On Formally Undecidable Propositions of Principia Mathematica and Related Systems (Dover Books on Mathematics)

Theorem 15 is missing a bar over the left side of the formula. The bar is also missing in the second half of the undecideability proof for proposition VI. This makes the argument incoherent and super confusing to follow! Obviously, Meltzer didn't understand the argument when he made the translation, otherwise he would have caught such a major ommission. Fortunately, Braithwaite's introduction has a formally identical but slightly simplified version of Godel's argument. From that it is possible to spot the typo on page 59. You can also verify the missing bars from Godel's original German article.Upside, spotting the error is a great exercise in making sure you understand Godel's argument!

As indicated in two other reviews of mine here, my comprehension of Goedel's work is opposite to the general one. My marking three stars regardless for this book is motivated by his extensive influence, but also by his fair admission later in life that his thesis could amount to hocus-pocus.Indeed, I see it as one of the prominent mistakes in logical history, and I shall endeavor to explain as best I can. It should suffice to consider his Section 1, an outline of his proposed proof.Although that section is brief, it already foreshadows an oppressingly complex logical symbolism for statements that in my view can be made much clearer using ordinary language. The symbolism, to be sure, is intended to establish a formal language, whose meaning is to be decided separately. This will be seen one of the problems.For now, let me give the principal statement Goedel contended to be true but undecidable (neither provable nor disprovable):"This statement is unprovable."He symbolized it (p.40) as: "~Bew[R(n);n]". Font limitations made me slightly change it; the tilde "~" means "not", "Bew" is a German abbreviation for "provable", and within brackets "R(n)" says "Statement n" and "n" stands for the full statement.Goedel proceeds: "...supposing...~Bew[R(n);n] were provable, it would also be correct; but that means...that...~Bew[R(n);n] would hold good, in contradiction to our initial assumption. If, on the contrary, the negation of ~Bew[R(n);n] were provable, then [its provability] would hold good. ~Bew[R(n);n] would thus be provable [in contradiction to the unprovability it states], which again is impossible." (I corrected some errors within brackets.)So since both ~Bew[R(n);n] and its negation are unprovable, it is undecidable, and Goedel continues (p.41): "...it follows at once that ~Bew[R(n);n] is correct, since...certainly unprovable (because undecidable). So the proposition which is undecidable in the system...turns out to be decided by metamathematical considerations.""Metamathematical", in excusing the contradiction, designates the above formal system void of assigned meaning, whereas the statement discussed is to have meaning. Not quite a lucid argument. Overlooked, furthermore, is a contradiction using the same reasoning as in the preceding.Coupled with the preceding finding that ~Bew[R(n);n] CANNOT be proved unprovable (for if so proved, it would be contradicted), can in contradiction be that it CAN be proved unprovable. For if it were instead provable, it would again be contradicted. The statement in question thus becomes a paradox, rather than true, similar to paradoxes like the "liar", mentioned by Goedel (p.40).He strangely adds to it the footnote: "Every epistemological [paradox] can likewise be used for a similar undecidability proof." The "liar", however, is, like all paradoxes, not a true statement, as required, but one harboring a contradiction. (I deal in my book with, and offer solutions to, paradoxes more fully, including Goedel's resulting one, without naming him.)There occurs, further, another huge blunder in the alleged proof. The undecidability is said to apply to some of mathematics; in the above formula, ~Bew[R(n);n], the "n" refers to a number, with this justification by Goedel (p.38): "For metamathematical purposes it is naturally immaterial what objects are taken as basic signs, and we propose to use natural numbers for them." Adding (p.39): "Metamathematical concepts and propositions thereby become concepts and propositions concerning natural numbers..."How so? In one breath he proposes using natural numbers as immaterial signs, and in the next breath the material concerns natural numbers!The fallaciousness can indeed be made clear by considering our statement, ~Bew[R(n);n], interpreted as "This statement is unprovable." As noted, in ~Bew[R(n);n] the "n", now a number, is to name the whole statement, inside which it is also used in "Statement n..." But whether or not the statement is named by a number, the point is that the name must refer to the intended content of the statement to correspondingly function, not to the usual number possibly represented. Therefore the statement, or anything else similarly used, has nothing to do with numbers, or mathematics generally.

I wish I had been exposed years ago to the philosophy of mathematics and the inseparableness of mathematics and logic. Seems obvious now. I could blame the education I received and the focus it placed on how instead of why. But perhaps that’s too easy an excuse and the why was always there and I was too immature to see it. Either way, it’s a good read...

This book is quite short, but it is also very deep. Kurt Gödel was a mathematician back in the 1930s that had an idea. He grew up during a time where it was thought that everything could be explained through mathematics and that mathematics itself would be "complete." However, Kurt Gödel comes up one fine day in 1931 or so and publishes this little paper explaining that there are ideas that can't be expressed in the language of mathematics. Using the language developed by Bertrand Russell and Alfred North Whitehead, Kurt Gödel establishes basic math and then proceeds to tear it down. A tour de force of logic.

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.

I had to read it a few times but I thought it was worth reading.

This is not a simple read for non-mathematicians, but it is outstanding due to the extended explanatory introduction for others with backgrounds in natural sciences, logical, or philosophical matters.

In my humble opinion one of the most important discoveries of the 20th century. In fact it means that to describe the world with a consistent and complete logical theory, we need an infinit number of hypothesis!

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.

ゲーデルの証明を読むつもりは毛頭なかった。そもそも理解出来る頭だったら密林にジットリ潜伏してしみったれたレビューなぞ量産していない。ゲーデルの証明だけならば現在はネットでまるまるゲットが可能…ま、アテクシには関係ないが。アテクシが本書を買ったのはR.B. ブレイスウェイス教授が序文を寄せてると発見して「へー！」となったから。ブレイスウェイス教授はゲーデルと同時代人で科学哲学と宗教哲学専門のイギリス人哲学者。教授の『An Empiricist's View of the Nature of Religious Belief』は現在も引用されるくらい一部で有名。ブレイスウェイス教授がどんな解説をしてくれるのかと序文だけ読んでみた次第。うわ、何が何だか分からない。序文が本書の五割を占めているというのに。歴史的アプローチとしても不親切で、三十年代の西欧スーパーエリート世界に「数学者と自然科学者に『不可知』はない！」とヒルベルトに高笑いの宣言をさせた（ホントに高笑いしたらしい）「驚異的な知的楽観主義」の空気があったことには触れねばならないのでは、先生。それでゲーデルの証明を見た肝心のラッセルが「意味分からん」言い、「こんな分野から足洗って良かったぜ」となり、ウィトゲンシュタインが「これは数学ではない」言ってゲーデルに今生では消えない怒りを残したとかハイドラマがあるんだが、ここらへんはブレイスウェイス先生の執筆年代が1960年らしいんで色んなネタが揃ってなかったのだろうが、それにしても、シロートを誘う要素が皆無の序文。バカで申し訳ないが、教授の解説文から得た僅かな知識は、足し算だけの自然数体系の完全性は既に証明されており、ゲーデルが「不完全性定理」でなさったのは「掛け算を加えるとひっくり返るんだよね」ってことだと？うーん、この理解でオッケ？取り敢えずここから得た物凄く大枠の知見があるとすれば、本書はシロート向けに書かれたゲーデル解説本としてはごく初期のものらしいが、初期の解説は恐ろしく不親切だったとかか？おそらく呻吟しつつ本書を読んだ優秀な学徒の皆さんが後年「俺の方が上手く解説出来る」と自負して新たなゲーデル解説本を書いたりしたんだろうなと。現在はシロートにも取っ掛かり部分だけは提供してくれるような読み易く興趣を喚起して下さるゲーデル解説が見つかる訳だが、六十年代にはシロートが立ち入れるようなドアはなかったということか。時代の変遷とネット誘導の知の民主化といにしえのインテリ書籍文化についてちょいと思い馳せることは出来たんで、三つ星。

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

This is a copy bought from amazon.in. The Indian copy printing is slanted, it cuts off the title itself on the cover page. And then inside, it cuts off number indices for referenced equations. When I tried to return it on Amazon, it offered me only a replacement copy. I have no reason to believe that the replacement copy is going to be better printed than the one I got, so I'm choosing to leave this review instead.This book is a classic and I was intending to keep the copy for perpetuity but the shoddy quality of it does a great injustice to what the book represents to me.

Testo molto compatto, chiede forti conoscenze già acquisite e grande concentrazione nella lettura. Ma possibile che nessuno lo traduca in italiano? Consigliato