On Thu, Mar 3, 2011 at 1:58 PM, Richard O'Keefe <o...@cs.otago.ac.nz> wrote:
> > I can't think of an approach that doesn't require all but one of > the tuple elements to have Bounded types. It's not possible. Such an enumeration could potentially have an uncomputable order-type, possibly equal to the order-type of the rationals. (In other words, there could be countably infinitely many elements between any two elements) It's possible to define a computational system where you can do arithmetic on countable ordinals, but it has the expressive power of Turing machines with oracles (where an oracle is a thing that correctly guesses the "right" answer for a computation that does not halt in finite time (consider a sequence approaching pi as a limit). We can re-interpret the oracle's guess as passing to a limit ordinal. In any case, TMs+ oracles are strictly stronger than just TMs.
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