Proofs and Algorithms An Introduction to Logic and Computability

From the reviews:

“This work examines when the application of an algorithm can replace the construction of a proof. … focuses on establishing that provability is undecidable in predicate logic (Church’s theorem). The text generally consists of propositions followed by proofs, with commentary, examples, and exercises interspersed. … The book would be of interest to those with adequate background. Summing Up: Recommended. Graduate students and above.” (J. R. Burke, Choice, Vol. 49 (1), September, 2011)

“Mathematical logic is a challenging subject for many students. … this book, with its focus on the nature of proofs and algorithms and their relationship, appears to be targeted precisely for such an audience and should appeal to computer scientists and philosophers … . this book remains an introductory book on mathematical logic suited for a beginning graduate course in logic. … Its conciseness makes it well suited for a one-semester graduate course.” (Burkhard Englert, ACM Computing Reviews, February, 2012)

From the reviews:
"This work examines when the application of an algorithm can replace the construction of a proof. ... focuses on establishing that provability is undecidable in predicate logic (Church's theorem). The text generally consists of propositions followed by proofs, with commentary, examples, and exercises interspersed. ... The book would be of interest to those with adequate background. Summing Up: Recommended. Graduate students and above." (J. R. Burke, Choice, Vol. 49 (1), September, 2011)
"Mathematical logic is a challenging subject for many students. ... this book, with its focus on the nature of proofs and algorithms and their relationship, appears to be targeted precisely for such an audience and should appeal to computer scientists and philosophers ... . this book remains an introductory book on mathematical logic suited for a beginning graduate course in logic. ... Its conciseness makes it well suited for a one-semester graduate course." (Burkhard Englert, ACM Computing Reviews, February, 2012)