On Formally Undecidable Propositions of Principia Mathematica and Related Systems - Kurt Gödel  ## Metadata - Author: **Kurt Gödel** - Full Title: On Formally Undecidable Propositions of Principia Mathematica and Related Systems - Category: #books - Tags: #epistemology #logic #mathematic ## Highlights - Gödel established metamathematical results about the strings of his formal system by considering numbers co-ordinated with the strings. ([View Highlight](https://read.readwise.io/read/01gkr5y3pgdmbft58pdkwgsf4c)) - Gödel’s rule of arithmetization ensures that to every class of strings there corresponds a unique class of Gödel numbers, and vice versa. And that to any relation R between strings there corresponds a unique relation R′ between Gödel numbers, and vice versa: i.e. the n-adic relation R′ holds between n Gödel numbers if and only if the n-adic relation R holds between the n strings. For example, the metamathematical statement that the series s of formulae is a ‘proof’ of the formula f is true if and only if a certain arithmetical relation holds between the Gödel numbers of s and of f which corresponds to the relation: being a ‘proof’ of ([View Highlight](https://read.readwise.io/read/01gkr62sj92nw9v9j4nan4ct4y)) - The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica (PM)[2](#ftn_fn2) and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann).[3](#ftn_fn3) These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference ([View Highlight](https://read.readwise.io/read/01gkr74kwyp6kgrqysaqt7rwa3))