@misc {IOPORT.03859105, author = {Li, Xiang}, title = {Effective immune sets, program index sets and effectively simple sets - generalizations and applications of the recursion theorem.}, howpublished = {Logic, Proc. Southeast Asian Conf., Singapore 1981, Stud. Logic Found. Math. 111, 97-106 (1983).}, year = {1983}, abstract = {[For the entire collection see Zbl 0532.00005.] In this paper, A is called an effectively immune set if ($\exists$ partial recursive $\psi)$ $(\forall x)[W\sb x\inf inite \Rightarrow \psi(x)\downarrow \&\psi(x)\in W\sb x-A],$ A is called a program index set if for every unary partial recursive function $\psi$ there exists a unique $x\in A$ such that $\psi =\phi\sb x$, and A is called an effectively simple set if A is an r.e. set, $\bar A$ is infinite and $(\forall i)[W\sb i\subset \bar A\Rightarrow \vert W\sb i\vert \le f(i)],$ where f is a recursive function, $\vert W\sb i\vert$ is the cardinality. It is shown that, (1) there are effectively immune sets, (2) there are $2\sp{\aleph\sb 0}$ immune sets which are not effectively immune, (3) the program index set $M\sb 0=\{x:(\forall y)[\phi\sb x=\phi\sb y\Rightarrow x\le y]\}$ is an effectively immune set, (4) program index sets $M\sb{n+1}=\{x:x\not\in \cup\sb{i\le n}M\sb i\&(\forall y)[y\not\in \cup\sb{i\le n}M\sb i\&\phi\sb x=\phi\sb y\Rightarrow x\le y]\}$ are immune sets, (5) there exists a recursive function S such that $(\forall x)[\phi\sb x$ total \& $\phi\sb x$ nondecreasing \& $\lim\sb{n\to +\infty}(\phi\sb x(n)-n)=+\infty \Rightarrow W\sb{s(x)}$ is an effectively simple set with bound $\phi\sb x]$, and (6) let f be a nondecreasing recursive function and $\lim\sb{n\to +\infty}(f(n)-n)=+\infty,$ then there exists an effectively simple set such that it is not effectively simple with bound f.}, identifier = {03859105}, }