Bird's Proof is a theorem by Chris Bird that states that:[1][2][3]

\[\forall a \geq 3, b \geq 2, c \geq 1, d \geq 2: \{a, b, c, d\} > \underbrace{a \rightarrow a \rightarrow \ldots \rightarrow a \rightarrow a}_d \rightarrow (b - 1) \rightarrow (c + 1)\]

using array notation and chained arrow notation.

In other words, four entries of array notation are comparable to chained arrow notation, but five entries far surpass it.

It was published in his paper "Array Notations for Super Huge Numbers", and was named by Jonathan Bowers on his website.


  1. Bird's proof: Note that, at the time of the writing of the proof, Bowers' function defined {a,b} as the sum of a and b, while Bird redefined it as \(a^b\), considering it to be a different function. Bowers later revised his own notation to match Bird's, so the notations are identical.
  3. Array Notation

See also