This book provides an introduction to Gödel's theorem. Gödel's theorem states that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic. The content of the theorem is elucidated and distinguished from that of other claims with which it is often confused. The significance of the theorem is also discussed. Particular emphasis is laid on the appeal of axiomatization and on attempts that were made, in the half century preceding Gödel's proof, to provide the very thing that the theorem precludes. This includes discussion of Hilbert's programme, part of which was to provide a consistent foundation for mathematics and to demonstrate its consistency by mathematical means. Two proofs of Gödel's theorem are given. The second and more elaborate proof is also shown to yield Gödel's second theorem: that no consistent axiomatization of arithmetic can be used to prove a statement corresponding to a statement of its own consistency. The final two chapters of the book explore the implications of Gödel's results: for Hilbert's programme; for the question whether the human mind, in its capacity to think beyond any given axiomatization of arithmetic, has powers beyond those of any possible computer; and for the nature of mathematics.

Bibliographic references

Includes bibliographical references and index.

Contents

Preface

List of illustrations

1 What is Gödel's theorem?

2 Axiomatization: its appeal and demands

3 Historical background

4 The key concepts involved in Gödel's theorem

5 The diagonal proof of Gödel's theorem

6 A second proof of Gödel's theorem, and a proof of Gödel's second theorem

7 Hilbert's programme, the human mind, and computers

8 Making sense in and of mathematics

Appendix: A sketch of the proof of Gödel's theorem(s)

References

Further reading

Index.

ISBN

0-19-194317-7

Doi

10.1093/actrade/9780192847850.001.0001

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