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I this blog post I'll present some new arrow notations.

Weak arrow notation

This is based on down arrow notation, but this is really weak.

Rules

1. \(a \uparrow_{weak} b = a^b\)

2. \(a \uparrow_{weak}^c 1 = a\)

3. \(a \uparrow_{weak}^c b = (a \uparrow_{weak}^c b-1) \uparrow_{weak}^{c-1} b\)

Values and bounds

\(a \uparrow_{weak}^2 b = a^{b!}\)

\(a \uparrow_{weak}^{c+1} b = a^{b!c}\)

\(a \uparrow_{weak}^{b+1} b = a^{T(b)}\)

\(a \uparrow_{weak}^{a+1} a = a^{T(a)} < a \uparrow\uparrow 4\)

Mixed Arrow Notation 2.0

The normal rules of chained arrow notation remains unchanged.

a1. \(X \rightarrow\uparrow^{c} a = X \rightarrow\uparrow^{c-1} a \rightarrow\uparrow^{c} a-1\)

a2. \(X \diamond\uparrow Y = X \diamond Y\)

a3. \(X \diamond 1 = X\)

Of course we can extend.

a4. \(X \diamond a \diamond\rightarrow\uparrow^{c} b = X \diamond a \diamond\rightarrow\rightarrow b \diamond\rightarrow\rightarrow c\)

Analysis

\(a \rightarrow\uparrow b \approx f_{\omega \cdot b}(a)\)

\(a \rightarrow\uparrow b \rightarrow 2 \approx f_{\omega^2}^b(a)\)

\(a \rightarrow\uparrow b \rightarrow c+2 \approx f_{\omega^2+c}^b(a)\)

\(a \rightarrow\uparrow a \rightarrow a \rightarrow 2 \approx f_{\omega^2+\omega+1}(a)\)

\(a \rightarrow\uparrow a \rightarrow\uparrow a  \approx f_{\omega^22}(a)\)

\(a \rightarrow\uparrow^{2} b  \approx f_{\omega^2 \cdot b}(a)\)

\(a \rightarrow\uparrow^{c} b  \approx f_{\omega^c \cdot b}(a)\)

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