In-depth · Turing and Gödel
The Machine and the Limit
Turing, Gödel and the mathematical foundation of the incalculable remainder
"The computable functions are a countable island in an uncountable ocean."
The computable is an island - the incalculable is the ocean
From the book «Beyond Turing - The Incalculable Remainder»
Inside the article
The three knots
The halting problem
The boundary proved
Not «not yet found»: it cannot exist. Rice's theorem generalizes: every non-trivial property of programs is undecidable.
Gödel
Truth exceeds proof
Every consistent formal system contains truths it cannot prove: every system has an outside, and mathematics proved it by looking at itself.
The Turing test
A measure of performance, not existence
Language models pass the imitation game: they prove conversation is computable, not that someone has appeared behind the surface.
