Mathematical Logic and the Foundations of Mathematics An Introductory Survey

by Kneebone, G. T.

• ISBN13: 9780486417127
• ISBN10: 0486417123
• Format: Paperback
• Copyright: 2001-04-25
• Publisher: Dover Publications
• More Book Details
• Purchase Benefits

Summary

Graduate-level historical study is ideal for students intending to specialize in the topic, as well as those who only need a general treatment. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics, emphasizing Hilbert's metamathematics. Part III focuses on the philosophy of mathematics. Each chapter has extensive supplementary notes; a detailed appendix charts modern developments.

Table of Contents

Preface	v
Acknowledgements	viii
Part I MATHEMATICAL LOGIC
Traditional Logic
Introduction: mathematics and logic	3	(1)
The nature of mathematics	4	(1)
The nature of logic	5	(3)
Terms and propositions	8	(6)
Syllogistic inference	14	(4)
The reduction of syllogisms	18	(2)
Deductive arguments of more complex form	20	(1)
Supplementary notes on Chapter 1
Aristotle's Organon	21	(1)
Transitional logic	22	(2)
Venn's diagrams	24	(1)
Additional reading	25	(1)
The Development of Logic	25	(259)
Symbolic Logic I---The Propositional Calculus
Propositional logic	26	(1)
Propositional variables and the basic connectives	27	(4)
Equiveridicity of formulae	31	(3)
Elementary transformations in the propositional calculus	34	(1)
Normal forms	35	(5)
Axiomatic treatment of the propositional calculus	40	(9)
Supplementary notes on Chapter 2
The term `semantic'	49	(1)
The logical notation of Lukasiewicz	49	(1)
The sixteen truth-functions of two propositional variables	50	(1)
Boole's application of algebraic symbolism to logic	51	(2)
Logical sum and logical product	53	(1)
C. I. Lewis's modal logic of strict implication	54	(1)
Many-valued logics	55	(1)
Books on symbolic logic	56	(2)
Symbolic Logic II---The Restricted Calculus of Predicates
Propositional functions	58	(2)
Quantification	60	(4)
Axiomatic treatment of the calculus of predicates	64	(6)
Completeness of the restricted calculus of predicates	70	(8)
The deduction theorem	78	(8)
Supplementary notes on Chapter 3
Some points concerning symbolism	86	(2)
A strong sense of `complete'	88	(1)
Further Development of Symbolic Logic
The relation of identity	89	(2)
Descriptions and the ι-symbol	91	(7)
Characteristic functions of formulae	95	(3)
Formalized mathematical theories	98	(2)
Hilbert's &epsis;-symbol	100	(7)
Symbolic resolution of existential axioms	104	(3)
Classes and relations	107	(3)
The extended calculus of predicates	110	(8)
Russell's antinomy	112	(4)
Axiomatization of the extended calculus of predicates	116	(2)
The logical calculus of Bourbaki's Elements de Mathematique	118	(3)
Gentzen's calculus of natural deduction	121	(6)
Supplementary notes on Chapter 4
The vicious-circle paradoxes	127	(1)
Tarski's formalization of semantics	128	(2)
The &epsis;-symbol as a selection operator	130	(1)
Additional reading	130	(3)
Part II FOUNDATIONS OF MATHEMATICS
The Critical Movement in Mathematics in the Nineteenth Century
Symbolic logic in relation to the foundations of mathematics	133	(1)
Greek mathematics	134	(3)
The beginnings of modern mathematics	137	(2)
The first phase of the critical movement	139	(2)
Peano's Formulaire de Mathematique	141	(9)
The symbolic language of Leibniz	150	(2)
Supplementary notes on Chapter 5
Peano's space-filling curve	152	(1)
Ideal numbers and ideals	153	(1)
Sources of historical information	154	(2)
The Logistic Identification of Mathematics with Logic
Russell's conception of mathematics	156	(2)
Dedekind's analysis of number	158	(3)
Cantor's definition of cardinal and ordinal number	160	(1)
Principia Mathematica	161	(12)
The logical calculus of Principia Mathematica	163	(3)
Russell's theory of types	166	(2)
The formal development of Principia Mathematica	168	(5)
Frege's logical analysis of arithmetic	173	(9)
The Begriffsschrift	174	(4)
The Grundgesetze der Arithmetik	178	(4)
Supplementary notes on Chapter 6
Non-euclidean geometry	182	(1)
Sense and denotation	183	(1)
The algebra of logic	184	(4)
`⊢' as a symbol for derivability	188	(1)
Introductory books on the foundations of mathematics	189	(1)
Books on the logistic conception of mathematics	189	(1)
Sources	190	(1)
Formalized Mathematics and Metamathematics
Hilbert's new approach to the foundations of mathematics	191	(2)
Hilbert's Grundlagen der Geometrie	193	(7)
Axiomatic theories and their significance	200	(4)
The domain of numerals, treated by finitary means	204	(3)
The metamathematics of formalized theories	207	(1)
Consistency of the restricted calculus of predicates	208	(1)
Consistency of arithmetic	209	(15)
Supplementary notes on Chapter 7
Foundations of projective geometry	224	(3)
Models of non-euclidean geometry	227	(1)
The literature of Hilbert's metamathematics	228	(1)
Godel's Theorems on the Inherent Limitations of Formal Systems
Godel's new metamathematical method	229	(2)
Godel's heuristic argument	231	(1)
The formal system P and its arithmetized metamathematics	232	(5)
The central argument of Godel's paper	237	(2)
The impossibility of an `internal' proof of consistency	239	(2)
Supplementary notes on Chapter 8
Gentzen's proof of the consistency of arithmetic	241	(1)
Alternative accounts of Godel's investigation	242	(1)
Intuitionism
The intuitionist outlook	243	(5)
Primary intuition	248	(2)
Intuitionist analysis	250	(4)
Heyting's formalization of intuitionist logic	254	(2)
Supplementary notes on Chapter 9
Accounts of Brouwer's intuitionism	256	(1)
Weyl's intuitionism	256	(2)
Recursive Arithmetic
The class of natural numbers as a progression	258	(2)
Recursive definitions	260	(1)
Systematic development of recursive arithmetic	261	(4)
Recursive arithmetic as a formal system	265	(2)
Representability of recursive arithmetic in the system (Z)	267	(3)
Primitive recursive and other recursive schemata	270	(2)
General recursive functions	272	(2)
The mathematical ideal of constructivity	274	(9)
Church's calculus of λ-conversion	275	(2)
Turing's conception of computability	277	(1)
The decision problem for formalized theories	278	(2)
Post's canonical form for formal systems	280	(3)
Supplementary notes on Chapter 10
Books on recursive arithmetic	283	(1)
Books on constructivity in general	283	(1)
The Axiomatic Theory of Sets	284	(95)
Pure mathematics as an extension of the theory of sets	284	(1)
The native theory of sets	285	(2)
Zermelo's axiomatic theory of sets	287	(5)
Von Neumann's new approach to the theory of sets	292	(2)
Bernays's unification of symbolic logic and the theory of sets	294	(7)
The theory of sets in Bourbaki's Elements de Mathematique	301	(2)
Limitations of the axiomatic treatment of sets	303	(4)
Supplementary notes on Chapter 11
Historically important papers on the theory of sets	307	(1)
Additional reading	307	(4)
Part III PHILOSOPHY OF MATHEMATICS
The Epistemological Status of Mathematics
Retrospect	311	(1)
The logistic conception of mathematics	312	(7)
Mathematics as an edifice of propositions	315	(1)
Logic and objective reality	316	(3)
The relativism of Brouwer's intuitionist outlook	319	(3)
Hilbert's twenty-three mathematical problems	322	(3)
Bourbaki's interpretation of mathematics	325	(6)
The Application of Mathematics to the Natural World
Pure mathematics and applied mathematics	331	(2)
The spatial structure of the world	333	(2)
Naive realism and its inadequacy	335	(3)
The a priori form of the physical world	338	(3)
Whitehead's theory of natural knowledge	341	(11)
The method of extensive abstraction	346	(2)
Definition of time and space by extensive abstraction	348	(2)
Objects	350	(2)
Mathematics and the logical analysis of the natural world	352	(2)
Supplementary notes on Chapter 13
Zeno's paradoxes of motion	354	(1)
The theory of games	354	(1)
The mathematical theory of crystalline structure	355	(1)
Books on the philosophy of science	356	(1)
Logic and the Activity of Thinking
The limitations of formal logic	357	(2)
The logic of concrete thought	359	(7)
Inductive reasoning	366	(7)
The element of judgement in induction	368	(2)
The calculus of probabilities	370	(3)
The genetic method in philosophy	373	(2)
The outcome of the philosophy of mathematics	375	(4)
Appendix Developments since 1939 in the Study of Foundations of Mathematics	379	(22)
1 Mathematical logic	381	(4)
2 Consistency of analysis	385	(2)
3 Constructive treatment of ordinal numbers	387	(1)
4 Modern views on proof of consistency	388	(1)
5 Models	389	(3)
6 Decision procedures	392	(2)
7 Effectiveness and constructivity	394	(1)
8 Intuitionism	395	(1)
9 The theory of sets	396	(1)
10 Many-valued logics	397	(1)
11 Metamathematics and algebra	397	(1)
12 Modal logic	398	(1)
13 Formalization of semantics	398	(1)
14 Probability and induction	398	(1)
15 Nominalism and platonism	399	(1)
16 Logic and computing machines	400	(1)
Bibliography	401	(23)
The Greek and German Alphabets	424	(1)
Index of Symbols	425	(2)
General Index	427