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\(\omega^{CK}_1\) is the ordinal strength of Turing Machines, and the smallest non-recursive ordinal. What about \(\omega^{CK}_2\) or \(\omega^{CK}_n\)? If \(\omega^{CK}_1\) is the limit of A(0), B(0), C(0), D(0), and so on, where A, B, C, D... are normal functions, is the limit of \(A(\omega^{CK}_1+1)\), \(B(\omega^{CK}_1+1)\), \(C(\omega^{CK}_1+1)\), \(D(\omega^{CK}_1+1)\)... \(\omega^{CK}_2\) or it is smaller. Also is the limit of oracle machines with access to the halting problem \(\omega^{CK}_2\)?

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