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The first edition of Herbert B. The author died in , but his webpages live on, including one with his own comments on his second edition. A final chapter covers second-order logic and some other matters. A Mathematical Introduction to Logic eventually became part of a logical trilogy, with the publication of the wonderfully lucid Elements of Set Theory and Computability Theory The first volume, by contrast, is not the most approachable first pass through its material.
Some might think this chapter to be slightly odd. For the usual motivation for separating off propositional logic and giving it an extended treatment at the beginning of a book at this level is that this enables us to introduce and contrast the key ideas of semantic entailment and of provability in a formal deductive system, and then explain strategies for soundness and completeness proofs, all in a helpfully simple and uncluttered initial framework. So what does happen in this chapter?
We also get an exploration which can be postponed of the idea of proofs by induction and the Recursion Theorem, and based on these we get proper proofs of unique readability and the uniqueness of the extension of a valuation of atoms to a valuation of a set of sentences containing them perhaps not the most inviting things for a beginner to be pausing long over.
We get a direct proof of compactness. And we get a first look at the ideas of effectiveness and computability. The core Ch. It starts with an account of first-order languages, and then there is a lengthy treatment of the idea of truth in a structure. This is pretty clearly done and mathematicians should be able to cope quite well but does Enderton forget his officially intended audience on p. Enderton then at last introduces a deductive proof system pages into the book.
The chapter ends with a little model theory — compactness, the LS theorems, interpretations between theorems — all rather briskly done, and there is an application to the construction of infinitesimals in non-standard analysis which is surely going to be too compressed for a first encounter with the ideas. The discussion then turns to the notions of definability and representability.
We are taken through a long catalogue of functions and relations representable in Robinson-Arithmetic-with-exponentiation, including functions for encoding and decoding sequences. As is revealed by that quick description there really is a lot in Ch. To be sure, the material here is not mathematically difficult in itself indeed it is one of the delights of this area that the initial Big Results come so quickly.
However, I do doubt that such an action-packed presentation is the best way to first meet this material. It would, however, make for splendid revision-consolidation-extension reading after tackling e. The final Ch. This goes very briskly at the outset.
However, if you have already read an entry-level presentation of first order logic e. And if you have already read an entry-level presentation on incompleteness e. Your email address will not be published. Save my name, email, and website in this browser for the next time I comment. Logic Matters : Book Notes. Skip to content.
A theory with successor and the order relation also shown to admit elimination of quantifiers and to be complete. Presburger arithmetic shown to be decidable by a quantifier elimination procedure, and shown not to define multiplication Robinson Arithmetic with exponentiation.
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A Mathematical Introduction to Logic
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Enderton (2001) A Mathematical Introduction to Logic with Solutions