Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing (in theory of computability, particularly in the theory of recursive functions, there is a fundamental theorem stating that there are semi-decidable sets (sets which can be effectively generated), that are not fully decidable. Moreover, there are a couple of areas there there is really a bit of too much hand-waving (for example, I would have loved a much more detailed treatment of the critical Correspondence Lemma, and of fundamental concepts such as that of model of a theory), but I must say that this book achieves the remarkable result of condensing the core of Godel's theorems in a succinct but very readable, approachable, meaningful and informative way, even including important technical details that allow the user to get a better understanding than what normally offered by most other semi-technical books on the subject. Moreover, while there are several global properties that a formal system may have (such as completeness, consistency, and the existence of an effective axiomatization), the incompleteness theorems only show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties at the same time. In particular, it is important to highlight that, far from stating that every system is or will be found inconsistent, Godels incompleteness theorems merely place limits on what consistent systems can prove (including the intriguing matter of their own consistency, of course). - last but not least: the ultimate confirmation that formal systems such as ZFC provide a consistent and normally sufficiently powerful formal system able to generate adequate mathematical structures for the satisfactory representation of the patterns ultimately characterizing the physical world, is available in the very experimental results of the physical sciences, as famously condensed by Wigner in his treatise about the unreasonable effectiveness of mathematics. And we should also bear in mind that, realistically speaking, it is safe to say that 99% of contemporary mathematics follows from a small, stable subset of ZFC Considering all the above, it is my strong view that, rather than proving any supposed epistemological limitations of mathematics, Godel's theorems actually clearly highlight the infinite, inexhaustible richness of the patterns, structures and truths that mathematics can offer structures and patterns that are also reflected in, and can faithfully represent, the inner core of physical reality.
A true statement whose unprovability resulted precisely from its truth! (i) Godel constructed a formula G of PM that represents the meta-mathematical statement: The formula G is not demonstrable using the rules of PM. However, if a formula and its own negation are both formally demonstrable, then PM is not consistent. (iii) Godel showed that though G is not formally demonstrable, it nevertheless is a true arithmetical formula.
So, in a word, the sentences, even though correctly built, will be INDEPENDENT OF the set of axioms.
As for its readability, math forever has a big problem in that it's *designed* to simplify reason, so when it itself becomes very complex, it can be difficult to impart. By the same reasoning, if you try to simplify a very complex proof, leaving out a lot of the messy details while trying to explain the rest in plain English, you're necessarily going to miss a lot and there will be important details that are missing. On the plus side, it's pretty readable for a math book, and the main conclusions are pretty easy to follow.
My thanks to AC for convincing me to take the plunge and purchase this little gem: who'd have thought that one-hundred-and-thirteen pages of mathematical logic could have been so entertainingly informative?
Premièrement un texte de vulgarisation visant à présenter le théorème d'incomplétude de Kurt Gödel, et à expliciter autant que possible pour le tout venant comme moi les grandes étapes du raisonnements, ainsi que les techniques employées par Gödel. C'est pas mal, les notions sont introduites par degré, les grandes étapes sont exposées, ainsi que les techniques employées. Ce genre de démonstration, on ne les comprends correctement qu'à partir du moment où non seulement on parvient à les refaire soi-même, mais en plus on a très bien assimilé les fondements sur lesquels il s'appuient: ici, ma patience ne va pas jusque là, et j'ai plutôt parcouru rapidement le texte que mis ma cervelle à l'alambic. Ce qui le font invoquent le doit à la métaphore, mais on peut aussi à bon droit les soupçonner d'imposture si la notion prise pour image est non pas une chose claire et évidente pour tous, mais un concept difficile qui en impose par son obscurité.
Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmeticunless the proof itself employs rules of inference in certain essential respects different from the Transformation Rules used in deriving theorems within the system. To express what is intended by this latter sentence, one must write: Chicago is tri-syllabic. Godel devised a method of representation such that neither the arithmetical formula corresponding to a certain true meta-mathematical statement about the formula, nor the arithmetical formula corresponding to the denial of the statement, is demonstrable within the calculus. Gödels method of representation also enabled him to construct an arithmetical formula corresponding to the meta-mathematical statement The calculus is consistent and to show that this formula is not demonstrable within the calculus. It follows that the meta-mathematical statement cannot be established unless rules of inference are used that cannot be represented within the calculus, so that, in proving the statement, rules must be employed whose own consistency may be as questionable as the consistency of arithmetic itself.
"Gödel's Proof" was one of the first books in my fall "Learn Math" program. Some areas I thought I'd understood before this book: - What a "formal system" is - What it means for a system to be "sufficiently powerful" - What the hell the phrase "foundational crisis of mathematics" really means; - How mechanical theorem provers work, and why it's a reasonable goal to try and write software that can assist with and verify proofs - What phrases like "theorem", "proof", "demonstration", "decidable", "complete" etc really MEAN when discussing formal systems. I read "The Little Prover" (https://www.goodreads.com/book/show/2...) right before "Gödel's Proof". If you want some practice in using a language like Scheme to play with some of the concepts of formal proof, and try to implement a language like Principia Mathematica, I'd recommend cracking "Little Prover" next.
It ultimately describes the 1931 paper Kurt Godel published in German entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems.