Feb 19, 2025
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My recent post notwithstanding, there is a bit more to Godel’s theorems than the diagonalization magic of Lawvere’s theorem. It has to do with statements of consistency being Π_1 sentences, and thus inductively but not recursively computable. So if you allow your axiomatization of arithmetic to be Π_1 instead of recursively enumerable, Godel’s theorem fails (source).
The Barber Paradox
A barber shaves all men that do not shave themselves. Must he shave himself?
Barber paradox that shows the relativity of the logic/rationality that is so revered by science. Together with, among others, Goedel’s incompleteness theorem this pleads for room for intuition in science (what Kurt Goedel also seems to have pursued).
This paradox has its own Wikipedia page and on that page it is argued that this is not so much a paradox as an untenable assumption about the village in question. Contrary to what the poser of the question claims this `paradox’ shows the power of first-order logic: one can translate the statement into a logical formula and verify that the statement is never satisfiable, that its negation is always satisfied and thanks to Gödel’s Completeness theorem we know that that negation has a formal derivation.
The original paradox, due to Russell, shows, ironically, that it was the intuition that let down the initial developers of Set Theory. That paradox caused a thorough revision of Set Theory’s foundation in logical terms.
Many books and web pages have been written about Gödel’s Incompleteness Theorems and I am not going to add another long essay to this collection. The content of these theorems, especially the first one, is much more technical than many people think; my advice to anyone who really wants to know what these results say is: take a book on Mathematical Logic and study it up to and including the incompleteness theorems; I learned logic from an earlier edition of this book.
An Important, relevant, sassy, scholarly, and awesome article by the late great Stanley Jaki.
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It is on the ultimate success of such a quest [for a Theory of Everything] that Gödel's theorem casts the shadow of judicious doubt. It seems on the strength of Gödel's theorem that the ultimate foundations of the bold symbolic constructions of mathematical physics will remain embedded forever in that deeper level of thinking characterized both by the wisdom and by the haziness of analogies and intuitions. For the speculative physicist this implies that there are limits to the precision of certainty, that even in the pure thinking of theoretical physics there is a boundary present, as in all other fields of speculations.
Stanley Jaki, The Relevance of Physics
Hi, I'm auditioning for the role of Kurt Gödel, and I'll be singing "Incomplete" by Backstreet Boys
I don’t consider my work a 'facet of the intellectual atmosphere of the early 20th century,' but rather the opposite. It is true that my interest in the foundations of mathematics was aroused by the 'Vienna Circle,' but the philosophical consequences of my result, as well as the heuristic principles leading to them, are anything but positivistic or empiricistic.
Kurt Gödel on his incompleteness theorems