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Dislike the textbook. Not very helpful ways to approach the certain problems. There needs to be more examples in the book, so future students are able to learn better to solve problems. Never go for this textbook. Instead, go for the different logic textbook, like Mendelson's textbook, if you are taking Math Logic course.
Review: A Mathematical Introduction to LogicUser Review - John Chilton - Goodreads
I love terse books, but even for me this book is too terse. It could really benefit from additional samples and explanations. Read full review
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Assume atomic formulas axiom group axiomatizable theory binary relation Boolean function cardinality Cn A E compactness theorem computable consistent constant symbols COROLLARY countable decidable deduction definable deﬁned definition effectively enumerable elementarily equivalent example Exercise expression fact ﬁrst first-order first-order logic formal language function f function symbol G¨odel numbers Hence infinite isomorphic lemma logical axioms many-sorted modus ponens n-place function natural numbers notation number theory obtained occur free one-to-one parameters partial function PrbT PROOF quantifier-free quantifiers real numbers recursive function recursive partial function recursively enumerable register machine representable in Cn satisfiable satisﬁes second-order second-order logic Section 3.5 sentence symbols sentential logic set of sentences set theory Show soundness theorem structure suppose translated true truth assignment two-place predicate symbol universe valid variable Z-chains