1st Edition
by George Tourlakis (Author)
A comprehensive and user-friendly guide to the use of logic in mathematical reasoning
Mathematical Logic
presents a comprehensive introduction to formal methods of logic and
their use as a reliable tool for deductive reasoning. With its
user-friendly approach, this book successfully equips readers with the
key concepts and methods for formulating valid mathematical arguments
that can be used to uncover truths across diverse areas of study such as
mathematics, computer science, and philosophy.
The book develops
the logical tools for writing proofs by guiding readers through both
the established "Hilbert" style of proof writing, as well as the
"equational" style that is emerging in computer science and engineering
applications. Chapters have been organized into the two topical areas of
Boolean logic and predicate logic. Techniques situated outside formal
logic are applied to illustrate and demonstrate significant facts
regarding the power and limitations of logic, such as:
- Logic can certify truths and only truths.
- Logic can certify all absolute truths (completeness theorems of Post and Gödel).
- Logic
cannot certify all "conditional" truths, such as those that are
specific to the Peano arithmetic. Therefore, logic has some serious
limitations, as shown through Gödel's incompleteness theorem.
Numerous
examples and problem sets are provided throughout the text, further
facilitating readers' understanding of the capabilities of logic to
discover mathematical truths. In addition, an extensive appendix
introduces Tarski semantics and proceeds with detailed proofs of
completeness and first incompleteness theorems, while also providing a
self-contained introduction to the theory of computability.
With its thorough scope of coverage and accessible style, Mathematical Logic
is an ideal book for courses in mathematics, computer science, and
philosophy at the upper-undergraduate and graduate levels. It is also a
valuable reference for researchers and practitioners who wish to learn
how to use logic in their everyday work.