How Gödel's Proof Works
In 1931, Kurt Gödel stunned the mathematical world by proving that any axiomatic system for mathematics is either inconsistent or incomplete. His incompleteness theorems demonstrated that there will always be true mathematical statements that cannot be proven within the system, and no system can prove its own consistency. This article explains Gödel's ingenious method of mapping mathematical statements to numbers (Gödel numbering), how he constructed a self-referential statement that asserts its own unprovability, and why this shattered the dream of a complete, consistent foundation for mathematics.
Gödel’s proof killed the search for a consistent, complete mathematical system.
- MathMonkeyMan
"Gödel's Proof" by Ernest Nagel and James R. Newman helped me to get it at some point.
On Amazon: <https://www.amazon.com/Godels-Proof-Ernest-Nagel-ebook/dp/B0...>
I might even pick up an ebook version if I can find it somewhere else. Been a while.
- gregfjohnson
Show HN: I recently gave a talk on the incompleteness theorem, specifically expressed in the language of software. It starts with a bit of historical background and a discussion of some of the philosophical context in which he carried out his work. The second half of the talk is my attempt to show the beautiful essential idea at the core of Godel's idea, pitched to a technically knowledgeable general audience. These are the slides from the talk, not translated into web pages; YMMV.
- gavinsyancey
If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
- matherial
> However, although G is undecidable, it’s clearly true.
That's... not really true; it's surprising to see it in Quanta, of all places.
Godel's (separate) completeness theorem says that in first-order logic, anything that's semantically true in all possible scenarios can be syntactically proved. So, if G is "clearly true", that ought to make it provable.
The theorems don't contradict each other because in FOL, G is not guaranteed to be true. Its truth is independent of the machinery Godel put in place.
It's not something you really need to get into an introductory text, but it actually makes the whole outcome easier to grasp, and leads to many more counterintuitive results, such as Skolem's paradox.
- the-mitr
Of possible interest
Godels Incompleteness Theorem (Little Mathematics Library)
by V. A. Uspensky