# Creative, recursively enumerable

I'm trying to show that the set $K$ is creative. $K$ has to do something with $\phi_x$ and the only thing I can get out of creative is if there is a total recursive $f$ s.t. $f(e)$ is an element of $A$ iff $f(e)$ is an element of $W_e$.

Can someone also explain recursive enumerable vs. creativity? Thanks in advance.

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## 1 Answer

To show $K$ is creative: let $f(e)=e$ be the function.

Not every r.e. set is creative, a counterxample is the simple set.

It's known that a r.e. set is creative iff it's m-complete iff it's 1-complete. More about it refers to Robert I. Soare Recursively Enumerable Sets and Degrees, Chapter 2.

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Can you please explain why we select f(e) = e to be the functino –  mary Mar 7 '13 at 2:38
$K=\{x:x\in W_{x}\}$, so $e\in K\leftrightarrow e\in W_{e}$, so we choose $f(e)=e$. –  Chao Chen Mar 7 '13 at 3:56
How does this example show that K is creative? Is it by the s-m-n theorem? I'm trying to better understand this definition. thanks –  mary Mar 7 '13 at 6:27
Your definition of creative set is there is a total recursive f s.t. f(e) is an element of A iff f(e) is an element of We, so f(e)=e satisfies the definition. –  Chao Chen Mar 7 '13 at 14:32