BOCF VS veblen函数：修订间差异

2025年7月6日 (日) 21:30的版本

本条目展示BOCF和veblen函数的列表分析。


BOCF	veblen函数
$ψ (0)$	1
$ψ (0) \times 2$	2
$ψ (1)$	$ω$
$ψ (1) \times 2$	$ω \times 2$
$ψ (2)$	$ω^{2}$
$ψ (3)$	$ω^{3}$
$ψ (ψ (1))$	$ω^{ω}$
$ψ (ψ (1) + 1)$	$ω^{ω + 1}$
$ψ (ψ (1) \times 2)$	$ω^{ω \times 2}$
$ψ (ψ (2))$	$ω^{ω^{2}}$
$ψ (ψ (ψ (1)))$	$ω^{ω^{ω}}$
$ψ (ψ (ψ (ψ (1))))$	$ω^{ω^{ω^{ω}}}$
$ψ (Ω)$	$φ (1, 0)$
$ψ (Ω) + 1$	$φ (1, 0) + 1$
$ψ (Ω) \times 2$	$φ (1, 0) \times 2$
$ψ (Ω) \times 3$	$φ (1, 0) \times 3$
$ψ (Ω + 1)$	$ω^{φ (1, 0) + 1}$
$ψ (Ω + ψ (1))$	$ω^{φ (1, 0) + ω}$
$ψ (Ω + ψ (ψ (1)))$	$ω^{φ (1, 0) + ω^{ω}}$
$ψ (Ω + ψ (Ω))$	$ω^{φ (1, 0) \times 2}$
$ψ (Ω + ψ (Ω) + 1)$	$ω^{φ (1, 0) \times 2 + 1}$
$ψ (Ω + ψ (Ω) + ψ (1))$	$ω^{φ (1, 0) \times 2 + ω}$
$ψ (Ω + ψ (Ω) \times 2)$	$ω^{φ (1, 0) \times 3}$
$ψ (Ω + ψ (Ω) \times 3)$	$ω^{φ (1, 0) \times 4}$
$ψ (Ω + ψ (Ω + 1))$	$ω^{ω^{φ (1, 0) + 1}}$
$ψ (Ω + ψ (Ω + 1) + 1)$	$ω^{ω^{φ (1, 0) + 1} + 1}$
$ψ (Ω + ψ (Ω + 1) \times 2)$	$ω^{ω^{φ (1, 0) + 1} \times 2}$
$ψ (Ω + ψ (Ω + 2))$	$ω^{ω^{φ (1, 0) + 2}}$
$ψ (Ω + ψ (Ω + 3))$	$ω^{ω^{φ (1, 0) + 3}}$
$ψ (Ω + ψ (Ω + ψ (1)))$	$ω^{ω^{φ (1, 0) + ω}}$
$ψ (Ω + ψ (Ω + ψ (Ω)))$	$ω^{ω^{φ (1, 0) \times 2}}$
$ψ (Ω + ψ (Ω + ψ (Ω) \times 2))$	$ω^{ω^{φ (1, 0) \times 3}}$
$ψ (Ω + ψ (Ω + ψ (Ω + 1)))$	$ω^{ω^{ω^{φ (1, 0) + 1}}}$
$ψ (Ω + ψ (Ω + ψ (Ω + ψ (Ω + 1))))$	$ω^{ω^{ω^{ω^{φ (1, 0) + 1}}}}$
$ψ (Ω \times 2)$	$φ (1, 1)$
$ψ (Ω \times 2 + 1)$	$ω^{φ (1, 1) + 1}$
$ψ (Ω \times 2 + ψ (Ω))$	$ω^{φ (1, 1) + φ (1, 0)}$
$ψ (Ω \times 2 + ψ (Ω + ψ (Ω)))$	$ω^{φ (1, 1) + ω^{φ (1, 0) \times 2}}$
$ψ (Ω \times 2 + ψ (Ω \times 2))$	$ω^{φ (1, 1) \times 2}$
$ψ (Ω \times 2 + ψ (Ω \times 2 + 1))$	$ω^{ω^{φ (1, 1) + 1}}$
$ψ (Ω \times 2 + ψ (Ω \times 2 + ψ (Ω \times 2)))$	$ω^{ω^{φ (1, 1) \times 2}}$
$ψ (Ω \times 3)$	$φ (1, 2)$
$ψ (Ω \times 4)$	$φ (1, 3)$
$ψ (Ω \times ω)$	$φ (1, ω)$
$ψ (Ω \times ω + 1)$	$ω^{φ (1, ω) + 1}$
$ψ (Ω \times ω + ψ (Ω))$	$ω^{φ (1, ω) + φ (1, 0)}$
$ψ (Ω \times ω + ψ (Ω \times 2))$	$ω^{φ (1, ω) + φ (1, 1)}$
$ψ (Ω \times ω + ψ (Ω \times ω))$	$ω^{φ (1, ω) \times 2}$
$ψ (Ω \times ω + ψ (Ω \times ω + 1))$	$ω^{ω^{φ (1, ω) + 1}}$
$ψ (Ω \times ω + Ω)$	$φ (1, ω + 1)$
$ψ (Ω \times ω \times 2)$	$φ (1, ω \times 2)$
$ψ (Ω \times ω^{2})$	$φ (1, ω^{2})$
$ψ (Ω \times ψ (Ω))$	$φ (1, φ (1, 0))$
$ψ (Ω \times ψ (Ω) \times 2)$	$φ (1, φ (1, 0) \times 2)$
$ψ (Ω \times ψ (Ω + 1))$	$φ (1, ω^{φ (1, 0) + 1})$
$ψ (Ω \times ψ (Ω + ψ (Ω)))$	$φ (1, ω^{φ (1, 0) \times 2})$
$ψ (Ω \times ψ (Ω + ψ (Ω + ψ (Ω))))$	$φ (1, ω^{ω^{φ (1, 0) + 1}})$
$ψ (Ω \times ψ (Ω \times 2))$	$φ (1, φ (1, 1))$
$ψ (Ω \times ψ (Ω \times 3))$	$φ (1, φ (1, 2))$
$ψ (Ω \times ψ (Ω \times ω))$	$φ (1, φ (1, ω))$
$ψ (Ω \times ψ (Ω \times ψ (Ω)))$	$φ (1, φ (1, φ (1, 0)))$
$ψ (Ω \times ψ (Ω \times ψ (Ω \times ψ (Ω))))$	$φ (1, φ (1, φ (1, φ (1, 0))))$
$ψ (Ω^{2})$	$φ (1, 1)$

@@ 第1行： / 第1行： @@
 本条目展示[[OCF#BOCF|BOCF]]和[[veblen函数]]的列表分析。
+{| class="wikitable"
-<math>\psi(0)=1</math>
+|+
+!BOCF
+!veblen函数
+|-
+|<math>\psi(0)</math>
+|1
+|-
+|<math>\psi(0)\times2</math>
+|2
+|-
+|<math>\psi(1)</math>
+|<math>\omega</math>
+|-
+|<math>\psi(1)\times2</math>
+|<math>\omega\times2</math>
+|-
+|<math>\psi(2)</math>
+|<math>\omega^2</math>
+|-
+|<math>\psi(3)</math>
+|<math>\omega^3</math>
+|-
+|<math>\psi(\psi(1))</math>
+|<math>\omega^{\omega}</math>
+|-
+|<math>\psi(\psi(1)+1)</math>
+|<math>\omega^{\omega+1}</math>
+|-
+|<math>\psi(\psi(1)\times2)</math>
+|<math>\omega^{\omega\times2}</math>
+|-
+|<math>\psi(\psi(2))</math>
+|<math>\omega^{\omega^2}</math>
+|-
+|<math>\psi(\psi(\psi(1)))</math>
+|<math>\omega^{\omega^{\omega}}</math>
+|-
+|<math>\psi(\psi(\psi(\psi(1))))</math>
+|<math>\omega^{\omega^{\omega^{\omega}}}</math>
+|-
+|<math>\psi(\Omega)</math>
+|<math>\varphi(1,0)</math>
+|-
+|<math>\psi(\Omega)+1</math>
+|<math>\varphi(1,0)+1</math>
+|-
+|<math>\psi(\Omega)\times2</math>
+|<math>\varphi(1,0)\times2</math>
+|-
+|<math>\psi(\Omega)\times3</math>
+|<math>\varphi(1,0)\times3</math>
+|-
+|<math>\psi(\Omega+1)</math>
+|<math>\omega^{\varphi(1,0)+1}</math>
+|-
+|<math>\psi(\Omega+\psi(1))</math>
+|<math>\omega^{\varphi(1,0)+\omega}</math>
+|-
+|<math>\psi(\Omega+\psi(\psi(1)))</math>
+|<math>\omega^{\varphi(1,0)+\omega^{\omega}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega))</math>
+|<math>\omega^{\varphi(1,0)\times2}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega)+1)</math>
+|<math>\omega^{\varphi(1,0)\times2+1}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega)+\psi(1))</math>
+|<math>\omega^{\varphi(1,0)\times2+\omega}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega)\times2)</math>
+|<math>\omega^{\varphi(1,0)\times3}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega)\times3)</math>
+|<math>\omega^{\varphi(1,0)\times4}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+1))</math>
+|<math>\omega^{\omega^{\varphi(1,0)+1}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+1)+1)</math>
+|<math>\omega^{\omega^{\varphi(1,0)+1}+1}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+1)\times2)</math>
+|<math>\omega^{\omega^{\varphi(1,0)+1}\times2}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+2))</math>
+|<math>\omega^{\omega^{\varphi(1,0)+2}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+3))</math>
+|<math>\omega^{\omega^{\varphi(1,0)+3}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+\psi(1)))</math>
+|<math>\omega^{\omega^{\varphi(1,0)+\omega}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+\psi(\Omega)))</math>
+|<math>\omega^{\omega^{\varphi(1,0)\times2}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+\psi(\Omega)\times2))</math>
+|<math>\omega^{\omega^{\varphi(1,0)\times3}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+\psi(\Omega+1)))</math>
+|<math>\omega^{\omega^{\omega^{\varphi(1,0)+1}}}</math>
+|-
+|<math>\psi(\Omega+\psi(\Omega+\psi(\Omega+\psi(\Omega+1))))</math>
+|<math>\omega^{\omega^{\omega^{\omega^{\varphi(1,0)+1}}}}</math>
+|-
+|<math>\psi(\Omega\times2)</math>
+|<math>\varphi(1,1)</math>
+|-
+|<math>\psi(\Omega\times2+1)</math>
+|<math>\omega^{\varphi(1,1)+1}</math>
+|-
+|<math>\psi(\Omega\times2+\psi(\Omega))</math>
+|<math>\omega^{\varphi(1,1)+\varphi(1,0)}</math>
+|-
+|<math>\psi(\Omega\times2+\psi(\Omega+\psi(\Omega)))</math>
+|<math>\omega^{\varphi(1,1)+\omega^{\varphi(1,0)\times2}}</math>
+|-
+|<math>\psi(\Omega\times2+\psi(\Omega\times2))</math>
+|<math>\omega^{\varphi(1,1)\times2}</math>
+|-
+|<math>\psi(\Omega\times2+\psi(\Omega\times2+1))</math>
+|<math>\omega^{\omega^{\varphi(1,1)+1}}</math>
+|-
+|<math>\psi(\Omega\times2+\psi(\Omega\times2+\psi(\Omega\times2)))</math>
+|<math>\omega^{\omega^{\varphi(1,1)\times2}}</math>
+|-
+|<math>\psi(\Omega\times3)</math>
+|<math>\varphi(1,2)</math>
+|-
+|<math>\psi(\Omega\times4)</math>
+|<math>\varphi(1,3)</math>
+|-
+|<math>\psi(\Omega\times\omega)</math>
+|<math>\varphi(1,\omega)</math>
+|-
+|<math>\psi(\Omega\times\omega+1)</math>
+|<math>\omega^{\varphi(1,\omega)+1}</math>
+|-
+|<math>\psi(\Omega\times\omega+\psi(\Omega))</math>
+|<math>\omega^{\varphi(1,\omega)+\varphi(1,0)}</math>
+|-
+|<math>\psi(\Omega\times\omega+\psi(\Omega\times2))</math>
+|<math>\omega^{\varphi(1,\omega)+\varphi(1,1)}</math>
+|-
+|<math>\psi(\Omega\times\omega+\psi(\Omega\times\omega))</math>
+|<math>\omega^{\varphi(1,\omega)\times2}</math>
+|-
+|<math>\psi(\Omega\times\omega+\psi(\Omega\times\omega+1))</math>
+|<math>\omega^{\omega^{\varphi(1,\omega)+1}}</math>
+|-
+|<math>\psi(\Omega\times\omega+\Omega)</math>
+|<math>\varphi(1,\omega+1)</math>
+|-
+|<math>\psi(\Omega\times\omega\times2)</math>
+|<math>\varphi(1,\omega\times2)</math>
+|-
+|<math>\psi(\Omega\times\omega^2)</math>
+|<math>\varphi(1,\omega^2)</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega))</math>
+|<math>\varphi(1,\varphi(1,0))</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega)\times2)</math>
+|<math>\varphi(1,\varphi(1,0)\times2)</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega+1))</math>
+|<math>\varphi(1,\omega^{\varphi(1,0)+1})</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega+\psi(\Omega)))</math>
+|<math>\varphi(1,\omega^{\varphi(1,0)\times2})</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega+\psi(\Omega+\psi(\Omega))))</math>
+|<math>\varphi(1,\omega^{\omega^{\varphi(1,0)+1}})</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega\times2))</math>
+|<math>\varphi(1,\varphi(1,1))</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega\times3))</math>
+|<math>\varphi(1,\varphi(1,2))</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega\times\omega))</math>
+|<math>\varphi(1,\varphi(1,\omega))</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega\times\psi(\Omega)))</math>
+|<math>\varphi(1,\varphi(1,\varphi(1,0)))</math>
+|-
+|<math>\psi(\Omega\times\psi(\Omega\times\psi(\Omega\times\psi(\Omega))))</math>
+|<math>\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))</math>
+|-
+|<math>\psi(\Omega^2)</math>
+|<math>\varphi(1,1)</math>
+|-
+|
+|
+|}
 [[分类:分析]]