A few self-referential, meta and recursive and inductive bits. Lots of lazy Wikipedia links thrown about for more info.
An Englishman, an Irishman and a Scotsman walk into a bar. The bartender turns to them, takes one look, and says, ‘What is this - some kind of joke?’
What do you get when you cross a joke with a rhetorical question?
Hofstadter’s Law: Complicated tasks always take longer than you expect, even when you take into account Hofstadter’s Law.
Some self referential quizzes here.
I also came up with the following mini-maths-quiz before finding the above, uses only basic algebra:
- Find the mode of the following numbers: 3, 6, 9, 12, 6, 14
- Solve for x: 1+2x=9
- What is the mean average of the answers to these three questions?
Self referential acronyms are pretty common in computing.
- MUNG Until No Good (originally Mash Until No Good)
- GNU’s Not Unix
- PHP: Hypertext Preprocessor
- Curl URL Request Library
- JACK Audio Connection Kit
- TLA: Three Letter Acronyms
Some say that the ‘B’ in Benoit B. Mandelbrot stands for ‘Benoit B. Mandelbrot’
Wikipedia list of examples
An image with a copy of itself in the image creates an infinite effect called the Droste effect:
Baba is you is a very cool computer game in which you push around blocks Sokoban-style, but some of those blocks have words on them. Those words can be arranged into sentences, (and those sentences can be broken up again) to change the rules of the game.
- My meta sudokus
- Similarly, some Sudoku puzzles are designed to be solved by uniqueness logic, very meta!
Exploding Heads Puzzles
Here is a fun extension to the “World’s Hardest Logic Puzzle” involving self-referential paradoxes to extract possibly-infinite information from a single yes-or-no question. Uses Coercive Logic to blow-up the brains of higher beings. Ouch.
Some things in maths are defined self-referentially or “impredicatively”. Only included in here since some cool transfinite ordinals are defined this way. For example, the Fefermann-Schutte ordinal can be defined as the smallest ordinal Γ such that φ(Γ,0) = Γ, where φ is the Veblen function.
On a similar note, there’s techniques of ’transfinite induction’ and ’transfinite recursion’. Transfinite induction works similarly to normal proof-by-induction but with a third case for limit ordinals.
Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves.