<br><br><div class="gmail_quote">On Thu, Mar 3, 2011 at 1:58 PM, Richard O'Keefe <span dir="ltr"><<a href="mailto:ok@cs.otago.ac.nz">ok@cs.otago.ac.nz</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
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I can't think of an approach that doesn't require all but one of<br>
the tuple elements to have Bounded types. </blockquote><div><br></div><div>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)</div>
<div><br></div><div>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.</div>
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