User blog comment:Deedlit11/Keith Ramsay on the largest definable number/@comment-5150073-20130322164719/@comment-1605058-20130322174038

If it was that simple, we could "reach" w_1 long time before - we can extend oracle Turing machines as far into countable ordinals as we want, just by defining all weaker oracle machines (what we can do in theory, as we proceed by transfinite induction) and diagonalizing oracle from them. Once I had an idea of extending it to uncountable ordinals, no matter how odd this sounds. Once there appeared concept of a-machines (for ordinal a) which were working on tape of length a. My concept was w_1-machine working on uncountable tape for uncountable amount of time with uncountable oracle diagonalizing through uncountably many countable ordinal oracle machines. This super oracle machine would overcome any countable definition. But we don't have to stop here.