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\f0\fs24 \cf0 We define two successive extensions of Kleene's O using infinite time Turing machines. The first extension, O^+, is proved to code a tree of height \uc0\u955 , the supremum of the writable ordinals, while the second extension, O^++, is proved to code a tree of height \uc0\u950 , the supremum of the eventually writable ordinals. Furthermore, we show that O^+ is computably isomorphic to
\f1\i h
\f0\i0 , the lightface halting problem of infinite time Turing machine computability, and that O^++ is computably isomorphic to
\f1\i s
\f0\i0 , the set of programs that eventually write a real. The last of these results implies by work of Welch that O^++ is computably isomorphic to the \uc0\u931 _2 theory of
\f2\b L
\f0\b0 _\uc0\u950 , and by work of Burgess that O^++ is complete with respect to the class of the arithmetically quasi-inductive sets. }