A Survey of Mathematical Logic | Date: 30 April 2011, 11:41
|
A Survey of Mathematical Logic.
By Hao. WANG
Title Page
Preface
Contents
Part One General Sketches
Chapter I the Axiomatic Method
Chapter II Eighty Years of Foundational Studies
Chapter III on Formalization
Chapter IV: The Axiomatization of Arithmetic
Chapter V: Computation
Part Two Calculating Machines
Chapter VI a Variant to Turing's Theory of Calculating Machines
References
References
Chapter VII Universal Turing Machines: an Exercise in Coding
Chapter VIII: The Logic of Automata
Chapter IX Toward Mechanical Mathematics the Gallant Tailor: Seven (flies) in One Blow. Ibm 704: 220 Theorems (in the Propositional Calculus) in 3 Minutes
Appendices
References
Chapter X: Circuit Synthesis by Solving Sequential Boolean Equations
Chapter XI the Predicate Calculus
Chapter XII Many-Sorted Predicate Calculi
Chapter XIII the Arithmetization of Metamathematics
Chapter XIV Ackermann's Consistency Proof
Chapter XV Partial Systems of Number Theory
Part Four Impredicative Set Theory
Chapter XVI Different Axiom Systems
References
Chapter XVIII: Truth Definitions and Consistency Proofs
Chapter XIX: Between Number Theory and Set Theory
References
Part Five Predicative Set Theory
Chapter XXI Certain Predicates Defined by Induction Schemata
Chapter XXII: Undecidable Sentences Suggested by Semantic Paradoxes
Chapter XXIII the Formalization of Mathematics
Chapter XXIV Some Formal Details on Predicative Set Theories
Chapter XXV: Ordinal Numbers and Predicative Set Theory
* Publisher: -
* Number Of Pages:
* Publication Date: 1963
* ISBN-10 / ASIN: B0014IQ48Y
* ISBN-13 / EAN:
*****
The bulk of this book consists of a collection of the author's papers up to about 1959, and thus does not include his striking recent results on $\forall\exists\forall$ formulae of predicate logic. Some of the original papers are expanded, e.g., more detail is given on the formalisation of predicative set theory than in J. Symbolic Logic 19 (1954), 241--266 ; some are modified, e.g., the material on finite axiomatisability in Math. Ann. 125 (1952), 56--66, is corrected. As the author points out, these papers do not constitute by themselves a survey of logic. Moreover, several topics first considered by him were later taken up by others and developed in a more systematic and definitive form. Thus a good deal of his early material is now more accessible elsewhere. Nevertheless, the title of the book is quite reasonable because a number of general expository sketches are included, covering such basic results as Godel's completeness and incompleteness theorems, and results on decidability and undecidability. Also more specialised matters are expounded, such as Ackermann's use of the $\varepsilon$-substitution method in arithmetic or Friedberg's construction of recursively enumerable sets of incomparable degree. There are non-technical sections on adequacy of formalisation and evidence in mathematics, in which these topics are treated quite pragmatically (and in a very engaging style). No coherent philosophic position is presented, i.e., whenever possible, the author comments on the formal results in homely language or lets them `speak for themselves', and does not attempt to formulate general philosophic conceptions in terms of which these formal results are to be interpreted. This seems quite successful when applied to finitistic proof theory, but much less so with intuitionistic mathematics (touched on occasionally), or set theory. Thus a reader of this book, despite getting a lot of very interesting historical information on the paradoxes or mathematical curiosities on Quine's set theory, might overlook the remark (after Theorem 10 on p. 407) that Zermelo's set theory and the theory of finite types with the integers as type 0 objects have the same intuitive model (namely, the cumulative type structure). In short, he is not told in so many words what notion of set is under discussion.
Among the offshoots of logic (beyond its original aim of the analysis of reasoning), the author concentrates particularly on the theory of computing machines and neglects almost wholly the theory of models. The book has a very informative table of contents (much more useful than an index would have been), and out-of-the-way references in the text: thus on p. 317 (bottom) he mentions Godel's construction of maximal consistent extensions for arbitrary, i.e., possibly non-denumerable, consistent sets of propositional formulae in 1930/31 (published in Ergebnisse Math. Kolloq. Heft 3 (1932), 20--21), which is rarely quoted. Most of these historical remarks are intrinsically interesting, and, of course, none is used as a mere pretext for personal priority claims.
|
DISCLAIMER:
This site does not store A Survey of Mathematical Logic on its server. We only index and link to A Survey of Mathematical Logic provided by other sites. Please contact the content providers to delete A Survey of Mathematical Logic if any and email us, we'll remove relevant links or contents immediately.
|
 |
|
Comments (0)
All
