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On Formally Undecidable Propositions of Principia Mathematica and Related Systems

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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.

80 pages, Paperback

First published April 1, 1992

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About the author

Kurt Gödel

51 books187 followers
Kurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

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Displaying 1 - 30 of 33 reviews
Profile Image for Matt.
752 reviews609 followers
November 17, 2019
Kurt Gödel was a genius and his paper is proof of that fact.

Even after reading it twice I cannot say with certainty I understood everything from these 26 pages, but I believe I got the gist of it. The title, in English “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, sounds a little bulky, but the text (the words that surround the formulas) is quite accessible. As for the formulas: They look kind of nice on paper, and they carry a certain kind of nostalgic charm (coming from almost 90 years ago when mathematical formulas looked a lot different than today), but, frankly, I skipped many of those.

To explain what Gödel found, one has to first understand what a formal system is. A formal system is a closed system in which mathematical statements can be proven. Each formal system consists of:
a) a “language” and a set of characters with which well-formed formulas (called “statements”) can be build,
b) a finite (preferably small) set of axioms,
c) a finite set of inference rules that can be used to derive new statements from previously proven ones (or from the axioms).

The axioms themselves are also statements of the system, but they cannot be proven. They are believed to be true, or better yet, the axioms are true by definition. [For instance in the formal system of arithmetic, there are only five of those axioms, the so-called Peano axioms ) which basically describe the set of natural numbers]. It’s important to note that every well-formed statement of the system (=every word of the system’s “language”) is either true or false – it cannot be both, nor can it be neither. This implies that each statement is either true or its negation is true.

If you set up a formal system there are two desirable properties you want it to have:
1. Each true statement is provable, in other words “If S is true, then S is also provable (by only using the system’s axioms and inference rules)”. A system with this property is called complete.
2. Each provable statement is true, in other words “If S is provable, then S is also true”. This means there are no contradictions in the system. Such a system is called consistent.

For a formal system that is complete and consistent, the terms truth and provability are one and the same. Although there are statements, for example in the system of arithmetic (like the famous Goldbach conjecture), that are notoriously hard to prove, mathematicians believed this system to be complete as well as consistent. It would only be a matter of time before someone will come up with a valid prove of those hard statements, and there’s no such thing as ignorabimus in the field of mathematics, as David Hilbert put it in his program on mathematics: “a proof that all true mathematical statements can be proven in the formalism.” That was the state of affairs when Godel entered the stage.

In his paper, published in 1931, Gödel demonstrated that in a formal system of sufficient complexity (like the one on arithmetic), there will be necessarily statements which are true, but cannot be proven within the system. Such a system must therefore be incomplete! What Gödel did, was to assign numbers to statements (in general to each mathematical object of the system), and through this kind of “Gödelisation” (or Gödel-numbering as it is now known) he was able to phrase statements about statements within the system itself. At the core of his proof Gödel constructed a statement (a Gödel-number) S that basically says about itself “I am not provable (within my system).” Since S is a valid statement it must necessarily be either true or false, like any other statement. If S were false: that would mean the opposite of S is true, or, in other words, S would be provable. So there’s a false statement that is provable and this renders the formal system inconsitent. On the other hand if S were true, it would mean that we now have a true statement which is not provable and that makes the system incomplete. In the case of the formal system of arithmetic, Gödel actually constructed a true statement of the above kind, and by that showed that this system must be incomplete! Adding such a statement S to the axioms in order to make the system complete won’t help, because then one could simply start over an construct another statement S' with the same properties. The resulting systems will never be complete.

I know this all sounds pretty out of touch with the real world for most people, but Gödel’s incompleteness theorems (there are two of those, the second one stating that a formal system cannot prove its own consistency) had a huge impact in the field of mathematics and beyond, and I personally like the fact that someone was able to put things into perspective and that there are indeed things which can never be known.
_____________________

PS: The above mentioned Goldbach conjecture is still not proved until today. It states that every even integer number N greater than two can be written as the sum of two prime numbers p and q (like 4=2+2, 6=3+3, 8=3+5, or 444=131+313 etc.), and it occurred to me that, if it would be possible to prove that this conjecture cannot be proven (within its formal system), the conjecture must be true, because otherwise there would only be finitely many examples for the equation N=p+q, and those can surely be formulated within the system of arithmetic. Strange.
_____________________

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Profile Image for Nick Black.
Author 2 books875 followers
November 14, 2008
The greatest achievement in human thought to date, IMHO. Every family ought have a copy on their coffee table. I read this the summer after high school, and have gone back a few times. In less than one hundred pages, that silent madman Gödel will shatter your mind, put it back together with turing tape, and leave you changed forever. Minimal mathematical background is required.
Profile Image for Francisco Barrios.
630 reviews47 followers
December 7, 2016
A few words of advice:

1. If you are NOT a mathematician, please, read (and re-read) thoroughly the introduction by R. B. Braithwaite until you feel comfortable with Gödel’s notation, specially in those cases where it parts from everything you have read or been taught before. It will pay off heavily later on.

2. If you are a mathematician, you can go straight into Gödel’s paper starting on page 37. Pay attention to the definition of class-sign on page 43 —it will come back on page 57!—, and work on each of the 45 functions (and relations) on pages 49-55. It will save you a lot of time when you are reading the proof to the famous undecidability theorem.

3. Formula (15) on page 59 is wrong (in my copy): the antecedent needs to be overlined to denote negation. The same correction needs to be made in this page, line 17.

All these been said... Gödel wrote a fine piece of mathematics which deals with the limits to certain formal systems. He was not trying to demolish or tear down Hilbert’s formalistic standpoint (as some people suggest) and you will become aware of this fact towards the very end of the paper (page 71). In coming up with two beautiful results (the undecidability theorem and the one stating that the consistency of a formal consistent system is unprovable within itself), he displayed an astonishing sharpness and unbelievable insight. You will find both in this book.

Happy readings.
Profile Image for Kunal Sen.
Author 31 books60 followers
February 9, 2015
It was much harder than I thought, and it is best suited for professional mathematicians. For the rest of us Godel's Proof by Nagel is a better option. Still, had to try the original for one of the most audacious ideas humankind can be proud of.
Profile Image for Bonnie.
77 reviews13 followers
Read
May 7, 2025
Sorry if it’s gauche to mark this as “read” on Goodreads.
Profile Image for David Joseph.
100 reviews
September 7, 2012
It does necessarily follow from the valid statement "2+2=4" that some other statement is equally valid, but not necessarily so.

Weird.

One can apply a variety metalogical thinking strategies to deductive sytems and... blah blah blah.

For sure, hard thinkin' leads straight to this proof, but thinking hard alone will not really make it clear.

Also weird.

This is the most difficult text I've ever read. I picked it up thinking I could develop an understanding of it through a careful reading. I didn't realize that a careful reading was going to require the mastery of such a counter-intuitive language.

In the end it required of me, a lot of humility and a bunch of sharp pencils and notebook paper. )Oh ,and flashcards.

And Hofstadter!






This entire review has been hidden because of spoilers.
Profile Image for sologdin.
1,834 reviews833 followers
February 20, 2016
Snarky assholes should present these arguments to their seventh grade algebra teachers: but this is all impossible! why bother? formal axiomatics is undecidable!
Profile Image for Alamir.
10 reviews
August 7, 2015
I read this book in 2011 and four years later I have come to realize it's probably one of the top ten most influential books I've ever read (not that I have a list). Judging by some of the other reviews on here, I'm afraid that readers may feel overwhelmed by his more in-depth equations and may give up on the book. If that's the case, my advice would be to try to first understand his greater argument and then you can always return to his equations to drill down further.
Profile Image for Mohamed Hasn.
67 reviews4 followers
June 15, 2019
بغض النظر اني مفهمتش غير ال intro
وحاجات بسيطة بعدين
الكتاب جامد
Profile Image for Melita Mihaljevic.
50 reviews4 followers
February 27, 2017
A challenging exercise of reading this in English and following the proof. Amazing how significant exposure to even mathematics in a native language makes the difference
Profile Image for William Schram.
2,310 reviews95 followers
April 19, 2015
I thought it was quite interesting, but I don't feel I have the necessary background in math to completely appreciate this work. Gödel makes up his own notations, but follows some standards for mathematical logic. It helped that the book had references to what it was talking about in the book itself though, and it is also extensively footnoted. I though it was a very interesting paper, but like I said, I need a more thorough grounding in logic to fully appreciate it.

I will be reading this again when I can understand it better. Five out of five for turning the establishment on it's head though. Before this paper, mathematicians assumed it was possible to go and explain everything in math. But you can't explain everything in math using math, so there are some things that are just unexplainable. I probably didn't really get that right, but it matters not.
Profile Image for Vladimir.
4 reviews1 follower
July 11, 2012
This is the hardest text that I have ever read in my life. The result is revolutionary, and it is worth all the effort. Because of the conceptual density of the book, I needed to read it many times to get acquainted with the notation and to realize indeed how beautiful the proof is: all concepts introduced are so cleverly used that the proof is minimalistic and inevitable and its main result is compelling in its force and implications.
Profile Image for Kory.
26 reviews
June 3, 2011
Godel gives one of the most interesting proofs I have seen. The proof itself is hard to read and understand, to say the least. However, the consequences of his proof are incredible.
Profile Image for D.H. Bernhardt.
Author 9 books12 followers
November 27, 2018
I first heard of Gödel's theorem from a recorded talk by Alan Watts given at IBM in the late 60's. The theorem piqued my interest because, apart from the impact it has had on mathematics, I was impressed by the profound ontological implications (as Watts so eloquently expressed).

The math in Gödel's paper was well beyond me and I found myself lost in a sea of strange symbols. I take it at the word of the mathemeticians that Gödel's set up works and does in fact prove the limitations of principal mathematics. I frequently found myself on Youtube looking for more digestable forms of the thereom to little avail.

There was one video, however, that expressed the theorem (thankfully) in lingusitic terms; imagine you have a sheet of paper. On one side is written the phrase "The statement on the other side of this paper is false." If you flip the paper over, you would find written "The statement on the other side of this paper is true." The paradox is immediately evident and, as the matemetician in the video states, Gödel expresses this same paradox in mathematical terms.

The general application of this theorem, from my position of relative ignorance, seems to imply that you can never understand the entirety of a system without being able to step outside of the system. As Watts so concisely said "you can't see the back of your own head with your naked eye." I think this concept represents a massive obstacle in the path of artificial intelligence - never will we be able to recreate our conciousness artificially without being able to step outside the axioms of our conciousness. So too, the implications this has when one considers the origin of language leads one to rather interesting ontological musings.

In summary, this along with similarly categorized concepts I've learned from David Hilbert (Hilbert Space), Benoit Mandelbrot (Mandelbrot's set), and Newton (1st Law of Thermodynamics) are all, in my estimation, substantial proofs for an unbounded exisitence contained and governed by a force much greater than ourselves.

On a brief side note, this paper also introduced me to the concept of metamathematics. A mind-blowing concept and a subject I look forward to exploring further.


Profile Image for Infinite Jen.
96 reviews844 followers
April 20, 2024
You grab the bastard by the head, screaming, "Whether in number theory, or space-time cosmology, Gödel’s method was to advance the formalization of the system under consideration and then test it to destruction upon the ‘strange loops’ it generated (paradoxes of self-reference and time-travel). In each case, the system was shown to permit cases that it could not consistently absorb, opening it to an interminable process of revision, or technical improvement. It thus defined dynamic intelligence, or the logic of evolutionary imperfection, with an adequacy that was both sufficient and necessarily inconclusive. What it did not do was trash the very possibility of arithmetic, mathematical logic, or cosmic history - except insofar as these were falsely identified with idols of finality or closure!" The knuckles of your large hands pop out like anal beads woven from spider silk (why did I do this?), your fingertips blanch Lugosian as all blood is chased from local capillaries, causing your victim to exclaim: "Dear YHWH! Would you look at that intravascular volume depletion! You must have quite a grip!" Causing the muscles of your forearms to stand out like corded steel as you grit your teeth. "WOULD YOU JUST LOOK AT THAT VOLUME CONTRACTION OF INTRAVASCULAR FLUID!" They scream as you continue to ratchet up the pressure. "Contrary to what you've witnessed in Game of Thrones, it would be impossible for even the strongest human to explode the skull through compressive forces exerted by any means (either with their hands bilaterally or by stepping on it). Any portion of the skull. Explosionless, bruv." Causing your nostrils to flare like giant intake fans and your forehead to crease with superhuman exertion. "This is quite uncomfortable, but I'm serious, mate. To shatter the skull using this method would require 500 kgF, or the force that 500 kilograms (1,100 pounds) would exert in standard gravity. A person would have to weigh 235 500 kilograms (519188 pounds) to do that by stepping on the head, and it would be impossible to shatter it with their hands even if 90 percent of the 235,500 kg were biceps muscles." Still undeterred you begin to screw your palms into their skull as if uncorking a particularly difficult jar of pickles. "Take for instance if you were to shift your thumbs to my eyes and attempt to 'Mountain' me. Extreme pressure on the eyes would lead to rupture of the globes with leaking of its content. However, it would probably not be exciting because it would only lead to a leakage of a clear fluid between the warrior’s fingers. Probably a small fracture of the inferior orbital wall - the thinnest portion of the orbital wall – would occur (i.e. a common fracture called "blowout fracture"). No explosion would be seen. The eyes of the victim (eg. myself) would be pushed backward some few inches. That’s it." You HNNNNNNNNNNNNNNNGGGGGG and WALURRGGHGHHHHHHHHHHHHH and MMMMMMMMMPHHHHHHHHH to no avail, eventually capitulating to the methods of science you so passionately defended before trying to rupture a skull like Khan Noonien Singh (played by Benedict Cumberbatch), the dangerous product of a eugenics war, and all around Kirk-thwarting, genetically engineered giga-chad.

Interlocutor: Now, as I was saying: The result of our previous discussion is that our axioms, if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent.

Executioner: I just tried to crush your head over this!

Monocular: In the early 20th century, cosmological physics was returned to the edge of time, and the question: what ‘came before’ the Big Bang? For cosmology no less than for transcendental philosophy - or even speculative theology - this ‘before’ could not be precedence (in time), but only (non-spatial) outsideness, beyond singularity. It indicated a timeless non-place cryptically adjacent to time, and even inherent to it. The carefully demystified time of natural science, calculable, measurable, and continuous, now pointed beyond itself, re-activated at the edges.

Executioner: NO! DON'T GIVE HIM THE STICK!

Rock Butter: One of the hallmarks of evolutionary epistemology is the notion that empirical testing alone does not justify the pragmatic value of scientific theories, but rather that social and methodological processes select those theories with the closest "fit" to a given problem. The mere fact that a theory has survived the most rigorous empirical tests available does not, in the calculus of probability, predict its ability to survive future testing. Karl Popper used Newtonian physics as an example of a body of theories so thoroughly confirmed by testing as to be considered unassailable, but which were nevertheless overturned by Einstein's insights into the nature of space-time. For the evolutionary epistemologist, all theories are true only provisionally, regardless of the degree of empirical testing they have survived

Executioner: That's exactly what I said before I commenced to murdering!

Vegemite on toast w/cheese: The only thing that makes the modern sciences elevated beyond epistemic procedures seen in other times and other cultures is the fact that there is a mechanism beyond human political manipulation for the elimination of defective theories. Karl Popper is on that level just totally right. If it’s politically negotiable, it’s unscientific by definition. You don’t trust scientists, you don’t trust scientific theories, you don’t trust scientific institutions in so far as they have integrity, what you trust is the disintegrated zone of criticism and the criteria for criticism and evaluation in terms of repeated experiments, in terms of the heuristics that are built up to decide whether a particular theory has been defeated and eliminated by a superior theory. It’s that mechanism of selection that is the only thing that makes science important and makes it a system of reality testing. And this is obviously intrinsically directed against any kind of organic political community aiming to internally determine—through its own processes—the negotiation of the nature of reality. Reality has to be an external disruptive critical factor.

Review: In any reasonable mathematical system there will always be true statements that cannot be proved. Awesome.
53 reviews
December 14, 2022
Very annoying typo in prop VI proof

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!
43 reviews1 follower
September 1, 2020
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.
Profile Image for Rhizomal Ennui.
55 reviews1 follower
September 23, 2022
I *read* this book. As to mean I followed the ideas and propositions presented here but to be frank could not distinguish the connection in parts of the proposition and not assess the accuracy of them. Regardless it remains an okay pamphlet and an elaborate explanation of the simple categorical impotency of rigid mathematical collections. True to the tradition of Philosophical Investigations of Wiggy the elusive signs are meant for those that are already enthralled by them. The myth of one genius german mathematician eschewing and ruining any ground for mathematics to stand on is also too sweet for most to pass.
Profile Image for Julien Michel.
25 reviews
March 6, 2023
The existence of this book is a cause for celebration. Not exactly light reading for your beach holiday or daily commute. If you want to read the derivation be prepared to invest efforts. The introduction is very accessible.
19 reviews
September 17, 2017
Straightforward and clear but otherwise hampered by a needless and often disorganized introduction in the first half.
Profile Image for Mack .
1,497 reviews56 followers
February 11, 2018
I understood little of it, but I wanted to find out what I needed to learn in order to understand. Godel’s theorems are stunning.
Profile Image for Didier "Dirac Ghost" Gaulin.
102 reviews23 followers
August 21, 2022
The most important result in logic's history. The introduction is a great explication of the thought process behind Godel's result.
Profile Image for Aleksandar Janjic.
147 reviews25 followers
December 24, 2015
И тако сам се ја у једном тренутку намјерио да почнем да читам класичне математичке текстове и кренуо сам са Њутном, али испоставило се да је Њутн прилично гадан ако не знаш добро Еуклида (а ја сам иначе за геометрију тараба), тако да сам однио Еуклида на штампање, а док сам чекао да се то доврши бацио сам поглед на ово лагано штиво од седамдесетак страна. При чему од тих седамдесетак страна половина отпада на предговор који објашњава оно што слиједи.

Елем, значај овог Геделовог дјела није нам сасвим непознат, међутим ја сам преглуп да бих то објашњавао, па вас упућујем да прочитате опис ове књиге овде негдје горе изнад. Ваљда. У сваком случају, никад нисам доживио овако интензивно спаљивање мозга као у ових Геделових тридесетак страна и иако ми је огромна већина тога у првом пролазу профурала високо изнад главе, успио сам у одређеном тренутку да ухватим неки трачак поенте, што ми даје наду да бих некад у далекој будућности могао да спознам како све ово функционише. Иако сам прочитао да је овај енглески превод који сам имао критикован у неким круговима и да (наводно) постоје бољи, нисам видио гдје би тачно били проблеми с њим. И превод и излагање самог Гедела изгледају ми сасвим прихватљиво за нешто што је овако брутално тешко. Главнина проблема су ознаке (пошто све оно што нам је познато од раније, логичке и аритметичке операције и сл., мора да се пребаци у потпуно другачији запис), које су језиве, не зато што је Гедел садиста, него зато што ситуација тако налаже. Без њих можете из самог текста понешто да провалите (али то на крају крајева можете и из предговора), али сваки иоле ситнији детаљ ће да вам промакне. Ово моје прво читање могло би се назвати само хватањем контура комплетне конструкције, онда на другом очекујем да мало дубље уђем у саму процедуру, а након сто педесетог мислим да ћу успјети да разумијем читаву ствар.
12 reviews3 followers
December 31, 2013
The symbols of Godel's formal logic are outdated and bizarre to the modern reader, but it's still an absolutely essential text. Other books describe the findings of what is one of the most remarkable discoveries of modern times in Godel's incompleteness proofs, but actually watching it unfold over the course of 40 or so pages is remarkable.
104 reviews
June 29, 2025
I only understand maybe half, def less than half of this book. I’ve made multiple attempts but I’m not a trained logician. I think it’s funny such a small book tore down the three volume Principia Mathematica.
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