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

## Bird's old array notation

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 [] that diagonalizes over negation signs like  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

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 notations

Notation Approximation
BEAF $$\{7,\{7,\{7,\{7,7(X,X(1)2)2\}(X,X(1)2)2\},2(X,X(1)2)2\},3(X,X(1)2)2\}$$
Hyperfactorial array notation $$7![1()2]![1()2]![2()2]![3()2]$$
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

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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