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I'm reading the Hrbaceck and Jech's book and they say that recursion until $\omega$ does not work for constructing the Borel sets

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But I haven't been able to find such sequences, do you have any suggestions?

Selena
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    Can you prove $\Sigma^0_{n+1} \ne \Sigma^0_n$ ? If so, choose $B_n \in \Sigma^0_{n+1} \setminus \Sigma^0_n$ on disjoint intervals $(n,n+1)$. The union is not in any $\Sigma^0_n$. – GEdgar Mar 11 '24 at 00:33

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