Remark
15
[25C]This fact holds true:
\[ β yββ, yβ β
β β xββ , S(x)=y \]
this can be proven by induction, as in [1YP], or by proving that, if
\[ β yββ, yβ β
β§ β xββ , S(x)β y \]
then \(β ⧡ \{ y\} \) would be an S-saturated set smaller than \(β\), a contradiction. In particular by [1YM] we get that the successor function
\[ S:β β β⧡\{ 0\} \]
is bijective.
If \(n\neq 0\), we will call \(S^{-1}(n)\) the predecessor of \(n\).