Intuitive Ordinal Notations (IONs)

In "Measuring the Intelligence of an Idealized Mechanical Knowing Agent", we presented an intuitive ordinal notation system and argued that the task of notating computable ordinals is a task that gets directly at the heart of what it means to be intelligent. This library contains some examples of ordinal notations. Currently, the largest example in the library is ω^ω.

Think of this library as a meditation on automated code generation. The ordinal notations in this library are themselves computer programs which output ordinal notations (except for 0.py, which outputs nothing). The notation for ω^ω is a computer program which could, by its lonesome, be used to programmatically reproduce computer programs equivalent to all the lesser ordinal notations in the library.

Reading Order

We recommend the ordinal notations in this library be read in the following order. Take time to reflect on each program and understand it before proceeding to the next. If you want to actually run these programs yourself and see their output, you can do so by running the "Standalone" versions linked below (using Python).

Level 1 (up to ω)

• 0 With Commentary or Compiled and Uncommented or Standalone
• 1 With Commentary or Compiled and Uncommented or Standalone
• 2 With Commentary or Compiled and Uncommented or Standalone
• 3 With Commentary or Compiled and Uncommented or Standalone
• ω With Commentary or Compiled and Uncommented or Standalone

Level 2 (up to ω²)

• ω With Commentary or Compiled and Uncommented or Standalone
• ω+1 With Commentary or Compiled and Uncommented or Standalone
• ω+2 With Commentary or Compiled and Uncommented or Standalone
• ω+3 With Commentary or Compiled and Uncommented or Standalone
• ω·2 With Commentary or Compiled and Uncommented or Standalone
• ω·2+1 With Commentary or Compiled and Uncommented or Standalone
• ω·2+2 With Commentary or Compiled and Uncommented or Standalone
• ω·3 With Commentary or Compiled and Uncommented or Standalone
• ω² With Commentary or Compiled and Uncommented or Standalone

Level 3 (up to ω³)

• ω With Commentary or Compiled and Uncommented or Standalone
• ω² With Commentary or Compiled and Uncommented or Standalone
• ω²+ω With Commentary or Compiled and Uncommented or Standalone
• ω²+ω·2 With Commentary or Compiled and Uncommented or Standalone
• ω²·2 With Commentary or Compiled and Uncommented or Standalone
• ω²·3 With Commentary or Compiled and Uncommented or Standalone
• ω³ With Commentary or Compiled and Uncommented or Standalone

Level 4 (up to ω⁴)

• ω With Commentary or Compiled and Uncommented or Standalone
• ω² With Commentary or Compiled and Uncommented or Standalone
• ω³ With Commentary or Compiled and Uncommented or Standalone
• ω³+ω² With Commentary or Compiled and Uncommented or Standalone
• ω³+ω²·2 With Commentary or Compiled and Uncommented or Standalone
• ω³·2 With Commentary or Compiled and Uncommented or Standalone
• ω³·2+ω² With Commentary or Compiled and Uncommented or Standalone
• ω³·2+ω²·2 With Commentary or Compiled and Uncommented or Standalone
• ω³·3 With Commentary or Compiled and Uncommented or Standalone
• ω⁴ With Commentary or Compiled and Uncommented or Standalone

Level ω (up to ω^ω)

• 1 With Commentary or Compiled and Uncommented or Standalone
• ω With Commentary or Compiled and Uncommented or Standalone
• ω² With Commentary or Compiled and Uncommented or Standalone
• ω³ With Commentary or Compiled and Uncommented or Standalone
• ω⁴ With Commentary or Compiled and Uncommented or Standalone
• ω^ω With Commentary or Compiled and Uncommented or Standalone