Gödel's Proof

If Godel proved that no sufficiently complex system, i.e one that is capable of arithmetic, can prove its own consistency or if you assume the system is consistent there will always exist (infinitely many) true statements that cannot be deduced from its axioms, in what system did he prove it in? Is that system consistent? In what sense is the Godel statement true if not by proof? You'll have hundreds of questions popping in your mind every few minutes, and this short book does a very good job of tackling most of them. Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a "model" of the axioms. The hard part of it is to do this by avoiding the "circular hell". Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like "Set of all elements which do not belong to the set". Godel's proof tries hard to avoid more complicated paradoxes like this : Let p = "Is a sum of two primes" be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = "A number which satisfies the property described by its reverse ASCII representation". Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel's proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it. 1. Construct a meta-mathemtical formula G that represents "The formula G is not demonstratable". ( Like Richardian ) 2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian) 3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ). 4. Finally, Godel showed that the meta-mathematical statement "if `Arithmetic is consistent' then G follows" is demonstrable. Then he showed that "Arithmetic is consistent" is not demonstrable. It took me a while to pour over the details, back and forth between pages. I'm still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key. When I first came across Godel's theorem, I was horrified, dismayed, disillusioned and above all confounded - how can successively applying axioms over and over not fill up the space of all theorems? Now, I'm slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them - one as more fundamental than the other. I'm very slowly beginning to understand why Godel's discovery was a shock to me. You see, I'm good at rolling with it while I'm working away, but deep down, I don't believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify. Kindle Edition - I would advise against the Kindle edition as reading anything with a decent bit of math content isn't a very linear process. Turning pages, referring to footnotes and figures isn't easy on the Kindle.

I remember my excitement when I read the first edition of this little gem back in 1999 (actually it was its Turkish translation). Being a young student of mathematics, it was impossible to resist reading a popular and clear account of maybe the most important theorem related to the fundamentals of axiomatic systems. After that came Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid which introduced more questions related to symbolic logical reasoning, artificial intelligence, cognitive science, and the consequences of Gödel's work in those ares. With that background and ten years after the second edition, it was truly an exciting second reading, a refresher that was both fun and putting lots of things into perspective. Hofstadter's foreword to this edition is a delight to read and ponder upon. On the other hand, I don't think this is a point strong enough to persuade most of the people who own the first edition anyway. But if you don't have the first edition and want a concise and clear explanation of what Gödel's work is all about then this book is definitely for you.

Bought this to get a little more insight into the philosophy behind Godel's proof, and it's exactly what I wanted. It's helpful to have read a more formal account, and be pretty well versed in the rules of inference in first and second order logic, but only for the purposes of coming to some deep insights on the development of computability theory. Very readable to a layperson who is interested in mathematical history, logic, computation, and philosophy of science. Also, nice sized font, so no squinting (paperback version).

I have been a big fan of the issue of formal mathematics and the theory of computation but I always missed a full grasping of the goedel theorem. The book presents the line along which Goedel moved: mostly formal systems and the most interesting issue of "calculating grammar" by powers of primes: an outstanding example, to me, was how using that instrument one could know whether a sentence was introduced by a "not" or not (in the case simply by checking if the figure expressing the formula was even or odd). In that way the full system of mathematics turned into a sort of a computer program that of course could not calculate every function. For that matter I came to wonder why the demonstration of goedel theorem could not be carried out simply by showing a formal system has the same power as a universal Turing machine and thus transferring to it the (much easier) results obtained on that issue - like for example to problem of the stop or the one of finding any semantic information in a program without actually executing it.

Although some details of the proof are missing, and some theorems just given without further details, the most fundamental aspects are explained in the text, and some details given in the appendix. With the "vocabulary", grammar and logic principles taught by the book it is even sometimes possible to find the deduction ("proof") by just experimenting with what you have learned. Additional literature is not necessary, but after reading this book highly recommended, as it is a good foundation to seek further understanding. For those who like think about the connection to other discplines of science, "Goedel, Escher, Bach" is also recommended - although you may find some redundancy; which might also mirror the special nature of the given topic.

I bought this based on praise from Doug Hofstadter. It helped him understand Godel's theorem. If you're already familiar with the intuition of Godel's theorem and not able or willing to actually read his proof, this book is a great middle ground between the two levels of understanding. I also discovered it is an excellent sleep aid. Through use of this text I am able to fall asleep within minutes at virtually any time of day.

Los autores de este libro lograron desarrollar los dos teoremas de incompletitud de Gödel con una simpleza tal que cualquiera con el suficiente interés en el tema los puede entender. Los aportes de Gödel a las matemáticas y a la lógica son consideradas como de los más importantes en la historia de la humanidad, así que considero, al menos para los universitarios que estudian materias de matemáticas, que es una lectura que puede rendir muchos frutos.

A must read for anyone practicing mathematics or computer science. It changes the way you think about the field and it's important to recognize

Great well paced tutorial however it will exercise your logic skills. This book is worth the effort though. Fast delivery and in good condition