Mathematical Logic
This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic
eBook, English, 1994
Second edition View all formats and editions
Springer New York, New York, NY, 1994
1 online resource (x, 290 pages)
9781475723557, 1475723555
851760204
Print version:
A
I Introduction
II Syntax of First-Order Languages
III Semantics of First-Order Languages
IV A Sequent Calculus
V The Completeness Theorem
VI The Löwenheim-Skolem and the Compactness Theorem
VII The Scope of First-Order Logic
VIII Syntactic Interpretations and Normal Forms
B
IX Extensions of First-Order Logic
X Limitations of the Formal Method
XI Free Models and Logic Programming
XII An Algebraic Characterization of Elementary Equivalence
XIII Lindström's Theorems
References
Symbol Index
English