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I'm stuck on a problem in Sack's Higher Recursion Theory (#2.4)- any hints are welcome. He defines Kleene's O in the usual way, and the corresponding order $<_O$. A path through O is a linearly ordered subset Z s.t. $w<_Ov\in Z\rightarrow w\in Z$. A path can be continued if there is a $w\in O$ s.t. $\forall z\in Z z<_O w$. The problem is find a path that can't be continued of order type $<\omega_1^{ck}$. I believe I can show that there is such a path using a counting argument, but I can't find one explicitly.

Thank you.

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