نیسٹڈ ہائپر فیکٹوریل (چھوٹے گاما) فنکشن نوٹیشن میں تمام بنیادی باتوں کی تمام نظرثانی پر نظر ثانی جس کے بارے میں مجھے بہت افسوس ہے

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Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).

  —— Concrete Mathematics, page 3 margins.

actually we are now back to \(\zeta_0\) even though as you know our analysis has reached \(\psi(\Omega_2^\Omega)\), this analysis is very important because we will build the foundation again and now we know how to get to a high level, we have to update it from the beginning to support the above, this is a big revision

(|2) = psi(Ω^2)

(1|2) = psi(Ω^2)+1

(2|2) = psi(Ω^2)+2

(3|2) = psi(Ω^2)+3

((|1)|2) = psi(Ω^2)+psi(Ω)

((|2)|2) = psi(Ω^2)+psi(Ω^2)

((|2)(|2)|2) = psi(Ω^2)+psi(Ω^2)+psi(Ω^2)

((|2)!^(|0)|2) = psi(Ω^2)w

((|2)!^((|0)|0)|2) = psi(Ω^2)w^w

((|2)!^(((|0)|0)|0)|2) = psi(Ω^2)w^w^w

((|2)!^((((|0)|0)|0)|0)|2) = psi(Ω^2)w^w^w^w

((|2)!^(|1)|2) = psi(Ω^2)psi(Ω)

((|2)!^(|2)|2) = psi(Ω^2)psi(Ω^2)

((|2)!^(|2)|2) = psi(Ω^2+2)

((|2)!^((|2)!^(|2)|2)|2) = psi(Ω^2+3)

((|2)!^(((|2)!^(|2)|2)!^(|2)|2)|2) = psi(Ω^2+4)

((|2)!^((((|2)!^(|2)|2)!^(|2)|2)!^(|2)|2)|2) = psi(Ω^2+5)

((1|2)|2) = psi(Ω^2+w)

((2|2)|2) = psi(Ω^2+w+1)

((3|2)|2) = psi(Ω^2+w+2)

(((|0)|2)|2) = psi(Ω^2+w2)

((((|0)|0)|2)|2) = psi(Ω^2+w^2)

(((((|0)|0)|0)|2)|2) = psi(Ω^2+w^w^w)

((|1)|2)|2) = psi(Ω^2+psi(Ω))

((|2)|2)|2) = psi(Ω^2+psi(Ω^2))

(((|2)|2)|2)|2) = psi(Ω^2+psi(Ω^2+psi(Ω^2)))

((((|2)|2)|2)|2)|2) = psi(Ω^2+psi(Ω^2+psi(Ω^2+psi(Ω^2))))

(((((|2)|2)|2)|2)|2)|2) = psi(Ω^2+psi(Ω^2+psi(Ω^2+psi(Ω^2+psi(Ω^2)))))

(<|1>|2) = psi(Ω^2+Ω)

(<|1>|2) = psi(Ω^2+Ω)

(1 <|1>|2) = psi(Ω^2+Ω)+1

((|1) <|1>|2) = psi(Ω^2+Ω)+psi(Ω)

((|2) <|1>|2) = psi(Ω^2+Ω)+psi(Ω^2)

((|2) <|1>|2) = psi(Ω^2+Ω)+psi(Ω^2)

(((|2)|2) <|1>|2) = psi(Ω^2+Ω)+psi(Ω^2+psi(Ω^2))

((((|2)|2)|2) <|1>|2) = psi(Ω^2+Ω)+psi(Ω^2+psi(Ω^2+psi(Ω^2)))

((<|1>|2)<|1>|2) = psi(Ω^2+Ω+psi(Ω^2+Ω))

(((<|1>|2)<|1>|2)<|1>|2) = psi(Ω^2+Ω+psi(Ω^2+Ω+psi(Ω^2+Ω)))

((((<|1>|2)<|1>|2)<|1>|2)<|1>|2) = psi(Ω^2+Ω+psi(Ω^2+Ω+psi(Ω^2+Ω+psi(Ω^2+Ω))))

( <1|1>|2) = psi(Ω^2+Ω2)

( <2|1>|2) = psi(Ω^2+Ω3)

( <3|1>|2) = psi(Ω^2+Ω4)

( <(|0)|1>|2) = psi(Ω^2+Ωw)

( <((|0)|0)|1>|2) = psi(Ω^2+Ωw^w)

( <(((|0)|0)|0)|1>|2) = psi(Ω^2+Ωw^w^w)

( <(|1)|1>|2) = psi(Ω^2+Ωpsi(Ω))

( <( <|1>|2)|1>|2) = psi(Ω^2+Ωpsi(Ω^2+Ω))

( <( <( <|1>|2)|1>|2)|1>|2) = psi(Ω^2+Ωpsi(Ω^2+Ωpsi(Ω^2+Ω)))

( {1} |2) = psi(Ω^2*2)

( {2} |2) = psi(Ω^2*3)

( {3} |2) = psi(Ω^2*4)

( {4} |2) = psi(Ω^2*5)

( {(|0)} |2) = psi(Ω^2*w)

( {((|0)|0)} |2) = psi(Ω^2*w^w)

( {(((|0)|0)|0)} |2) = psi(Ω^2*w^w^w)

( {((((|0)|0)|0)|0)} |2) = psi(Ω^2*w^w^w^w)

( {(|1)} |2) = psi(Ω^2*psi(Ω))

( {(|2)} |2) = psi(Ω^2*psi(Ω^2))

( {(|2)} |2) = psi(Ω^2*psi(Ω^2))

( {( {(|2)} |2)} |2) = psi(Ω^2*psi(Ω^2)*psi(Ω^2))

( {( {( {(|2)} |2)} |2)} |2) = psi(Ω^2*psi(Ω^2)*psi(Ω^2*psi(Ω^2)))

( {( {( {( {(|2)} |2)} |2)} |2)} |2) = psi(Ω^2*psi(Ω^2)*psi(Ω^2*psi(Ω^2*psi(Ω^2))))