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Very Short Introductions #718
Gödel's Theorem: A Very Short Introduction
Very Short Introductions: Brilliant, Sharp, Inspiring
Kurt Gödel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago. The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions. Gödel's Theorem has since established itself as a landmark intellectual achievement, having a profound
impact on today's mathematical ideas. Gödel and his theorem have attracted something of a cult following, though his theorem is often misunderstood.
This Very Short Introduction places the theorem in its intellectual and historical context, and explains the key concepts as well as common misunderstandings of what it actually states. A. W. Moore provides a clear statement of the theorem, presenting two proofs, each of which has something distinctive to teach about its content. Moore also discusses the most important philosophical implications of the theorem. In particular, Moore addresses the famous question of whether the theorem
shows the human mind to have mathematical powers beyond those of any possible computer
ABOUT THE The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
Kurt Gödel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago. The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions. Gödel's Theorem has since established itself as a landmark intellectual achievement, having a profound
impact on today's mathematical ideas. Gödel and his theorem have attracted something of a cult following, though his theorem is often misunderstood.
This Very Short Introduction places the theorem in its intellectual and historical context, and explains the key concepts as well as common misunderstandings of what it actually states. A. W. Moore provides a clear statement of the theorem, presenting two proofs, each of which has something distinctive to teach about its content. Moore also discusses the most important philosophical implications of the theorem. In particular, Moore addresses the famous question of whether the theorem
shows the human mind to have mathematical powers beyond those of any possible computer
ABOUT THE The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
150 pages, Kindle Edition
Published November 10, 2022
26 people are currently reading
118 people want to read
About the author
A.W. Moore
24 books14 followersAdrian William Moore (born 1956) is a Professor of Philosophy and Lecturer in Philosophy at the University of Oxford and Tutorial Fellow of St Hugh's College, Oxford.
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Displaying 1 - 7 of 7 reviews
May 25, 2024
Čitanje ove knjige mogu uporediti sa intenzivnim treningom snage nakon duge pauze. Moj matematički dremež je naglo prekinut i to jednim od najvećih intelektualnih dometa formalne logike XX veka. Nakon Gedela matematika nije ista, ali ni ljudska misao. Najjednostavnije rečeno, a teško je ovom prilikom išta jednostavno reći, Gedel je otvorio vrata matematičkoj samorefleksivnosti, pokazujući da matematički iskaz ne mora da bude samo tačan ili netačan, već i da „ne može da bude dokazan”. Iskazi mogu da budu tačni ili netačni u odnosu na sistem aksioma, te je u ovoj knjizi bilo dosta reči o aksiomizaciji, od Euklida do danas. Aksiomi su tvrdnje koje su evidentne, koje se ne dovode u pitanje i kao takve predstavljaju polazište mišljenja. Međutim, Gedel dokazuje da, koliko god imali aksioma, ne možemo uspostaviti savršen matematički sistem, jer će uvek postojati nedokazivi tačni iskazi. Ovo znači da je moguće dokazati da se nešto ne može dokazati, a ta mogućnost sigurnosti je briljantna!
Kao što su mi briljatne i dve hipoteze o kojima se piše u knjizi. Goldbahova hipoteza koja glasi da svaki paran broj veći od dva može da se predstavi u obliku zbira dva prosta broja. Ova hipoteza je uspostavljena u 18. veku i do dana današnjeg nije dobila dokaz. Stoga predstavlja nerešeni problem. Ukoliko je valjano izvedena matematička formula ona i samo ona koja može imati dokaz, onda posebna pažnja treba da se pristupi onim iskazima koji deluju tačno, a i dalje nemaju dokaz (niti dokaz o tome kako se dokaz ne može uspostaviti). Kada dobiju dokaz, hipoteze postaju teoreme.
Tako je tzv. Poslednja Fermaova teorema dokazana vekovima nakon što je zapisana (iako je autor navodno imao predivan i jednostavan dokaz). Ona glasi ovako: Ne postoje pozitivni celi brojevi a, b i c, takvi da a^n + b^n = c^n.
U matematici postoji zaista nešto čudesno. Ne mislim pritom na nekakav mistični osećaj, nego na fascinantnu i nezavršivu težnju za preciznošću, koja je ujedno i apstraktna i sasvim konkretna. A retko nam pada na pamet da je i matematika oblik jezika. Ukoliko jezike delimo na formalne i prirodne, gde su prirodni oni kojima se svakodnevno služimo i čije je značenje uvek promenljivo, a formalni oni koji moraju da budu nedvosmisleni, onda postoji mogućnost da se na celinu znanja gleda drugačije. Verujem u uspostavljanje veza i zato me knjižice ove edicije ispunjavaju nadom: znatiželjni duhovi zaslužuju da znaju nešto što je van njihovog uskog okvira. To je svet, prilika za susret. Valja to iskoristiti.
Kao što su mi briljatne i dve hipoteze o kojima se piše u knjizi. Goldbahova hipoteza koja glasi da svaki paran broj veći od dva može da se predstavi u obliku zbira dva prosta broja. Ova hipoteza je uspostavljena u 18. veku i do dana današnjeg nije dobila dokaz. Stoga predstavlja nerešeni problem. Ukoliko je valjano izvedena matematička formula ona i samo ona koja može imati dokaz, onda posebna pažnja treba da se pristupi onim iskazima koji deluju tačno, a i dalje nemaju dokaz (niti dokaz o tome kako se dokaz ne može uspostaviti). Kada dobiju dokaz, hipoteze postaju teoreme.
Tako je tzv. Poslednja Fermaova teorema dokazana vekovima nakon što je zapisana (iako je autor navodno imao predivan i jednostavan dokaz). Ona glasi ovako: Ne postoje pozitivni celi brojevi a, b i c, takvi da a^n + b^n = c^n.
U matematici postoji zaista nešto čudesno. Ne mislim pritom na nekakav mistični osećaj, nego na fascinantnu i nezavršivu težnju za preciznošću, koja je ujedno i apstraktna i sasvim konkretna. A retko nam pada na pamet da je i matematika oblik jezika. Ukoliko jezike delimo na formalne i prirodne, gde su prirodni oni kojima se svakodnevno služimo i čije je značenje uvek promenljivo, a formalni oni koji moraju da budu nedvosmisleni, onda postoji mogućnost da se na celinu znanja gleda drugačije. Verujem u uspostavljanje veza i zato me knjižice ove edicije ispunjavaju nadom: znatiželjni duhovi zaslužuju da znaju nešto što je van njihovog uskog okvira. To je svet, prilika za susret. Valja to iskoristiti.
March 27, 2023
Written by a philosopher, this book demystifies and clarifies Gödel’s Theorem very well. Importantly, it discusses its historical and technical contexts so that the theorem will not be misunderstood or over-extrapolated. The proofs are on the whole comprehensible. A small portion of the book is too abstract or advanced for me (by trade I’m a medical doctor). The last pages on what meaning is and how we understand and acquire meanings are nsightful and easy to understand. However, some sentences are convoluted and on first encounter appear inscrutable. So overall three stars.
October 8, 2023
A nice short that serves it purpose. It requires a large degree of focus to work through the actual set-up and deployment of the theorems, but is nonetheless very interesting. A nice reference to have to jump and out of when a reminder is needed of the formal language and associated concepts. The final chapter was very difficult for myself and something I'll return to at a later point.
October 30, 2024
Very well-written. The mathematics is pretty understandable and is accurate as far as I can tell, given I am not an expert on logic or set theory. Can only recommend this. It poses some vague questions implied by Gödel's theorems, but it doesn't expand on these meaningfully, I find. That is my only qualm with it.
December 31, 2024
We are so ficking fucked and math is igly and stupid but at least humans are more goated than computers. This shit ugly as Hell bro I can’t believe it. Iya like you thought it was this beautiful woman but then when you get closer and look too hard it’s actually the wicked witch of being not pretty or sometning
June 20, 2025
An intriguing subject.
Where mathematics, logic and philology combine to explain something about human thought.
Moore makes the difficult path to an understanding of Godel's Theorem a little easier and in doing so opens up a range of possibilities. A worthy read but one that repays the necessary effort involved.
Where mathematics, logic and philology combine to explain something about human thought.
Moore makes the difficult path to an understanding of Godel's Theorem a little easier and in doing so opens up a range of possibilities. A worthy read but one that repays the necessary effort involved.
September 2, 2024
Very demystifying. Reccomended for those interested!
The middle part is hard, but we'll worth it to go through a couple of times to really get a grip of the argument.
The middle part is hard, but we'll worth it to go through a couple of times to really get a grip of the argument.
Displaying 1 - 7 of 7 reviews