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Kurt Godel

B. Meltzer

On Formally Undecidable Propositions of Principia Mathematica and Related Systems Paperback – April 1 1992

by Kurt Gödel (Author)

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In 1931, a young Austrian mathematician published an epoch-making paper containing one of the most revolutionary ideas in logic since Aristotle. Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.
The present volume reprints the first English translation of Giidel's far-reaching work. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. B. Braithwaite (Cambridge University}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument.
This Dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics. Translated by B. Meltzer, University of Edinburgh. Preface. Introduction by R. B. Braithwaite.

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Publisher ‏ : ‎ Dover Publications; Revised ed. edition (April 1 1992)
Language ‏ : ‎ English
Paperback ‏ : ‎ 80 pages
ISBN-10 ‏ : ‎ 0486669807
ISBN-13 ‏ : ‎ 978-0486669809
Item weight ‏ : ‎ 135 g
Dimensions ‏ : ‎ 13.34 x 1.27 x 20.32 cm

Best Sellers Rank: #37,335 in Books (See Top 100 in Books)
- #22 in Mathematical Logic
- #78 in Pure Mathematics Books
- #5,067 in Textbooks

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Ben Rothke

One of the most influential treatises of all time

Reviewed in Canada on December 12, 2000

This is indeed one of the most influential treatises of all time. In a nutshell, Godel's Undecidability Theorem proved that within a formal system, there are questions that exist which are neither provable nor disprovable on the basis of the axioms that define the system. He also showed that in formal system, a decidability of all questions is required. Given that, there will be contradictory and unfinished statements. This makes up his Incompleteness Theorem.
With these two theorems, Godel showed that there are problems that simply can't be solved by any set of rules or procedures.
This disproved a common belief at the time (primarily via Whitehead & Russell in Principia Mathematica) that mathematics could be completely integrated into a single foundation of logic.
For those that had attempted to make a deity out of logic and mathematics, Godel simply nailed their coffin closed. Mathematics indeed has its limits.
While I found the underlying logic and proofs often perplexing, the importance of the book can't be overemphasized.

2 people found this helpful

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John C. Landon

From the horse's mouth, 'le text'

Reviewed in Canada on August 2, 2003

Speaking not as a math specialist but one disposed to read a number of the popular explications of Godel's famous proof I can say that it was Godel's original text that did it for me. The reason is that it is the proof and not a lot of verbiage about the proof. And it is short and sweet. One problem is that the more common Turing Machine approach is actually 'easier', where Godel's approach is that of recursive functions which are more obscure, or at least less often discussed. If you can sort of glare at the recursive function issue and proceed with the basics of the proof it will stand out suddenly better than many of the popularizations. At least give it a try.

5 people found this helpful

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Juan Pablo Pira

Read the masters!

Reviewed in Canada on October 31, 2001

THE proof as Goedel wrote it (plus typos). I have seen modern proofs of this theorem which are much easier to follow (as an example, a Mir book on mathematical logic by a Russian mathematician whose name I cannot recall), but this one is the REAL thing.
Modern proofs can be much clearer, but the original always has an added value. The writing style is not the best, but by reading this version you get a clearer idea of how Goedel came up with his theorem and the many difficulties he faced. Remember, by the time most of us read or heard about this for the first time, mathematical logic had advanced quite a few decades.

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dan kandahl

Milarepa attained enlightenment through sheer force of will

Reviewed in Canada on December 24, 2000

Anyone who wants to trace this proof is free to do so. Though the formal logic can be formidable, and must be learned before tackling the proof, only the basic structure is necessary and it is not difficult to learn. It is also necessary to know a little about prime numbers, specifically that every composite number can be decomposed into some unique group of prime factors. Otherwise, all the technical aspects of the proof (barring the conclusion of theorem VI and a bit of the recursion) can be perfectly understood by someone outside of the world of formal mathmatics.
The proof itself is meant for a professional mathmatician. If you are interested and willing this will not dissuade you. To say Godel was not a master of exposition is misleading for he is ,if nothing else, just that. I have heard working through the proof compared to a mystical experience and the proof itself to a symphony. It is truly beautiful to even the mere math enthusiast. Godel is not, however, a college professor and does not wish to explain what need not be explained. This will not be of much consolation when he prefaces a statement with, "of course," for the twentieth time and you have no idea what he is talking about. But if you are not afraid to go ahead when you have tried and failed to understand, and are not afraid to return when you have gained some small piece of the puzzle and try again, everything will come clear. This is the original. All the commentaries are great, and some are even helpful before you get to the conclusion, but they are not the proof and should not be taken as a substitute. They do not suffice the way a generic drug does. There is no way to understand the full scope of the proof if you are not willing to immerse yourself in it and the language it uses. Everything in it is self-referential, you miss the reference when you skip the proof.
Don't worry if it seems to be going nowhere, because you'll get there soon enough yourself and it turns out everything matters(although nothing has meaning). If you want you can skip everything after theorem VI is proved up to the beginning of theorem XI. Then you will have everything you read about.
Perhaps, if you can, you should get a couple other people to work through it with; different perspectives make all the difference, even in math. It's nice to have someone to share your frustration with and sometimes to have it relieved. Plus the delight of watching the theorem build like a wave and crash down upon itself is best shared with others.
Last, and particularly directed toward anyone at DOVER: The proof as you have printed it is horribly mangled! There are countless misprints (more than fifty). This is bad enough in prose but absolutely disgusting in a math text. The second to last line of theorem VI (the proof of undecidability) has a misprint to the effect of changing a negative statement into a positive one. The proof is hard enough as it is. There is a much better translation (without any misprints) in "Frege and Godel" and "From Frege to Godel," which are sadly out of print but may be found at a library. The DOVER is cheap and it reads that way.
If you're interested, do it, and don't worry about it being too hard. You will realize the technical aspects are almost all quite easy when you plow through them. So wade in and enjoy!

2 people found this helpful

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Amazon Customer

<I>The</I> classic in Godel's Undecidablity Theorem...

Reviewed in Canada on April 17, 1999

Wow! If you don't already own this, and you know anything about advanced number theory, buy it now! You should also probably pick up a deciphering guide, as well, because the symbolism is quite intense in places. A true classic with a deserved place in any 'Mathmatical Recreationalists' shelf.

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John Harbord

If you are interested in the origins of formal logic ...

Reviewed in the United Kingdom on August 19, 2016

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If you are interested in the origins of formal logic this is a key text, but it is not the easiest of reads. It deals with some very complex formulations and the notion of logical substitution from one framework [the logical] to another [the mathematical] takes a lot of rereading to get clear in your head.

One person found this helpful

Nick

Excellent Product - Good Value

Reviewed in the United Kingdom on December 15, 2018

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Excellent Product - Good Value

mjheath

Five Stars

Reviewed in the United Kingdom on April 26, 2017

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excellent

Frank

“Leed a los maestros” —Niels Henrik Abel

Reviewed in Mexico on December 23, 2016

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La argumentación de Gödel en su famoso teorema de incompletitud es sencilla —al igual que en la prueba de la imposibilidad de demostrar la consistencia al interior de cualquier sistema formal que contenga a la aritmética—, lo verdaderamente escabroso de este artículo, es la parte sobre funciones y relaciones recursivas: el lector debe revisar concienzudamente todas las definiciones que presenta el autor ¡aun antes de llegar al teorema de incompletitud citado!

Sin embargo, leer la fuente primaria siempre es más gratificante para cualquier estudiante de matemáticas o ciencias de la computación: aprenderán como nunca.

One person found this helpful

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Manolo Sijé

Genio total el Gödel

Reviewed in Mexico on May 4, 2018

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El texto que cimbró las matemáticas para siempre. Es complicado para los que no estudiamos una licenciatura en matemáticas pero con algunos conocimientos se puede entender. Gran genialidad de Gödel

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