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In the book Computability theory (Rebecca Weber) I stumbled about Exercise 7.1.24 with the definition of the omega jump. The book says: The omega jump of $A$, $A^{(\omega)}$ is the join of all $A^{(n)}$ for $n \in \mathbb{N}$: $A^{(\omega)}=\{\langle x, n \rangle : x \in A^{(n)}\}$.
I tried to understand this definition but i couldn't. I think I know that the jump is but I dont understand the omega jump. Is it simply $A^{(\omega)} = \cup_{n \in \mathbb{N}} A^{(n)}$? Why (not) ?
Can someone give me an equivalent definition or explain it to me?

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The union is not a good idea because you cannot tell, if a number is in the union, whether the number got in because it is an element of $0^{(n)}$ or $0^{(n+1)}$, etc. Instead, if you take the union of the $<n,a>$ where $a$ is an element of $0^{(n)}$ then you can keep them separate.

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