Bird's number is a large number defined by Chris Bird that uses his old extended array notation.[1]

Bird's old array notationEdit

Up to Nested Arrays, Bird's old array notation is almost the same new array notation.

After that, Bird develops a notation using the negation sign. The old separator \([1\neg2]\) is the same as [1\2] and [1/2]. The old separator \([1\neg3]\) is the same as [1\3] and [1/3]. In general, the old separator \([1\neg\text{A}]\) is the same as [1\A] and [1/A] for some array A. It turns out that the old separator \([1\neg1\neg2]\) has the same level as [1\1\2] and the old separator [[2]] that diagonalizes over negation signs like [2] diagonalizes over commas is the same as the new separator \([1[2\neg2]2]\).

In general, the separator [[...[[A]]...]] with n brackets is the same as the new separator \([1[A\neg n]2]\). The < > notation is the angle bracket notation for these arrays.

Bird's Number Edit

We define

\(N = \{\underbrace{3<<<...<<<3>>>...>>>3}_{\{3<<<<<<<3>>>>>>>3\}}\}\)

\(\text{X}(0) = N\)

\(\text{X}(n) = \{\underbrace{3<<<...<<<3>>>...>>>3}_{\text{X}(n-1)}\}\)

Bird's number is \(\underbrace{\text{X}(\text{X}(...\text{X}(\text{X}(N))...))}_{\text{X}(N)}\)

Approximations in other notationsEdit

Notation Approximation
BEAF \(\{7,\{7,\{7,\{7,7(X,X(1)2)2\}(X,X(1)2)2\},2(X,X(1)2)2\},3(X,X(1)2)2\}\)[2]
Hyperfactorial array notation \(7![1([1])2]![1([1])2]![2([1])2]![3([1])2]\)
Fast-growing hierarchy \(f_{\vartheta(\Omega^{\omega})+2}(f_{\vartheta(\Omega^{\omega})+1}(f_{\vartheta(\Omega^{\omega})}(f_{\vartheta(\Omega^{\omega})}(7))))\)
Hardy hierarchy \(H_{\vartheta(\Omega^{\omega})(\omega^2+\omega+2)}(7)\)
Slow-growing hierarchy \(g_{\vartheta(\Omega_2^{\Omega}+1)}(g_{\vartheta(\Omega_2^{\Omega})}(g_{\vartheta(\Omega_2^{\omega})}(g_{\vartheta(\Omega_2^{\omega})}(7))))\)

Sources Edit

  1. Chris Bird's Array Notations for Super Huge Numbers
  2. Using particular notation \(\{a,b (A) 2\} = A \&\ a\) with prime b.

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