15 years
Damn!
I am not involved in this community anymore, but earlier this year I released a full-length album of experimental electronic music peppered with references to googology.
Ellipsis considered harmful
Please do not use "..." when defining notations. Ellipses are helpful for giving a visual demonstration of how an expression expands, but they are often unclear or ambiguous, especially for complex recursion as seen in array notations. For beginners, I recommend that the "..." symbol not be used at all except in examples.
Use recursion instead. Instead of writing
- {a,b,c+1} = a {c} a {c} ... (b times) ... {c} a {c} a
you should write
- {a,1,c} = a
- {a,b+1,c+1} = {a,{a,b,c+1},c}.
Kleene's JavaScript
In this post, I'm going to demonstrate an alternative to Kleene's O that might be easier to understand. It assumes some familiarity with programming.
Kleene's JavaScript is a textual notation for notating any recursive ordinal, defined like so:
- "0" is a Kleene's JavaScript program. It evaluates to the ordinal 0.
- Let P be a Kleene's JavaScript program that evaluates to ordinal a. Then if you append "+1" to P, the result is a Kleene's JavaScript program that evaluates to the ordinal a + 1.
- Let F be JavaScript code that consists of anonymous function f:
- f takes a single nonnegative integer as an argument. (This is a theoretical variant of JS where integers can be infinitely large.)
- f is deterministic.
- f returns a string for every valid argument. That string…