User blog:Luckyluxius/Luxius' Exploding Array Notation

(Credit to Johnathan Bower)

Abr. LEAN

might be ill-defined*

\({2,0,(2,3)} = 2\)

\({2,1,(2,3)} = 2 + 3 + 2 + 3\)

\({2,2,(2,3)} = 2 \cdot 3 \cdot 2 \cdot 3\)

\({2,3,(2,3)} = 2^{3^{2^{3}}}\)

\({2,4,(2,3)} = 2 \uparrow 3 \uparrow 2 \uparrow 3\)

\({2,5,(2,3)} = 2 \uparrow\uparrow 3 \uparrow\uparrow 2 \uparrow\uparrow 3\)

\({2,n,(2,3)} = 2 \uparrow^{n-3} 3 \uparrow^{n-3} 2 \uparrow^{n-3} 3\)


 * First variable defines how many times the function is done

Second variable defines what mathematical function it is

Third and forth are any positive integer.

(anything that is ω or more, you don't need the "a" variable and "c" variable (as in {a,b,(c,d)}). The B variable has to be any countable ordinal.)

{\(\omega\),3} = f_{2}(3)

{\(\omega2\),3} = f_{f_{2}(3)}(3)

{\(ω^{2}\),3} = \(f_{ω}(3)\)

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Anything above \(ω^{ω}\) is equal to \(f_{\text{ordinal}}(n)\)

for example:

{\({ζ_{255}}\),3} = \(f_{ζ_{255}}(3)

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The uncountable \(ω_{1}\) has a different property.

for example:

{\(ω_{1}\),n} = \underbrace{n^{2 \cdot n^{3 \cdot n^{\cdots^{n \uparrow \uparrow {2}}}}}_{((n \uparrow \uparrow n) - 1) \text{times}}

and

{\(ω_{n}\),m} =

\underbrace{m^{2 \cdot n^{3 \cdot n^{\cdots^{n \uparrow^{n+1} 2}}}}}_{((n \uparrow^{n+1} n) - 1) \text{times}}

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