Principles of Mathematical Logic by David Hilbert (PDF)

Description

David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. It lays the groundwork for his later work with Bernays. This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors’ original formulation of Gödel’s completeness proof for the predicate calculus has been updated. In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert’s important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.

User’s Reviews

Editorial Reviews: Review This book is unmistakably a mathematician’s book, but it goes far beyond the limits of mathematics and makes available to everyone interested in logic one of the most permanent results of the study of the foundations of mathematics, the satisfactory symbolic treatment of the basic relations that play a part in deductive reasoning. –Cambridge University Press

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book by Wilhelm Friedrich Ackermann (1896-1962) and his PhD advisor/supervisor David Hilbert (1862-1943), translated (and corrected) in 1950 from the 1938 second edition of their 1928 “Grundzüge der theoretischen Logik”, contains a concise and tidy propositional calculus and predicate calculus. As the editor explains in the editor’s preface, various gaps and errors in the completeness proofs in the original German-language book have been remedied in this English translation.This book arises from Hilbert’s lectures, written up by Paul Isaac Bernays (1888-1977) and then arranged and presented by Ackermann in 1928, some 3 years after he obtained his doctorate. According to wikipedia, this book “contained the first exposition ever of first-order logic”. However, this translation is clearly not a historical document of the state of mathematical logic in 1928 because of the revisions made in 1938 and the corrections and additions made in 1950. Ackermann’s 1938 preface contains the following interesting comment.”It was possible to shorten the fourth chapter inasmuch as it was no longer necessary to go into Whitehead and Russell’s ramified theory of types, since it seems to have been generally abandoned.”Oddly enough, many mathematical logic authors continued to follow the Whitehead/Russell approach long after it was “generally abandoned”.Since this book is not an accurate reflection of the original 1928 first edition, it must be judged on its own merits. The propositional calculus, here called “sentential calculus”, contains many observations which I have marked in the margin (in pencil) as “good!”. It is a clear presentation, well motivated. So this is a good preparation before reading more modern presentations which tend to be dry and formal. There is an interesting discussion on pages 27-30 of the pros and cons of various choices of axioms for propositional calculus. Then they show how to write proofs and discuss completeness and consistency.Chapter II presents first-order logic, here called “calculus of classes” or “monadic predicate calculus”. This delves into the numerous modes of Aristotelian logic. Chapter III is a systematic presentation of a predicate calculus in a quite modern fashion, including substitution rules. The principal inference rules are on page 70, which we would nowadays call “universal introduction” and “existential introduction” rules. Then some examples of the application of this predicate calculus are given. It’s a bit tricky because you need to carefully interpret some subtle distinctions between bound and free variables.The rest of Chapter III is taken up with questions of independence, consistency, completeness, and the decision problem. Chapter IV presents second-order predicate calculus, numbers, and sets.So all in all, this is a good book to read for motivation and background before reading a more modern presentation.

⭐A founding work on modern mathematical logic by one of the best minds of the modern era.

⭐Good basic reference to my Mathematical Logic library

⭐Classical and well-organized.

⭐I learned logic from this book, and so I am very fond of it. As a presentation of what was known at that time, it cannot be beaten. The only problem with it is that a lot more has been discovered, so a modern treatment of the subject is better for the beginner who wants to be properly informed. For this I suggest the books by Copi or even Kleene. But if you don’t care about modernity, or have an interest in the way things used to be done, I strongly recommend this book. Also, I might mention that Hilbert’s “Geometry and the Imagination” is good even for the modern mathematician.

⭐Fast delivery and good quality for a collector’s item.

⭐good

⭐L’intérêt de ce genre d’ouvrage est aujourd’hui plus de l’ordre de l’histoire des idées que de l’ordre de la référence technique. Mais si c’est ça qui vous intéresse, c’est vraiment plaisant de tenir entre ses mains un des premiers exposés structurés sur le calcul des prédicats.

⭐

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