查看“︁Veblen 函数”︁的源代码

'''Veblen函数(别名：<math>\varphi</math>函数)'''是一个<math>\rm Ord\rightarrow Ord</math>的序数函数，由美国数学家Oswald Veblen定义。

== 定义 ==

=== 二元Veblen函数 ===
Veblen函数的定义基于序数函数的[[不动点]].

二元Veblen函数<math>\varphi(\alpha,\beta)~(\alpha,\beta\in\mathrm{Ord})</math>的定义如下：

# <math>\varphi(0,\beta)=\varphi(\beta)=\omega^\beta</math>
# <math>\varphi(\alpha+1,\beta)</math>是函数<math>x\mapsto\varphi(\alpha,x)</math>的第<math>1+\beta</math>个不动点.
# 对于[[序数#极限序数|极限序数]]<math>\alpha</math>,<math>\varphi(\alpha,\beta)</math>为所有<math>x\mapsto\varphi(\gamma,x)(\gamma<\alpha)</math>的第<math>1+\beta</math>个公共不动点.

其基本列定义如下：
# <math>(\varphi(\alpha_1,\beta_1)+\varphi(\alpha_2,\beta_2)+\cdots+\varphi(\alpha_k,\beta_k))[n]=\varphi(\alpha_1,\beta_1)+\varphi(\alpha_2,\beta_2)+\cdots+\varphi(\alpha_k,\beta_k)[n]</math>
# <math>\varphi(0,0)=1</math>
# <math>\varphi(0,\beta+1)[n]=\varphi(0,\beta)\cdot n</math>
# 对于极限序数<math>\beta</math>,<math>\varphi(\alpha,\beta)[n]=\varphi(\alpha,\beta[n])</math>
# <math>\varphi(\alpha+1,0)[0]=0</math>
# <math>\varphi(\alpha+1,\beta+1)[0]=\varphi(\alpha+1,\beta)+1</math>
# <math>\varphi(\alpha+1,\beta+1)[n+1]=\varphi(\alpha,\varphi(\alpha+1,\beta+1)[n])</math>
# 对于极限序数<math>\alpha</math>,<math>\varphi(\alpha,0)[n]=\varphi(\alpha[n],0)</math>
# 对于极限序数<math>\alpha</math>,<math>\varphi(\alpha,\beta+1)[n]=\varphi(\alpha[n],\varphi(\alpha,\beta)+1)</math>

=== 有限元Veblen函数 ===

=== 序数元Veblen函数 ===