根序数和ε-序数

ROOT ORDINALS AND ε-ORDINALS

定义1 设ρ是一个大于1的固定的序数,x是待定的序数,那么(1) ρ~x=x称做以ρ为底的指数方程,简称ρ—方程。如果取x=α,(1)式成立,则α称做这个方程的根,指数方程的根称根序数。 命题1 设1【ρ【β,则β—方程的根必是ρ—方程的根。

Suppose that ordinal p>1 , then px = x is called a p-equation and if x = a satisfies this equation, then a is called a root ordinal. For every ordinal p> 1 , we denoteand call it an ε-ordinal.For any fixed ordinal π, the set Kπ of all the ordinals associated with cardinal (?)a has a proper subset Eπ consisted of all the root ordinals in Kπsuch thatin which εξ is an ε-ordinal when ξ has direct predecessor and otherwise isnot. The ω is ε-ordinal, but all the other initial ordinals are not.Furthermore, the first element ε0 in Eπ is ωπ,and SupEπ = SupKπ =ωπ+1.