Mathematical Logic

Undergraduate students with no prior instruction in mathematical logic will benefit from this multi-part text. Part I offers an elementary but thorough overview of mathematical logic of 1st order. Part II introduces some of the newer ideas and the more profound results of logical research in the 20th century. 1967 edition.

PART I. ELEMENTARY MATHEMATICAL LOGIC

The Propositional Calculus

3

(71)

Linguistic considerations: formulas

3

(5)

Model theory: truth tables, validity

8

(5)

Model theory: the substitution rule, a collection of valid formulas

13

(4)

Model theory: implication and equivalence

17

(3)

Model theory: chains of equivalences

20

(2)

Model theory: duality

22

(3)

Model theory: valid consequence

25

(3)

Model theory: condensed truth tables

28

(5)

Proof theory: provability and deducibility

33

(6)

Proof theory: the deduction theorem

39

(4)

Proof theory: consistency, introduction and elimination rules

43

(2)

Proof theory: completeness

45

(5)

Proof theory: use of derived rules

50

(8)

Applications to ordinary language: analysis of arguments

58

(9)

Applications to ordinary language: incompletely stated arguments

67

(7)

The Predicate Calculus

74

(74)

Linguistic considerations: formulas, free and bound occurrences of variables

74

(9)

Model theory: domains, validity

83

(10)

Model theory: basic results on validity

93

(3)

Model theory: further results on validity

96

(5)

Model theory: valid consequence

101

(6)

Proof theory: provability and deducibility

107

(5)

Proof theory: the deduction theorem

112

(4)

Proof theory: consistency, introduction and elimination rules

116

(5)

Proof theory: replacement, chains of equivalences

121

(4)

Proof theory: alterations of quantifiers, prenex form

125

(9)

Applications to ordinary language: sets, Aristotelian categorical forms

134

(6)

Applications to ordinary language: more on translating words into symbols

140

(8)

The Predicate Calculus with Equality

148

(27)

Functions, terms

148

(3)

Equality

151

(6)

Equality vs. equivalence, extensionality

157

(10)

Descriptions

167

(8)

PART II. MATHEMATICAL LOGIC AND THE FOUNDATIONS OF MATHEMATICS

The Foundations of Mathematics

175

(48)

Countable sets

175

(5)

Cantor's diagonal method

180

(3)

Abstract sets

183

(3)

The paradoxes

186

(5)

Axiomatic thinking vs. intuitive thinking in mathematics

191

(7)

Formal systems, metamathematics

198

(3)

Formal number theory

201

(14)

Some other formal systems

215

(8)

Computability and Decidability

223

(60)

Decision and computation procedures

223

(9)

Turing machines, Church's thesis

232

(10)

Church's theorem (via Turing machines)

242

(5)

Applications to formal number theory: undecidability (Church) and incompleteness (Godel's theorem)

247

(7)

Applications to formal number theory: consistency proofs (Godel's second theorem)

254

(6)

Application to the predicate calculus (Church, Turing)

260

(5)

Degrees of unsolvability (Post), hierarchies (Kleene, Mostowski)

265

(8)

Undecidability and incompleteness using only simple consistency (Rosser)

273

(10)

The Predicate Calculus (Additional Topics)

283

(88)

Godel's completeness theorem: introduction

283

(12)

Godel's completeness theorem: the basic discovery

295

(10)

Godel's completeness theorem with a Gentzen-type formal system, the Lowenheim-Skolem theorem

305

(7)

Godel's completeness theorem (with a Hilbert-type formal system)

312

(3)

Godel's completeness theorem, and the Lowenheim-Skolem theorem, in the predicate calculus with equality

315

(6)

Skolem's paradox and nonstandard models of arithmetic

321

(10)

Gentzen's theorem

331

(7)

Permutability, Herbrand's theorem

338

(11)

Craig's interpolation theorem

349

(12)

Beth's theorem on definability, Robinson's consistency theorem

361

(10)

Bibliography

371

(15)

Theorem and Lemma Numbers: Pages

386

(1)

List of Postulates

387

(1)

Symbols and Notations

388

(1)

Index

389

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