic, with its metatheory, is now a secure field of knowledge; it is not the whole of logic, but it is important, and it is a jumping-off point for most other develop ments in modern logic. There seemed to me to be no book that tried to make accessible to non-mathematicians complete proofs of the basic metatheory of standard logic: hence this one. Axiomless systems (so-called 'natural deduction systems') are nowadays getting more popular than axiomatic systems, for
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hunter_1 - Preface My main aim is to make accessible to...

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