Tanaka (Ann Pure Appl Log 84:41–49, 1997) proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \(({\mathcal {M}}, {\mathcal {A}})\) of the subsystem \(\mathrm {WKL}_{0}\) of second order arithmetic, and any element m of \({\mathcal {M}}\), there is a self-embedding j of \(({\mathcal {M}},{\mathcal {A}})\) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka’s work by establishing the following results for a countable nonstandard model \(({\mathcal {M}},{\mathcal {A}})\ \)of \(\mathrm {WKL} _{0} \) and a proper cut \(\mathrm {I}\) of \({\mathcal {M}}\):

Theorem A. The following conditions are equivalent:

(a) \(\mathrm {I}\) is closed under exponentiation.

(b) There is a self-embedding j of \(({\mathcal {M}},{\mathcal {A}})\) onto a proper initial segment of itself such that I is the longest initial segment of fixed points of j.

Theorem B. The following conditions are equivalent:

(a) \(\mathrm {I}\) is a strong cut of \({\mathcal {M}} \) and \(\mathrm {I}\prec _{\Sigma _{1}}{\mathcal {M}}.\)

(b) There is a self-embedding j of \((\mathcal {M },{\mathcal {A}})\) onto a proper initial segment of itself such that \( \mathrm {I} \) is the set of all fixed points of j.

田中（Ann Pure Appl Log 84：41–49，1997）证明了弗里德曼自嵌入定理的强大推广，该定理指出给定了一个可数的非标准模型\（（{{mathcal {M}}，{\ mathcal {A}} ）\）子系统的\（\ mathrm {WKL} _ {0} \）二阶算术，以及任何元件米的\（{\ mathcal {M}} \） ，有一个自嵌入Ĵ的\ （（{{\ mathcal {M}}，{\ mathcal {A}}}）\）到其本身的适当初始段上，以使j固定m的每个前任。在这里，我们通过建立以下结果可数非标准模式扩展田中的工作\（（{\ mathcal {M}}，{\ mathcal {A}}）\ \）的\（\ mathrm {WKL} _ {0} \）和适当的切割\（\ mathrm {I} \）的\（{\ mathcal {M}} \）：

定理A. 以下条件是等价的：

（a）\（\ mathrm {I} \） 在取幂时被关闭。

（B）有一个自嵌入 Ĵ 的 \（（{\ mathcal {M}}，{\ mathcal {A}}）\） 到一个适当的初始段 本身 ，使得 我 是固定点的最长的初始段的 j。

定理B。 以下条件是等价的：

（a）\（\ mathrm {I} \） 是\（{\ mathcal {M}} \}和\（\ mathrm {I} \ prec _ {\ Sigma _ {1}} {\ mathcal {M}}。\）

（B）有一个自嵌入 Ĵ的\（（\ mathcal {M}，{\ mathcal {A}}）\） 到其自身的适当的初始段，使得\（\ mathrm {I} \）是 j的所有不动点的集合。