Gödel's Proof

I once taught this book to a grad class. That was the only time in my teaching career that I got a standing O. Excellent, scholarly work. Belongs in the library of every computer scientist or mathematician,

This short book shows Godel's proof of the incompleteness of axiomatic systems that may be consistent. It reads easily - until it doesn't. It's very, very helpful to feel comfortable with "~p v q" to understand the ingenious proof based on numbers, but the first 45 pages plus introduction are understandable to just about everyone.The text may be too short to give non-fanatics much insight into what the fuss is about anyway. The book's summary is a mere four pages.Let me add this. Immanuel Kant was important back in the late 1700's. Really important. He showed how mental math had real meaning in the material world and that logic has remained fixed for millennia. Kant's statements marked the end of classicism. In the 1800's, Gauss, Riemann and others showed how math does not AUTOMATICALLY (a priori) describe the real world (synthetic), and Boole developed symbolic logic (extreme a priori), that seemed to subsume math until the Godel showed that this was a sterile end also. In short, math became extremely separated from reality and Godel then detached it from being a "simple" mental construction.The question remains what does math and other mental systems have to do with the real world. People interpret the proof differently. This book only slightly helps resolve that problem. The early twentieth century gave three big ideas that may be very limited or very pervasive in reality. It's not really clear: * Relativity Theory (Einstein) * Indeterminacy Principle (Heisenberg) * Incompleteness Theorem (Godel)To get into a modern proof, that is nothing like the simple stuff taught in high school, read this book. To get into Godel, read Goldstein.

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've read a few walkthrough's on Godel's work, but this is the classic and undisputed best work for the non-doctoral math fan. It took me a while, and a couple of re-readings of some points, but I can now say that I somewhat understand and definitely appreciate the foundation shaking work of Kurt Godel. Hoftstaeder's annotations add considerably to the discussion.

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.

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

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.

Um texto clássico sobre o famoso Lema de Gödel, editado pela Imprensa Universitária da Universidade de Nova Iorque. Trata de forma didática sobre as publicações feitas por Kurt Gödel na década de 1930.

Si vous vous intéressez au mathématiques et à la logique, la lecture de cet ouvrage, qui présente de manière (relativement) simple la démonstration du théorème d'incomplétude de Gödel, vous apportera beaucoup de satisfaction.

Well written