Friendly Introduction to Mathematical Logic, A, 1/e
Christopher C. Leary, SUNY at Geneseo
Coming December, 1999 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 240 pp.
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Introduction to Logic-Philosophy
For a one-quarter/one-semester, junior/senior-level course
in Mathematical Logic.
With the idea that mathematical logic is absolutely central
to mathematics, this tightly focused, elementary text discusses
concepts that are used by mathematicians in every branch of the subjecta
subject with increasing applications and intrinsic interest. It features
an inviting writing style and a mathematical approach with precise
statements of theorems and correct proofs. Students are introduced
to the main results of mathematical logicresults that are central
to the understanding of mathematics as a whole.
A focus on core topics.
Conversational, accessible, and accurate writing style.
- Allows instructors to reasonably cover the central
topics of first order mathematical logic in a single semester or quarter
course. Gives students a smaller, more focused and less expensive
Early coverage of First Order Logic rather than Propositional
LogicAssumes that most students have an intuitive understanding
of Propositional Logic.
- Invites students to explore the subject. Also makes for
a much less intimidating environment that encourages students to read
and use the book as they work through the course.
A mathematical approachWith precise statements
of theorems and correct proofs. An in-depth knowledge of computer
science is not assumed, although the applications to computer science
- Provides more time for students to digest the
material. Gives instructors more time later in the course to
spend on more abstract and difficult topics.
Coverage of major theoremse.g., Godel's Completeness
Theorem, the Incompleteness Results of Godel and Rosser, the Compactness
Theorem, and the Lowenheim-Skolem Theorem.
- Exposes students to the core of the field as they work
through challenging and technical results.
Helpful asidesHighlight a particular point, provide
a pause, restate difficult points, or emphasize important topics that
may get lost along the way.
- Introduces students to the strength and power of mathematics
as well as its limitations.
Exercises and problems of varying difficulty.
- Give students another view of a topic.
- Provide students with the opportunity to use and apply
the material learned.
1. Structures and Languages.
Naïvely. Languages. Terms and Formulas. Induction. Sentences.
Structures. Truth in a Structure. Substitutions and Substitutability.
Logical Implication. Summing Up, Looking Ahead.
Naïvely. Deductions. The Logical Axioms. Rules of Inference.
Soundness. Two Technical Lemmas. Properties of our Deductive System.
Non-Logical Axioms. Summing Up, Looking Ahead.
3. Completeness and Compactness.
Naïvely. Completeness. Compactness. The Löwenheim-Skolem
Theorems. Summing Up, Looking Ahead.
Introduction. Language, Structure, Axioms. Recursive Sets,
Recursive Functions. Recursive Sets and Computer Programs. CodingNaïvely.
Coding Is Recursive. Gödel Numbering. Gödel Numbers and N.
NUM and SUB Are Recursive. Definitions by Recursion Are Recursive.
The Collection of Axioms Is Recursive. Coding Deductions. Summing
Up, Looking Ahead. Tables of D-Definitions.
5. The Incompleteness Theorems.
Introduction. The Self-Reference Lemma. The First Incompleteness
Theorem. Extensions and Refinements. The Second Incompleteness Theorem.
Another Explanation of the Second Incompleteness Theorem. Summing
Up, Looking Ahead.
Appendix: Set Theory.